Units of Physical Quantities and Their Dimensions (Mir, 1972)

L A SENA
UNITS
OF
PHYSICAL
QUANTITIES
AND
THEIR
DIMENSIONS
MIR PUBLISHERS
MOSCOW
JI. A. CEHA
EflHHHIJbl OH3HHECKHX BEJIHUHII

H MX PA3MEPH0CTH
H3flATEJIbCTJ30 «1IAJ’KA»
L. £ SENA
UNITS OF
PHYSICAL QUANTITIES
AND THEIR
DIMENSIONS
Translated from the Russian

by G. LEIB
MIR PUBLISHERS * MOSCOW • 1972

UDC 531.7—20
The present book sets out in detail the prin

ciples of constructing systems of units, and also
the fundamentals of the theory of dimensions.
Together with detailed information on the SI
system, which is the preferable one at present,
a description is given of other systems of units,
and also of some non-system units having prac
tical significance.
The book is intended for students of technical
colleges and will also be helpful for physics in
structors in higher and secondary schools.
Ha amjiuiicKOM si3biKe
TO THE READER
Mir Publishers w ill welcome your com

ments on the content, translation and
design of this book. We would also be
pleased to receive any suggestions you
care to make about our publications.
Our address is:
Mir Publishers, 2 Pervy Rizhsky Pere-
ulok, Moscow, USSR.
Printed in the Union of Soviet Socialist Republics

CONTENTS
F o rew o rd ........................................................................ 7
Chapter One. General Concepts on Systems of Basic and Derived

U n i t s ......................................................................... 11
1.1. Physical Quantities and Their U n i t s ........................ 11
1.2. Direct and Indirect M easurements............................ 16
1.3. Basic and Derived U n i t s ............................................ 18
1.4. Constructing a System of U n i t s ................................ 24
1.5. Selection of Basic U n i t s ............................................ 34
1.6. Non-System Units ........................................................ 40
Chapter Two. Conversion of Units and Dimension Formulas 42

2.1. DimensionFormulas . . .............................................. 42
2.2. Conversion of Dimension When Using Different
Basic Units ..................................................................... 47
2.3. Conversion of Dimensions with Different Defining
Relationships ................................................................. 48
2.4. Determining the Relationship between Units of
Different Systems ........................................................ 52
2.5. Compilation of Conversion T a b le s ............................ 58
2.6. On the So-called Meaning of Dimension Formulas 59
2.7. Brief Conclusions on Chapters Oneand Two . . . 62
Chapter Three. Analysis ofD im ensions....................................... 65

3.1. Determining Functional Relationships by Comparing
D im en sion s......................................................................... 65
3.2. The II-Theorem and the Method of Similarity . . . 72
Chapter Four. Units of Geometrical and Mechanical Quantities 79

4.1. Introduction ..................................................................... 79
4.2. Geometrical Units ................................................ 80
4.3. Kinematic Units .................................................... 91
4.4. Static and Dynamic U n i t s ............................................ 96
4.5. Units of Mechanical and Molecular Properties of
a Substance .................................................................... 109
Chapter Five. Thermal U n it s .................... 121

5.1. T e m p e r a tu r e .................................................................... 121
5.2. Temperature S c a l e s ........................................................ 128
5.3. Fixed Temperature P o in t s ............................................ 130
5.4. Other Thermal U n i t s .................................................... 130
5.5. Units of Thermal Properties of Substances . . . . 135
6 CONTKNTH
Chapter Six. Acoustic U n it s ............................................ 142

6.1. Objective Characteristics of Mechanical Wave Pro
cesses ................................................................................ 142
6.2. Subjective Characteristics of S o u n d ........................... 147
6.3. Some Quantities Connected with the Acoustics of
B u ild in g s........................................................... 150
Chapter Seven. Electrical and Magnetic U n i t s ............................ 153

7.1. Introduction ....................................................................... 153
7.2. Possible Ways of Constructing Systems of Electrical
and Magnetic U n i t s ........................................................ 154
7.3. Units of the CGS S y s te m ............................................. 169
7.4. Units of the SI S y s te m ................................................. 182
7.5. On the So-called Wave Resistance of a Vacuum 196
7.6. International U n i t s ........................................................ 198
Chapter Eight. Units of R ad iation ............................................ 201

8.1. Scale of Electromagnetic W a v e s ................................ 201
8.2. Characteristics of Radiant E n e r g y ............................... 202
8.3. Illumination Engineering U n i t s ................................... 207
8.4. Relationship between Subjective and Objective Cha
racteristics of L ig h t ........................................................ 212
8.5. Units of Parameters of Optical Instruments . . . . 214
8.6. Units of Optical Properties of a Substance . . . . 217
Chapter Nine. Selected Units of Atomic P h y sic s........................ 218

9.1. Introduction ....................................................................... 218
9.2. Basic Properties of Atomic and Elementary Particles 218
9.3. Effective Interaction Cross S e c tio n s........................... 223
9.4. Units of Energy in Atomic P h y s ic s ........................... 225
9.5. Ionizing Radiation U n i t s ............................................... 229
9.6. Units of Radioactivity ................................................... 231
9.7. Ionization, Recombination and Mobility Coefficients 233
9.8. Natural Systems of U n i t s ............................................ 235
Appendix 1. Logarithmic Units ................................................... 238
Appendix 2. Measuring the Density of a Liquid with an Areo
meter ........................................................................... 241
Appendix 3. pH Index ................................................... 241
Appendix 4. Constants ................................................................ 242
Appendix 5. Tables ....................................................................... 245
B ib lio g r a p h y .................................................................................... 286
Index
FOREWORD
More than thirty years ago the author wrote a small book
named Units of Physical Quantities (in Russian). In 1948
a completely revised edition of this book was published,
and further editions with minor corrections appeared in
1949 and 1951.
Not counting the booklet by 0. D. Khvolson published
in 1887, this was probably the first attempt to systematize
the various systems of units and the methods of conversion
of units of one system to those of another. The need for
such a textbook lor students (and this was the purpose of
the book) was very great at that time. Students were forced
to master the cgs, mk(force)s, metre-ton-second and mks
systems, and in addition, when studying electricity and
magnetism, to understand the “contradictory” cgse and
cgsm systems of dimensions and units. Still more compli
cations were introduced by a great number of various non
system units.
Matters appreciably changed after the introduction of the
International System of Units, designated SI, which accord
ing to USSR State Standard GOST 9867-61 “has been
introduced as preferable in all fields of science, engineering
and the national economy, and also in education”. In this
connection the above book could have been republished,
nfter introducing the essential amendments and additions.
The present book, however, basically differs from the ori
ginal one and has other aims that will be considered below.
Unquestionably, the creation of a system that covers all
the fields of measurements and includes a great number
of units used in practice should be considered as a progres
sive step, but in the author’s opinion, however, it is not
necessary to run to extremes. While the existence of a great
diversity of systems of units leads to considerable incon
veniences, an attempt to confine all measurements to the
M FORK WOR D
Procrustean bed of a single system will hardly satisfy

everyone. Naturally the main, if not the single, require
ment for any system of units is its convenience.
The author completely agrees with the opinion of Aca
demician M. A. Leontovich (Vestnik Akademii Nauk S S S R ,
No. 6, 1964) that the cgs system should be retained, and not
as one whose “use is allowed’ , but as one enjoying abso
lutely all rights.
Other units, in particular some non-system ones, should
be retained for a number of purposes, and again for the
single reason of their practical convenience in the given
field.
If the question were only of introducing the SI system
and retaining or not retaining the other systems (in particu
lar, the cgs system), then the publication of this book would
hardly be justifiable, and a small article in a journal would
suffice, moreover since at present many reference books are
available in which the units of practically all the systems
and a great number of non-system units have been gathered.
The present volume has a different task reflected in its
title, of which special mention should be made. The word
“dimensions” included in the title underlines the circum
stance that together with problems of the systems of units,
much attention is given to systems of dimensions. From
the very beginning it must be underlined that the word
“dimensions” should be related only to the word “units”,
and not to the word “quantity”. In the author’s opinion,
it is shown quite convincingly in the following that the
concept of a “dimension of a physical quantity” is deprived
of any meaning whatsoever and can be used only as an
abbreviation of the concept “dimension of a unit of the
given physical quantity within the limits of the given
system of units”. Although for the sake of brevity (and also
taking into consideration its widespread usage) this abbre
viation is frequently used in the book, its true meaning
must always be kept in mind.
For the dimension formulas not to be abstract, brief
information is given on their use, in particular in the analy
sis of dimensions and the method of similarity. The use
fulness of this will be obvious if it is noted that these methods
are finding greater and greater application, while there
FOREWORD 9
arc very few publications on the subject, and they are not
always available.
Especially great attention is given to the general prin
ciples of constructing systems of units and the ways of
converting units from one system to another. In considering
how individual units are formed, it was found quite expe-
dient to explain the essence of the physical quantity being
measured and on the basis of what measurement the given
unit is established, rather than give the formal definition
of the unit.
In a book covering all the sections of physics, we have
to deal with a very great number of quantities. For this
reason the selection of the symbols used from among those
contained in various recommendations may perhaps have
a somewhat arbitrary nature. This selection has been deter
mined in the main by what symbols are used most frequent
ly in textbooks on physics. Since, in accordance with the
existing USSR standards and recommendations, capital
and small letters may be substituted for each other, those
have been selected that seemed to be more convenient*.
Finally, special mention should be given to the liberty
taken by the author in naming and designating the technical
unit of mass. Instead of designating it t.u.m. or using the
designation kgf-s2/m suggested by the dimension formula,
he has used the name “inerta” recommended by prof.
M. F. Malikov and the symbol “i”. If, as contemplated, the
technical system of units—mk(force)s—will with time be
completely abolished, the question of how to call the unit
of mass in this system will vanish. But as long as the system
is in use, it seems to be more convenient to have a special
name for the unit of such an important quantity as mass.
Without having the object of creating a reference book
on units, the author has nevertheless included a great
number of tables that may be useful in practical work.
L . A . Sena
* In the English edition use has been made wherever possible of

the symbols recommended by the International Union of Pure and
Applied Physics,— Translator's note,
CHAPTER ONE
GENERAL CONCEPTS ON SYSTEMS

OF BASIC AND DERIVED UNITS
L I. Physical Quantities and Their Units

Every day we have to deal with a variety of measure
ments. The measurement of such quantities as length, area,
volume, time and weight is encountered at every step and
is known to mankind from time immemorial. Without it
commerce, erection of buildings, division of land, etc.,
would be impossible.
Measurements are of especially great significance in
engineering and scientific research. It is owing to measure
ments that such sciences as mathematics, mechanics and
physics were begun to be called exact, since they acquired
the possibility of establishing exact quantitative relation
ships to express the objective laws of nature.
Quite often the results of measurements made in a scien
tific experiment gave a decisive answer to a question of
principle posed by science, allowed a selection to be made
between two hypotheses, and sometimes even led to the
appearance of a new theory or even a new branch of science.
Thus, measurement of the velocity of light in various media
facilitated the establishment of the wave theory of light,
an attempt to measure the absolute velocity of the Earth’s
motion led to the appearance of the theory of relativity,
measurement of the distribution of energy in the black
body spectrum gave birth to the quantum theory.
Not a single branch of engineering, from structural
mechanics to complicated chemical enterprises, from radio
engineering to nuclear power plants, could exist without
a developed system of measurements for determining the
dimensions and properties of the product, and for establish
ing conditions of adequate control of the machines and
production processes.
12 UNITS UF PHY SICAL QUANTITIES AND T11E1.U DIMENSIONS
The part played by measurements has especially grown

owing to the development of automatic control, since auto
matic systems and computers must receive as their input
data the information on various quantities that determine
how the process being controlled is proceeding, such as
temperature^ gas pressure, or flow rate of a fluid.
The tremendous diversity of phenomena encountered in
engineering and scientific research correspondingly expands
the range of quantities to be measured. The voltage in
electric mains, the viscosity of lubricating oil, the elasti
city of steel, the refractive index of glass, the power of an
engine, the luminous intensity of a lamp and the length
of an electromagnetic wave of a radio transmitter are only
a few of the countless numbers of quantities measured in
science and engineering.
The methods of measurement are also exceedingly diverse.
Simple measuring rules and complicated optical instruments
serve for measuring length; magnetoelectric, electromagnetic
and thermal instruments measure voltage and current;
pressure gauges of various types measure pressure, and so
on. Regardless of the method used, however, any measure
ment of a physical quantity consists in comparing the
measured magnitude of this quantity with that taken as
a unit. For example, when measuring the length of a table,
we compare this length with that of another body taken
as a unit of length (for example, a metre rule); when weighing
a loaf of bread, we find out how many times its weight is
greater than that of another body—a definite unit weight,
a “kilogram” or “gram”, or what fraction of this4unit it is.
By measuring the magnitude of a quantity is meant,
consequently, finding the relation between this magnitude
and the relevant unit. It is this relation, obviously, that
will show us the magnitude of the quantity we are interest
ed in.
Since the concept “greater-smaller” can be applied only
to homogeneous quantities, it is obvious that only such
quantities can be compared. The height of a building can
be compared with the distance between two towns, the
force tensioning a spring with the weight of a body (i.e.,
with the force of gravity), but there is absolutely no sense
in trying to find out whether the speed of a train is greater
S Y S T E M S OF BASIC AINU D E R I V E D UNITS 13
than the length of a pencil, or whether the volume of a cup

is greater than the weight of an inkpot. It is just as sense
less, of course, to try to measure speed in units of mass,
or area in units of force.
For a measurement to he single-valued, it is essential th a t’
the ratio between two homogeneous quantities he indepen
dent of the unit used to measure them. The overwhelming
majority of physical quantities satisfy this condition, which
is customarily called the condition of the absolute magnitude
of a relative value. This condition can be observed if there
exists a possibility, at least in principle, for such a quan
titative comparison of two homogeneous quantities as to
obtain a number showing the ratio between them.
Sometimes such properties are encountered, however,
that cannot be characterized by a quantity complying with
the above condition. In these cases certain conventional
numerical characteristics are introduced that already cannot
be considered as units. With the progress of measuring tech
niques, the possibility can sometimes appear of replacing
such conventional characteristics with genuine units. For
example, the velocity of wind was previously determined
with the aid of the conditional Beaufort wind scale based
on the “force of the wimr, which was later replaced by
measurement of the velocity of the wind in metres per
second. At present a definite range of wind velocities has
been made to conform to each number of the Beaufort scale.
The conventional quantities also include the hardness of
materials that is compared with the aid of various scales
between which, by the way, there does not even exist an
entirely single-valued relation. Although these conven
tional numerical characteristics of physical properties
are not units of measurement, for the reader’s convenience
the most widely used of them have nevertheless been inclu
ded in this book.
The question as to how to determine the unit of a quan
tity being measured, generally speaking, can be answered
only arbitrarily. And indeed, the history of material cul
ture knows a tremendous number of various units, espe
cially for measuring length, area, volume and weight.
This diversity of units still exists to a certain extent in
our time.
14 UNITS OF PHYSICAL QUANTITIES AND THEIR DIMENSIONS
The existence of a great number of diverse units natu

rally created difficulties in international commercial rela
tions, in the exchange of the results of scientific research,
etc. As a result scientists of various countries attempted
to establish common units that would be in force in all
countries. It can be understood, of course, that their aim
was not to establish only a single unit for each quantity.
Since both great and small values of quantities being
measured are encountered in practice, it was found expe
dient to have corresponding units of different magnitude,
both large and small ones, the condition being observed,
however, that the conversion of one unit into another be
as simple as possible. Such a system of units is the metric
system, created in the era of the French Revolution, a sys
tem that, as conceived by its authors, was intended to serve
“a tous les temps, a tous les peuples” (in all times, for all
peoples).
From the middle of the nineteenth century the use of
the metric system began to spread quite rapidly. It was
adopted in most countries and served as the basis for estab
lishing the units for measurement of various quantities
in physics and related sciences. A feature of the metric or,
as it is sometimes termed, the decimal system of weights
and measures, is that the ratio between the different units
of a quantity is equal to an integral (positive or negative)
power of ten.
Notwithstanding the obvious advantages and conve
niences of the metric system, other local units are used
together with it in a number of countries, while in the USA
and some other countries the metric system is not an official
one at present and is used, and even then not always, only
in scientific work. (In Great Britain the transition from
Imperial units to metric ones will most likely take place
in 1970/75.)
The circumstance that several units may be employed
for measuring a quantity leads to the necessity of being
able to convert one unit into another. In other words, it
is necessary to be able to determine the number measuring
the magnitude of a quantity in one unit if the number
measuring it in another unit is known. If a given magnitude
A of a quantity is measured using the unit that results
SYSTEMS OF BASIC AND DERIVED UNITS 15
in its numerical value being equal to au then we can write

that
If, when measuring the same magnitude A using the unit

a2, we get the value a2, then, correspondingly,
or
A— —CX<£L2
From the latter expression we get
g{ __ a2 ( 1 . 1)
Q>2 CCl
This formula expresses the well known tenet that the
numerical value of a physical quantity and the unit used
to measure it are in inverse proportion, i.e., the greater
the unit used to measure the given magnitude of the quan
tity, the smaller is the number expressing this magnitude.
Thus, if the height of a person measured in centimetres
is expressed by the number 175, then the same height
measured in decimetres will be expressed by 17.5. Many
people forget this when dealing with more complicated
and less familiar quantities.
For this circumstance not to be forgotten, one must
always remember that the symbols used in formulas repre
sent the numbers expressing the magnitudes of quantities
in the units used to measure them, rather than the quan
tities themselves. For a formula to be generally applicable,
the symbol of the unit used to measure a quantity is written
next to the number expressing its magnitude. For example,
we may write: “the height of that person is 17.5 dm” or
“the height of that person is 175 cm”. The expressions 17.5 dm
and 175 cm are equivalent designations of the same height.
For this reason we can write
17.5 dm = 175 cm
1.2. Direct and Indirect Measurements

As we have already noted, any measurement consists in
comparing the given quantity with another homogeneous
quantity taken as a unit. There are many instances, however,
when such a comparison cannot be made directly. In most
cases what we measure is not the quantity of interest to us,
but other quantities that are related to it by certain laws
and relationships. Not infrequently, to measure a given
quantity, it is first necessary to measure several others
whose values are used to compute that of the quantity
being sought. Thus, to determine the specific weight of a
body, its volume and weight must be measured, to find the
speed of a vehicle one has to measure the distance it has
covered and the time spent to cover it, etc.
In accordance with the above, all measurements are
classified as direct and indirect. The former are generally
considered to include all measurements that give the nume
rical value of the quantity being measured as a result of one
observation or reading (for example, on the scale of a mea
suring instrument). Actually, however, in the majority
of such cases we have not direct, but indirect measurements.
Indeed, various measuring instruments (voltmeters, amme
ters, thermometers, pressure gauges, etc.) give readings
in divisions of their scale, so that what we measure directly
are only linear or angular deviations of the pointer that
indicate the value of the quantity being measured by means
of a number of intermediate relationships connecting the
deviation of the pointer to the quantity being measured.
For example, in a magnetoelectric ammeter a magnetic
field determined by the shape and dimensions of a coil and
the current flowing through it (which is to be measured)
upon interacting with the magnetic field of a magnet induces
a torque; the latter is counteracted by the moment of a spring
that depends on its mechanical properties, and the coil
turns through a certain angle until it reaches a position
in which the torque and the moment are in equilibrium.
Thus the measurement of an electrical quantity—the inten
sity of a current—is reduced through a number of inter
mediate steps to an angular or linear measurement. If
ft shunt is used in the instrument, there will be an additional
intermediate step between the current to be measured and

the directly measured deviation of the pointer.
The reducing of the measurement of the most diverse
quantities to linear and angular measurements is characte
ristic of an overwhelming majority of measuring instru
ments. This is not accidental, since vision is the most
developed of our organs of sense, and hence a comparison
of the magnitudes of a quantity that we directly perceive
visually is the most clear and convenient for us. Such
quantities, naturally, are dimensional ones, first of all
lengths and angles. Where no especially high accuracy
is required, and with the exception of very small and very
great lengths, linear measurements are generally made by
directly comparing the length being measured with a unit
length and determining the number of times the unit is
contained in the given length. In the same way an angle
can be measured by superposing a suitable angular unit.
Lengths and angles are not the only quantities, however,
that can be measured directly. An area can be measured
by superposing on it a suitably selected unit of area, for
example, in the form of a square or a triangle. The volume
of a liquid can be found using a vessel whose volume is taken
as a unit. Time intervals can be measured by directly
counting the number of periods of a cyclic process (for
example, the oscillations of a pendulum or the changes
of day and night).
Indirect methods also are frequently used, however, for
measuring the above quantities, namely, the measurement
of areas and volumes is reduced to linear measurements,
time is read on the face of a time-piece (again linear or
angular measures!), etc. If we consider other quantities,
it will be easily seen that for most of them methods of direct
measurement are not available at present, and we use
either special instruments that convert changes in the
given quantity into changes in other quantities (mostly
lengths and angles), or a number of intermediate measure
ments from which the quantity being sought is obtained
by calculation.
The fact that practically all measurements can be reduced
to linear ones does not at all mean that the quantities being
measured lose their qualitative feature and are reduced to
3-1040
length. Actually this simply means that since all phenomena

observed in nature occur in space, each of them can be
reflected by a relevant spatial movement (expansion of the
mercury column in a thermometer, turning of the coil
of an electrical measuring instrument, deviation of a beam
of electrons in an oscillograph, etc.).
1.3. Basic and Derived Units

Most of the earlier units were established, as a rule,
absolutely independently of one another, only units of
length, area and volume being an exception in some instan
ces. Conversely, the main feature of modern units is that
relationships are established between units of different
quantities which are determined by the laws or definitions
relating the quantities being measured to one another.
Thus, several so-called fundamental or basic units are con
ventionally selected, and all the derived units are constructed
from them.
Since in indirect measurements the value of the quantity
being sought is determined from the values of other quan
tities related to it, the corresponding relationship can be
established between the relevant units. The relationships
and laws that determine the conditions of indirect measure
ment can obviously also serve for establishing relations
between basic and derived units.
To show how this is done, let us first consider the question
of what meaning should be given to the formulas expressing
the relationships between various physical quantities. Any
relationship between quantities, whether it is a law of nature
or the definition of a new quantity, shows how the given
quantity changes with a change in the other ones that it
is related to. Let us take as an example the relationship
between the areas of geometrical figures and their linear
dimensions, established by the theorem “the ratio between
the areas of geometrically similar figures is equal to the
square of the ratio between their corresponding linear
dimensions1’. This relationship can be written as follows:
( 1. 2)
If by the symbols A x, A 2, lx and l2 we understand the rele

vant quantities, then only the ratios A xlA 2 and l j l 2 will
have a concrete physical meaning. Expression (1.2) can
formally be rewritten as
A2 ~ l\ (1.2a)
but it will then lose the meaning contained in the original
formula (1.2). Indeed, if the square of the term IJL±, which
is the ratio between the lengths lx and Z2, is also a definite
number, then the square of a length, i.e., the product of two
lengths, has no meaning.
Matters will be different if we consider that the symbols
in formula (1.2) denote not the quantities themselves, but
the numbers by means of which these quantities are expres
sed when definite units are selected for the corresponding
quantities, in our case for length and area. Here each symbol
already represents the ratio between the given quantity
and another homogeneous quantity taken as a unit. With
such a conception of the symbols in the expression of any
physical law, they can be multiplied, divided, raised to
a power, etc., while the formulas themselves can be sub
jected to various transformations. In particular, formula
(1.2) can be written in a different form, for example
-f— J (l-2b>
For this reason in mathematical formulation and in the
exposition of various physical phenomena and the laws
they follow, and in their theoretical analysis, the symbols
of physical quantities are understood to denote the numbers
by means of which these quantities are expressed in the
units used. In the following we also shall always adhere
to such a meaning of the symbols.
From this viewpoint formula (1.2b) can be expressed in
words as follows: “for geometrically similar figures the
ratio between the number expressing the area of a figure
and the square of the number expressing the corresponding
linear dimension of the figure is a constant quantity”. Upon
denoting this constant by C, we can write the following
equation instead of expression (1.2b):
i4 = C7* (1.3)
20 UNITS OE PHYSICAL QUANTITIES AND THEIR DIMENSIONS
where the factor C depends, on one hand, on the shape of

the geometrical figure being measured, and, on the other,
on the units of length and area used*.
As mentioned earlier, these units can in principle be
selected absolutely independently of each other. The exis
tence of a relationship between the magnitudes of the area
and linear dimensions of the figure, however, makes it
possible to relate the units of area to the units of length,
i.e., to make the unit of area a derivative of the unit of
length. For this purpose it is essential to agree that the
unit of area will be the area of a certain figure whose linear
dimension is equal to a definite conventionally accepted
number of units of length. In geometry this is usually
done as follows. After faking any unit of length, for example,
a metre, as the basic one, the unit of area is taken equal
to the area of a square whose side is equal to the selected
unit of length, in our case one metre. This unit of area,
as is well known, is called a “square metre” (sq m). Assuming
in formula (1.3) that I = 1 m, we can write
1 sqm —C(I in)2 (1.3a)
whence
C ^ 1 sq m/m2
and formula (1.3) can accordingly be written as
A sq m = ^ 1 j (I m)2 (1.3b)
If, without changing the units of length and area, we
rewrite formula (1.3) for a circle, we get
A sq m = ( ^ — ^ ) (I m)2 (1.3c)
(where I is the diameter of the circle), since here the factor C
will be equal to — .
* Here and below we shall use the symbol C for the general desig
nation of a factor of proportionality in the formulas of physical laws
and definitions regardless of its concrete value. In separate instances,
when this will he found expedient, the factor C will he provided with
a subscript.
It is obvious that the accepted relationship between

a unit of area and one of length can be retained with any
other unit of length. Here formula (1.3) can be rewritten
for a square as
A=P (1.4)
and for a circle as
^l = — Z2 (1.4a)
Formulas (1.4) and (1.4a) can be expressed in words
asfollows: “ifthe area of a squarewhose side isequal to
a unit oflengthis taken as the unit of area, then the number
expressing the area of any square will be equal to the second
power of the number expressing the length of its side, while
the number expressing the area of any circle will be equal
to the product of ji/4 and the second power of the number
expressing its diameter”.
Of course such formulations are exceedingly cumbersome
and for this reason they are replaced by shorler ones, namely,
“the area of a square is equal to the second power of its
side”, and “the area of a circle is equal to the second power
of its diameter multiplied by ji / 4” , taking it as granted
that we are speaking of numbers used to express the rele
vant quantities writh their units appropriately selected.
The example considered above clearly shows the method
used to establish a derived unit. To do this it is necessary to:
1. Select the quantities whose units are to be taken as
the basic ones.
2. Establish the dimensions of the basic units.
3. Select the defining relationship between the quantities
measured by the basic units and the quantity for which
a derived unit is to be established.
4. Equate to unity (or any other constant number) the
factor of proportionality in the defining relationship.
It is obvious that the symbols of all the quantities in the
defining relationship should designate not the quantities
themselves, but their numerical values.
In the following, the quantities measured by basic units
will be called basic ones, and those measured by derived
units derived ones. It must be stressed here that these gene
rally used terms—-“basic quantities” and “derived quanti-
22 UNITS of physical quantities and th eir dimensions
ties”—should by no means be understood in the sense that

the former have some privileges or advantages of principle
over the latter. A quantity that in one selection has been
taken as a basic one may become a derived one in another
selection, and vice versa.
Defining relationships used to establish derived units
should be written, for convenience, in the form of an explicit
functional dependence of the derived quantity on the basic
ones.
Derived units established as described above can be
further used for developing new derived units. This is why
the defining relationships, in addition to basic quantities,
may include derived ones whose units were established
previously.
Let us explain the above with examples. The unit of
velocity can be established by using the relationship bet
ween distance and time, written as
/nr dl
(1.5)
For the particular case of uniform motion expression (1.5)

can be replaced by
p = <?4t (1.5a)
where, as previously, C is a factor depending on the units

of distance, time and velocity selected. As in the example
with the establishment of the unit of area, the unit of velo
city may be selected regardless of the units of distance and
time. In practice, however, the unit of velocity is determined
as a derivative of these units, which are taken as the basic
ones. The factor C is assumed to be equal to unity, so that
the'unit of velocity is determined as the velocity of such
uniform mo Lion when a distance equal to a unit of length
is covered in a unit of time. In a similar way the unit of
acceleration can be established with the aid of the formula
defining it, namely
dv
which for uniformly accelerated motion takes the form

a = C *=z2L (1.6a)
Here the difference v2 — vx denotes the change in velocity

during the time L Assuming, as previously, that C — 1,
we obtain the derived unit of acceleration, defined as the
acceleration of such uniformly accelerated motion in which
the velocity grows by one unit in a unit of time. In this
definition together with a basic unit (time) use is made of
a previously established derived unit (for velocity).
Let us consider another example—the establishing of the
unit of force. Like the units of any other quantities, the
unit of force can be established independently of others
and even be taken as a basic unit. Most frequently, however,
the unit of force is determined as a derived one on the basis
of Newton’s second law. Writing this law as
f = Cma (1-7)
(where m is the mass of a material point) and taking C
equal to unity, we obtain the definition of a unit of force
as such a force that imparts to a material point with a mass
equal to a unit of mass (taken as a basic unit) an accelera
tion equal to a unit of acceleration (previously determined
as a derived unit).
If, for example, we take the metre (m) as the unit of
length, the second (s) as the unit of time, and the kilogram
(kg) as the unit of mass, then a metre per second and a metre
per second per second will respectively be the derived units
of velocity and acceleration. The force imparting an acce
leration of one metre per second per second to a mass of one
kilogram is taken as the unit of force, called a newton (N).
Here, obviously, the factor C will have the value
c=_l N ^ s
kg-m
and Newton’s second law can be written in the following

form?
(1.7a)
If we take the centimetre (cm) as the unit of length and

the gram (g) as the unit of mass, retaining the second as
the unit of time, then the corresponding unit of force—
the dyne (dyn)—is determined as the force imparting an
acceleration of one centimetre per second per second to a
mass of one gram. Here the factor of proportionality is
q ^ d yirs-s
g-cm
and Newton’s second law can be written as follows:
« > ( . - £ ) < i 7 b >
In writing the formulas of physical relationships, such

a designation of the factor C, containing in essence the
definition of the derived unit, is usually omitted, so that,
for instance, Newton’s second law becomes
/ —ma (1.7c)
It should be remembered, however, that actually the factor
of proportionality is “invisibly” present in every such for
mula. Forgetting this often leads to misunderstandings and
serious errors.
The method of establishing a derived unit is reflected
in the designation of the unit, which is constructed by
grouping the units on which its determination is based in
accordance with the usual rules of algebra. The units of
area m2 (square metre), acceleration m/s2 (metre per second
per second), etc. are formed in this way.
An exception here are the units that are given their own
names, such as the dyne and newton. The symbols of these
units, the same as those of the basic units, may be included
in the compound designation of a derived unit. The unit
of pressure N/m2 (newton per square metre) can be mentioned
as an example.
1.4. Constructing a System of Units

A complex of basic and derived units forms a system
of units. For constructing a system of units it is necessary,
obviously, to select several basic units and establish the
derived units of all the other quantities we are interested

in with the aid of defining relationships. The latter may be
of two types. One includes relationships that are essen
tially a definition of the new quantity, as, for example, the
formula of acceleration (1.6) or that of work
/ \
dW = Cf-dUos{}, dl) (1.8)
The other type includes relationships between the quantities
being investigated that have been established experimen
tally or theoretically. These include the law of universal
gravitation and Coulomb’s law on the interaction of elec
trical charges. The division of relationships into “defini
tions” and “laws” is, however, not absolute and depends on
how the given concrete problem is approached. This, never
theless, does not play any significant part in the determi
nation of new units, since in both instances the relation
ships are given as formulas connecting the given quantity
to others for which units have already been established.
In connection with the outlined program for the forma
tion of derived units and constructing a system of units,
there naturally arises a question as to what extent we are
free in selecting the basic quantities (in particular, their
number), the defining relationships, and the factors of
proportionality. The arbitrariness with which the size
of the basic units is selected will hardly raise any doubts.
The existence of systems in which different units of length
(the metre and the centimetre), and also different units
of mass (the kilogram and the gram) are used as the basic
ones clearly illustrates that in principle this selection is
quite arbitrary.
It is easy to show further that there is also complete
arbitrariness in selecting the factors of proportionality
in the defining relationships. For this purpose let us return
to the example of establishing the unit of area considered
earlier. After selecting the metre as the unit of length, we
established as the unit of area the square metre—the area
of a square whose side is equal to one metre. This method
of establishing the derived unit of area, although it does
have certain practical advantages, is, however, not at
all compulsory. We can, for example, take the area of a
circle whose diameter is equal to one metre as the unit of
area. Let uscall this unit a “round metre” (rd m).

This method of establishing the unit of area is
equivalent to changing the factor in formula (1.3c) from
n sqm rdm and in formula ( 0 b ) f sqm 4 rdm
4 in- in- x ' m2 ji m2
Formulas (1.4) and (1.4a) will correspondingly become
A = I2 (the area of a square) (1.9)
and
A — I* (the area of a circle) (1.9a)
It should be noted that the measurement of area in round
metres instead of square ones is not unnatural or, moreover,
unlawful. Here we can only speak of the practical advan
tages of the relevant unit. (It should be noted in passing
that in the USA round units of area are employed for mea
suring the cross-sectional area of pipes, tubes, round beams
and girders, etc.) If the round metre were used as the unit
of area instead of the square metre, then the formulas
expressing the areas of various geometrical figures would
obviously change. For instance, the area of an equilateral
triangle would be expressed by the formula
A=^ -P (1.9b)
Regardless of how the unit of area has been determined,

a square metre or a round metre, its designation will be m2.
This shows that the symbol of a derived unit including
the symbols of basic units cannot in itself give any indi
cation of the magnitude of this derived unit. It is appro
priate to note here that the general use of square units of
area and correspondingly cubic units of volume has also
given rise to the names of the second and third powers of
numbers (the “square” and the “cube” of a number).
In the example considered above, the different defining
relationships (the area of a square and that of a circle)
led only to a change in the numerical coefficients in the
formulas, since in essence we used the same geometrical
law connecting the areas of similar figures with their linear
dimensions.
We shall now show that it is possible to select essentially

different defining relationships for establishing the derived
unit of a quantity, using as an example the establishing
of a unit of force. As mentioned earlier, for this purpose
use is generally made of Newton’s second law, which mathe
matically can be written as
f = Cma (1.7)
The factor of proportionality C that depends on the units
selected for the quantities in formula (1.7) will be called
the inertial constant and denoted by Ct. In all the systems
of units used in practice, the inertial constant is taken
equal to unity, owing to which the generally accepted
abridged formulation of Newton’s second law has become
possible, viz., “the force is equal to the product of the
mass and the acceleration”.
While retaining the units of length, mass and time as the
basic ones we, however, are not limited only to Newton’s
second law for determining the unit of force. We also have at
our disposal the law of universal gravitation, according
to which any two material points are attracted to each
other with a force directly proportional to the masses of
these points and inversely proportional to the square of
the distance between them. This law can be written as
follows:
f--=Cg ( 1. 10)
where r is the distance between the points being attracted,

and Cg is what is called the gravitational constant whose
numerical value depends on the units selected. It should be
noted that the generally accepted symbol for the gravita
tional constant is G. Here we have retained the symbol C to
stress that this constant belongs to the category of factors
of proportionality in expressions of physical laws. Expe
rience shows that if the kilogram, metre and second are
taken as the basic units, and the derived unit of force, the
newton, is determined from Newton’s second law, then
the gravitational constant Cg will be equal to 6.67 X
X 10"11 N -m2/kg2.
With the same basic units of length, mass and time
(m, kg and s), however, we can use formula (1.10) as the
defining relationship and, assuming Cg = 1, determine

the unit of force as the force of mutual attraction of two
material points whose masses are equal to unity with the
distance between the points equal to the unit of length*
It is obvious that if we follow this path we shall have to
retain an inertial constant differing from unity in the
expression of Newton’s second law. It is easy to see that
the new “gravitational” unit of force will equal 6.67 X
X 10~n N, and the inertial constant will be equal to Ct =
= 1.5 x 1010 grav, force unit-s2/(kg ^m).
Although the gravitational method of establishing the
unit of force is encountered quite rarely (mainly in astro
nomy), it is no less lawful as a result, however, than the
usual “inertial” method. In the following, when considering
the units of electrical and magnetic quantities, we shall
become acquainted with how the selection of different
defining relationships for establishing the unit of the same
quantity led in due course to the construction of different,
but fully equivalent systems of units.
It is thus evident that there are no strict limitations
in the selection of the defining relationships, as »n the
selection of the values of the basic units and the numerical
factors in the defining relationships. What is controversial
to the greatest extent and must be considered in greater
detail is the question of the number of basic units, on which
there exist two diametrally opposite opinions.
According to one of them the number of basic units has
been established for us by nature and is determined by the
character of the phenomena to be considered. Even “philo
sophical” considerations that each new quality must be
characterized and measured by a new basic unit are given
as a substantiation of such an opinion. It is stated that
to describe all the phenomena of mechanics it is necessary
and sufficient to have three basic units. In investigating
other physical phenomena it is necessary, in addition
to these three basic units, to introduce for each field of
physics at least one additional unit for a quantity charac
teristic of that field. For example, when dealing with heat
such a quantity may be the temperature, with electricity—
the electric charge (the quantity of electricity) or the cur
rent intensity, and so on.
The supporters of the opposite viewpoint, which the

author of this book belongs to, consider that the arguments
given above are not substantiated for the following reasons.
The qualities of the material world are infinitely
varied, and if we consider that each quality is characte
rized by a quantity whose unit should be a basic one, then
the number of such units will also be infinitely great.
Indeed, the concept of area cannot be deduced from that
of linear length and, consequently, the unit of area should
be a basic one. The same relates to the unit of volume. In
this case the units of electrical charge, of the induction
of a magnetic field, of force, of energy and, naturally, of
any other physical quantity should be independent, basic
ones. On the other hand, a supposedly “philosophical”
substantiation of the fact that only one unit should be the
basic one is also possible, since there exists a mutual rela
tionship between all phenomena of nature, reflecting the
unity of matter. Thus, attempts to substantiate the number
of basic units on the basis of “general philosophical con
siderations” lead to two diametrally opposite conclusions,
namely, the number of basic units should be infinitely
great or, on the contrary, there should be only one basic
unit.
Both these conclusions are erroneous. The matter is
that while the physical quantities reflecting the actual
properties of the world surrounding us are indeed infinitely
varied and cannot be reduced to one another, units of mea
surement are not by themselves objects of nature, and are
only an auxiliary apparatus for studying it. The laws of
nature do not in any way change their objective character
when one set of units is replaced with another one, in the
same way as no mathematical laws change when the decimal
system of counting is replaced with the binary one. For
lliis reason the main requirement which a system of units
should meet is that it must be convenient as possible for
practical calculations.
Assuming that the number of basic units is in principle
quite arbitrary and can be both increased and decreased,
we do not at all presume that qualitatively different phy
sical phenomena can be reduced to one another, in parti
cular to purely mechanical phenomena. The measurement
of different physical quantities, however, can be reduced

to the measurement of mechanical or even geometrical quan
tities, and, consequently, there is a possibility of making
the corresponding units derived ones.
To show more clearly the arbitrariness of the number
of basic units, let us turn again to the example on the estab
lishment of the unit of force considered above. We saw
that Newton’s second law and the law of universal gra
vitation can be used with equal right as the defining rela
tionship for this purpose. There is also a third possibility,
however, namely, by combining both laws, we can use
as the defining relationship the resulting equation expressing
the combined law. This equation can be expressed as
= ( 1 . 11)
The meaning of this equation consists in that the acceleration

acquired by a material point under the influence of attrac
tion to another fixed material point with a mass of m and
at a distance of r from the first one is proportional to the
mass m and inversely proportional to the square of the
distance r. The factor of proportionality C in formula (1.11)
is in essence the ratio between the gravitational constant
and the inertial one. The combination of Newton’s second
law and the law of universal gravitation into a single law
is not at all artificial, as it may seem to be at first sight.
It can be easily seen that formula (1.11) is equivalent to
Kepler’s third law (which is a law of nature discovered by
experiment, and, as should be noted in passing, was dis
covered before Newton’s laws). Indeed, let us assume for
the sake of simplicity that the planets move along circular
orbits with a constant angular velocity o) or a sidereal
period (period of revolution) T and substitute for the cen
tripetal acceleration in formula (1.11) its expression
a = co2r (1.12)
From the resulting expression, after the relevant transfor
mations, it is simple to see that the squares of the periods
of complete revolution of the planets around the Sun are
proportional to the cubes of the radii of their orbits. A more
general treatment taking into account that the planets
SYSTEMS OP BASIC AND DERIVED UNITS 31
move along ellipses leads to the same expression, the only

difference being that the square of the mean distance from
a planet to the Sun is substituted for the square of the radius
of its orbit. Kepler’s law, however, is true not only for the
motion of the planets about the Sun, but also for the motion
of satellites around their planets. Using the relationship
between the angular velocity and the period of revolution
2jx
to = y*, we can, by combining Newton’s second law and
the law of universal gravitation, write Kepler’s law as
follows
T2= 4n2CR* (1.13)
where the factor C is the same for all the planets moving
around the Sun; it is also the same, but of a different mag
nitude, for all the satellites of Jupiter; it is the same, but
again of a different magnitude, for all the satellites of
Saturn; it is the same, but also of a different magnitude,
for the Moon and artificial satellites of the Earth, etc.
In other words, this factor is the same for any bodies revol
ving about a single common centre, but is different for
different heavenly bodies serving as the centre of rotation.
It is easy to see that this factor is inversely proportional
to the mass of the body in the centre of the given system
(in the above examples the Sun, Jupiter, Saturn, and Earth,
respectively). By separating the mass from the factor C,
Kepler’s law can be rewritten as follows:
(1.14)
where C' is now a universal factor that depends only on
the units selected.
Formula (1.14) can serve as the defining relationship
for establishing the unit of mass as a derived unit if we
assume that C' = 1.
A system of units constructed in this way will have only
two, and not three, basic units, i.e., length and time. Of
great significance here is the fact that both the inertial and
the gravitational constants become dimensionless, and in
particular can be made equal to unity. This, of course, is
not accidental. If we analyse how we have found it possible
:\2 UNITS OK PHYSICAL QUANTITIES AND THEIR DIMENSIONS
to reduce the number of basic units, it will be easy to see

that this was achieved by making both constants dimen
sionless.
It thus follows that the number of basic units is closely
related to the number of factors in the expressions of phy
sical laws and definitions. These factors of proportionality,
such as the gravitational and inertial constants, determined
depending on the basic units and the defining relationships
selected, have been called universal constants. In this they
differ from the so-called specific constants characterizing
various properties of separate substances (these properties
include molecular weight, critical temperature, and electric
permittivity).
In principle, universal constants are always present in
the expressions of all physical laws and definitions, but by
appropriately selecting the units we can equate a certain
number of them to unity (or any other constant number).
Consequently, the greater the number of basic units used
for constructing a system, the greater the number of uni
versal constants that will be present in the formulas.
A reduction in the number of basic units is always accom
panied by a reduction in the number of universal constants.
It is natural to ask whether a further reduction in the number
of basic units to one (or even to none!) is possible in this
way.
Below, in dealing with the methods of constructing
a system of units of electrical and magnetic quantities, we
shall show that it is not difficult to reduce the number
of basic units to one. And what is more, one of the equations
of atomic physics, determining the so-called fine-structure
constant, makes it possible to construct a system without
any basic units whatsoever. At first sight this seems to be
a paradox. As we shall see in the following (Sec. 9.8), howe
ver, there actually exists such a possibility. It has been
found that if we equate certain constants to unity, we shall
thus rigidly fix the dimensions of the units of all physical
quantities.
Having analysed the fundamental principles of con
structing a system of units, we have become convinced that
there is almost unlimited arbitrariness in selecting the
ways of doing this.
This arbitrariness is such only theoretically, however.

Since a system of units is a sort of apparatus intended for
facilitating calculations in science and engineering, it must
comply with a number of practical requirements. From this
viewpoint the method of constructing a system of units
and, in particular, the number of basic units are not a matter
of indifference and to a certain extent are limited.
A too great number of basic units inevitably results
in a great number of universal constants in the physical
formulas, which makes it difficult to remember them and
leads to more complicated calculations. In addition, enor
mous work would be required to establish standard spe
cimens of all the basic units. The accuracy which these
standards would be established with would be different,
and as a result the universal constants in the formulas
expressing physical laws and definitions would also be of
different accuracy. On the other hand, a too small number
of basic units would limit the possibilities of constructing
the derived units to such an extent that a considerable part
of the latter would inevitably be either too great, or too
small, and, consequently, inconvenient for practical work.
It should be noted in passing, however, that the expression
“dimension of a unit convenient for practical work” has
at present become somewhat diffused owing to the fact
that the range of the dimensions of quantities encountered
in science and engineering is exceedingly broad. For in
stance, in nuclear physics lengths of the order of 10-15 m
are encountered, and in astronomy of the order of 1022-102(i m.
The capacities of electric power plants exceed 109 W, while
the power of a signal that can be picked up by a radar sta
tion is less than 10"16 W. In scientific research the values
of pressure range from below 10~15 to tens and hundreds
of thousands of atmospheres.
There are also other practical considerations that make
a system with a too small number of units quite unsuitable.
Some of them will be considered below when setting out
Ihe fundamental conceptions of the analysis oj dimen
sions*
With a view to what has been said above, it has been
found good practice to construct a system haviug about
three to six basic units.
;i-i0 4 o
34 UNITS OF PHYSICAL QUANTIT IES AN D T H E IR DIMENSIONS
1.5. Selection of Basic Units

It is expedient to select such quantities as the basic ones
that reflect the most general properties of mailer. Since
space and time are forms of existence of matter, it is natural
to include length and time in the basic units. Seeing that
mass is one of the most general characteristics of matter,
a unit of mass is taken as a third basic unit in most systems.
Such a system constructed on these three units (length,
mass, and time) was first proposed by Gauss and named the
absolute system by him.
Later this concept gradually lost its non-ambiguity. By
absolute were sometimes meant systems constructed on
quite definite units of length, mass and time (centimetre,
gram and second), sometimes, on the contrary, this name
was given a broader meaning, considering as absolute any
system having a certain limited number of basic units and
including all the remaining units from the field of geometry,
mechanics, electricity and electromagnetism as derived
ones. At present the term “absolute system'’ is used less
and less frequently, moreover since, as we have seen, there
is no criterion that would make it possible on the basis of
considerations of principle to give preference to any definite
system and attach such a committing name to it.
Among the existing systems of units the most favoured
ones are those based on units of the three quantities indi
cated above, some of the systems limiting the number of
basic units to only these three. The so-called technical
system covering only geometrical and mechanical measure
ments also has three basic units, but here the third unit
is that of force, and not mass. By decisions of the 10th and
11th General Conferences on Weights and Measures there
has been introduced the International System of Units,
designated SI, that covers the measurement of all mechani
cal, electrical, heat and light quantities. The basic units
of this system are length, time, mass, intensity of electric
current, temperature and luminous intensity. In accordance
with USSR State Standard GOST 9867-61, the International
System of Units has been introduced as preferable in all
fields of science, engineering and the national economy,
and also in educational institutions. Together with this
SYSTEMS OF BASIC AND D E R IV E D UNITS 35
system, however, the use of certain other systems, and also

of a number of non-system units, is also allowed.
Of great importance in establishing the basic units is the
possibility of retaining the constancy of a unit, of checking
and reproducing it, and if lost, of restoring it. For this
reason the trend appeared of relating the basic units to
quantities encountered in nature.
In the era of the Great French Revolution a special corn-
mil tee including the most renowned French scientists
of that time (Borda, Condorcet, Laplace and Monge), and
( routed in May 1790 by edict of the National Assembly,
proposed to take as the unit of length one ten-millionth
of a quarter of the Earth’s meridian. On March 30, 1791,
Ibo committee’s proposal was approved, and it began to
dclermine the accepted unit. As a result of its work, in
1799 there was introduced in France the “Metre vrai et
dilinitif” (“the genuine and final metre”) that served as the
basis of the metric system. The prototype of the metre
was a specially made platinum bar, which at present
is kept in the French National Archive (the “archive
metre”).
Together with the metre a unit of weight, the kilogram,
was introduced that was originally defined as the weight
of a cubic decimetre of water at 4°C. In the same way as
a sinndard bar was made for retaining the metre, a standard
weight, the prototype of the kilogram, was made for retain
ing the kilogram.
The second, determined as 1/86 400 of the mean solar day,
was taken as the unit of time.
I he increase in the accuracy of measurements as a result
«*l Ilie progress of measuring instruments and techniques
made it possible, however, to find that there was a slight,
bni quite measurable discrepancy between the selected
mnis and the prototypes made for them. The only exception
was |he second, which owing to the high accuracy of astro
nomical measurements remained practically unchanged and
nqiiiied only greater accuracy of the formulation.
In Ibis connection the problem arose of whether to make
iicw prototypes or to accept the existing discrepancies
mid lake as the legal units the measures determined by the
cm Iing prototypes.
3*
36 UNITS OF PHYSICAL Q U A N TITIE S A N D TH E ]K DIMENSIONS
Besides the fact that changing of the latter would in

itself involve enormous difficulties and inconveniences,
there would be no guarantees that a new more accurate
determination would not require them to be changed again.
For this reason it was found necessary to accept the pro
totypes as the basic legal standard units.
Thus the following units were established, which were
taken as the basic ones:
the unit of length—the metre (m)—defined as the distance
between the centres of marks made on a bar of a platino-
iridium alloy at 0°C. The platino-iridium alloy was selected
because if has a very low coefficient of thermal expansion,
while the shape of the cross section of
the bar (Fig. 1) met the requirement
of the smallest possible deflection,
the unit of mass—the kilogram (or
kilogram-mass) (kg)—the mass of a
plalino-iridium weight,
the unit of force—the kilogram-
force (kgf)*—the weight of the same
Fig. 1 prototype at its place of storage in
the International Chamber of Weights
and Measures at Sevre (near Paris);
the unit of time—the second, determined, as previously,
as 1/8G 400 of the mean solar day. In astronomy and related
fields the stellar second was taken defined as 1/80 400 of
a stellar day. Since owing to the rotation of the Earth
around the Sun the number of stellar days in a year is
greater by one than the number of solar days, then the
stellar second is 0.99720957 of the solar second.
The following systems were constructed on the basis of
the above units and their decimal fractions:
the cgs system, whose basic units are the unit of length—
the centimetre (cm), equal to one-hundredth of a metre;
the unit of mass—the gram or gramme (g), equal to one-
thousandth of a kilogram; and the unit of time—the second
(s).
* A number of other names were proposed for this unit, such as

kilograv and kilopond, but none of them were introduced to any
considerable extent. The name kilopond is used in Germany,
SYSTEMS OF BASIC ANJ) D E R IV E D UNITS 37
The mk(force)s system, whose basic units are the unit

of length—the metre (m); the unit of force—the kilogram-
force (kgf); and the unit of time—the second (s).
The mk(force)s system included only geometrical and
mechanical units, while the cgs system also covered elec
trical and magnetic measurements. The latter system
branched out into two independent ones, one of which was
based on electrostatic, and the other on electromagnetic
interactions. Correspondingly, the first of them was named
the electrostatic system (cgse), and the second the electro
magnetic system (cgsm). The units of both these systems,
however, were found to be inconvenient for practical work,
and as a result auxiliary practical units were introduced
for measuring quantities related to the process of flow of
a current (current intensity, potential difference and elec
tromotive force, resistance, work, power, etc.).
Originally these units did not form a harmonious system,
and could not even be used for solving problems involving
electrostatic and electromagnetic phenomena. In 1902 the
Italian engineer Giorgi proposed to so extend the system
of practical units as to make it just as universal as the cgs
system, i.e., covering all measurements in the fields of
mechanics, electricity and electromagnetism. Since this
system was to include practical units already being exten
sively used, it was found possible to retain the latter only
on condition of introducing at least one additional univer
sal constant, which is equivalent to including one of these
units in the number of basic ones. It was intended to use
in this capacity the unit of one of the following quantities:
the amount of electricity (electric charge), current inten
sity, potential difference, resistance, capacitance, induc
tance, magnetic flux, and magnetic permeability. A special
name “magif was even proposed for the latter unit.
After a number of discussions, with a view to the metro
logical considerations of the convenience and reliability
of reproducing the unit, the decision was reached to take
Ihe unit of current intensity, the ampere, as the fourth
basic unit. The definition of the ampere appreciably differs
in its nature from the definition of the other basic units—
Ihe metre, kilogram and second. The matter is that the
ampere was first introduced as one-tenth of the derived unit
38 UNITS OF PHY SICAL QUANT ITIES AND TH E IR DIMENSIONS
of current intensity in the cgsm system. For this reason,

although the ampere has been promoted to the rank of basic
units, it is in essence defined as a derived unit. According
to the accepted definition, “the ampere is the intensity of
an unchanging current which, upon flowing along two paral
lel conductors of infinite length and with a negligibly small
round section, arranged at a distance of one metre from
each other in a vacuum, would induce between these con-
ductors a force equal to 2 X 10 ~7 newtons per metre of
length”. We shall consider this definition in greater detail
and explain it in the chapter on the units of electrical andi
magnetic quantities.
In connection with the fact that the metre and kilogram'
were defined not as natural values, but according to proto
types, one of the advantages of the metric system, namely,,
its intactness and the possibility of exact reproduction of the-
units, has been lost. A further increase in the accuracy of
measurements made it possible to partly return to the
establishment of the basic units in accordance with the>
measurement of natural quantities. The determination of
the unit of mass, the kilogram, was retained according to>
the international prototype, while it was found possible
and most expedient to relate the length of the metre to the
wavelength of a definite line of the spectrum, for which
purpose the orange line of krypton was taken. Since natural
krypton contains six isotopes, whose spectral lines, although
very slightly, differ from one another, the definition of the
metre through wavelength is given more precisely by indi
cating that the isotope of krypton with a mass number of
86 (36Kr86) is taken as the source. The spectral line taken
corresponds to the transition of an electron in an atom of
krypton between the quantum states that in spectroscopy
are designated by the symbols 2p10 and 5d5. According to
.definition, a metre contains 1 650 763.73 wavelengths of
this spectral line in a vacuum.
The definition of the second has also been defined some
what more accurately, since the improvement in the accu
racy of measuring time made it possible to establish a cer
tain lack of constancy in the mean day. The new definition
of the second is based on the so-called tropical year—the
interval of time between two vernal equinoxes. According
SYSTEMS OF BASIC AND D E R IV E D UNITS 39
lo the new definition, a second is 1/31 556 925.9747th part

of the tropical year beginning at 12 noon on December 31,
1899. (According to the astronomical recording of time we
should write “12 noon on January 0, 1900”.) The indication
of a definite year has the aim of taking account of the cir
cumstance that the tropical year itself decreases by about
0.5 second in a century.
The progress of molecular and atomic radiation spectro
scopy has made it possible to establish a sufficiently accurate
relationship between the unit of time and the period of
oscillations corresponding to a definite spectral line. For
this reason a decision of the 13th General Conference on
Weights and Measures (1967) gave a new definition of the
second, according to which it is the duration of 9 192 631 770
periods of radiation corresponding to the transition between
two hyperfme levels of the principal state of an atom of
r)!iCs133 (the isotope of cesium with a mass number of 133).
It should be noted that in principle a unit of mass could
also be determined not by the mass of a prototype, but by
relating it to the mass of an atomic particle (for example,
a neutron). Unfortunately, the accuracy of determining
atomic masses is at present inferior to that of measuring
a mass by weighing.
With respect to the cgs system, then, since it is con
structed on three basic units (length, mass, and time), it
completely covers all geometrical, mechanical, electrical,
and magnetic measurements. It has been found most con
venient to use such a variant of the system in which electro
static quantities are measured in cgse units, and magnetic
ones in cgsm units. This system has been named the sym
metrical, or Gaussian system of units and designated cgs.
In principle this system could be applied, naturally, for
a ny other, in particular thermal and light, measurements,
lor which purpose the relevant quantities should be related
l>y defining relationships. The exceedingly widespread use
of temperature in science, engineering and everyday life,
however, makes it expedient from a practical viewpoint to
include it into the basic quantities. In light engineering
the quantities characterizing the subjective perception of
light (luminous intensity, illumination, and luminance)
a re significant. For this reason the use of only energy para-
40 UNITS OP PHYSICAL Q UANT ITIES A N D T H E IR DIMENSION S
meters in defining these quantities deprives them of the

most important property—a characteristic of their action
on our vision. Thus, in applying the cgs system to all phy
sical phenomena, not three, but five of its units should be
considered as basic ones.
As mentioned above, at present the International System
(SI) has been legally made the most preferable one. This
system is a development of the practical system of electrical
units, in which two more units—the degree, measuring the
thermodynamic temperature, and the candela, measuring
luminous intensity have been added to the basic units metre,
kilogram, second, and ampere. The definitions of these
units will be given below when considering the units relating
to the corresponding field of physics.
In 1919 in France there was adopted the metre-ton-second
system, in which the unit of mass is a ton (1 000 kilograms).
At one time this system was greatly advertized and was
even legalized by the relevant standards. It did not become
widespread, however, and at present it is practically com
pletely out of use, except for some of its units that have
become non-system ones.
1.6. Non-System Units

Notwithstanding certain advantages obtained by using
units determined by a system, up to the present wide use
is made of various units that do not fit into any of the
systems. Many of them cannot be discarded owing to the
convenience of using them in definite fields, while others
have been retained as a result of historical traditions.
In prerevolutionary Russia there was an old Russian
system of weights and measures that in 1924 was replaced by
the metric system. The names of the units of this system
have been retained at present only in Russian sayings and
proverbs, and only the unit of weight pood (40 Russian
pounds or 1G.3805 kg) is sometimes encountered in reports
on the production of agricultural goods. Even at present
in some countries units have been conservatively retained
whose inconvenience not only consists in that they are
not constructed according to the decimal system, but also
in that one name often hides several units (there are several
SYSTEMS OF BASIC AND D E R IV E D UNITS <1
miles, gallons, not completely accurately coinciding inches,

etc.).
Among the non-system units, decimal multiples and
submultiples of units should beset apart into a first group.
The names of these units are formed with the aid of the
relevant prefixes (deci-, centi-, milli-, deca-, hecto-, kilo-,
etc.). A list of these prefixes and the symbols used for them
are given in Table 52.
A second group of non-system units is formed by units
constructed from the units of a system without following
the decimal principle. These include first of all the units
of time, minute and hour.
A third group includes, finally, units that have no relation
with units of the established systems. They include, in
particular, the unit of length inch, the unit of the amount
of heat calorie, the units of pressure standard atmosphere and
millimetre of mercury, etc.
When considering the units of a quantity in the following
chapters, together with the units included in the cgs, SI
or mk(force)s systems, we shall give the most widespread
non-system units and show their relation to the system
units.
CHAPTER TWO
CONVERSION OF UNITS
AND DIMENSION FORMULAS
2.1. Dimension Formulas
The existence of different systems gives rise to the problem

of converting units of one system into those of another.
Obviously, a change in the basic units should lead to a
change in the derived ones. Thus, for example, if we take
the kilometre as the unit of length instead of the metre,
then the unit of velocity will be “kilometre per second”,
which is 1 000 times greater than the unit “metre per second”.
If we take the hour as the unit of time and retain the metre
as the unit of length, then the unit of velocity will be “metre
per hour”, which is 1/3 600th of the unit “metre per second”.
Finally, if we take the kilometre as the unit of length and
the hour as the unit of time, the unit of velocity will be
“kilometre per hour” , equal to 1 000/3 600 ^ 0.278 m/s.
Thus it can be seen that any change in the basic units cor
respondingly changes a derived unit.
It is obviously desirable to find such a relationship that
would make it possible to determine how the derived unit
of a quantity of interest to us changes with a change in each
of the basic units. Such a relationship is called the dimension
formula of a unit of the given quantity.
For acquaintance with how the dimension formulas are
constructed and applied for the mutual conversion of units,
let us first consider the case when the systems have the
same basic quantities and use the same defining relation
ships. For example, the cgs and SI systems are such systems
for mechanical measurements, since only three basic units
are used for this purpose from both systems, namely, length,
mass and time, and the systems differ from each other only
in the dimension of the basic units.
conversion op units and dimension formulas 43
It should be noted that if a derived unit changes nv

times with a change in the unit of length of n times, the
given derived unit is said to have the dimension p relative
to the unit of length. In the same way, if a derived unit
changes proportional to the q-tli power of a change in the
unit of mass and to the r-th power of a change in the unit
of time, then the derived unit is said to have a dimension q
relative to the unit of mass and r relative to the unit of
time. If the unit of a certain quantity A has the dimensions
p, q and r relative to the units of length, mass and time
respectively, then this is symbolically written as
[A] = LpM«Tr (2.1)
where the brackets enclosing the symbol of the quantity A
denote that we are dealing with the dimension of a unit of
this quantity, while the symbols L, M and T are generalized
designations of the units of length, time and mass without
indicating the concrete magnitude of the unit.
Formula (2.1) can be interpreted to have the meaning
that if the ratios between the units of length, mass and
time in two systems are equal respectively to L, M and 1\
then the ratios between the derived units will be LvM qTr.
Formula (2.1) is called the formula of the dimension of
a unit of a given quantity, or, as is frequently said for bre
vity, the dimension of a given quantity. It can be easily
seen that a dimension formula can be written only for
such quantities whose quantitative characteristic meets the
condition of the absolute value of a relative magnitude.
It has been found that with any selection of the basic units,
ihe dimension formula of a derived unit is a monomial con
sisting of the product of the symbols of the basic units to
certain powers, and the latter can be positive and negative,
integers and fractions*.
In compiling the dimension formulas of derived units
we shall use the following theorems.
1. If the numerical value of a quantity C is equal to the
product of the numerical values of the quantities A and B ,
* Those who desire w ill find a simple proof of this tenet in the
luiok Dimensional Analysis by P. W. Bridgman, New Haven, Yale
University Press, 1932.
44 UNITS OF PHYSICAL QUANT ITIES AND T H E I R DIMENSION S
then the dimension of C is equal to the product of the dimen

sions of A and B, he.
\C\ = [A]\B\ (2.2)
In other words, if
[A] = Lp«Mq«Tr*
and
IB] = L pt>Mq*Tr*
then
[C] = L p«+pbM9-'+g>>Tr*+rb (2.3)
2. If the numerical value of a quantity C equal to the
quotient of the numerical values of the quantities A and B ,
then the dimension of C is equal to the quotient of the
dimensions of A and B , i.e.
|C|= [ ^ ] “ W <2'4»
or
[C] - (2.5)
3. If the numerical value of a quantity C is equal to the
rc-th power of the numerical value of the quantity A, then
the dimension of C is equal to the n~th power of the dimen
sion of A, i.e,,
[C] = M*] = M]n (2.6)
Hence, if
[A] = L]}M qTr
then
[C] = LpnM qnTrn (2.7))
The proofs of all these theorems are very simple, and for
this reason we shall only give the first of them.
If the numerical value of a quantity C is equal to the'
product of the numerical values of the quantities A and
this means that when measuring these quantities by means,
of the units cl , a2 and bx we get
Ci = A iB l ( 2 . 8)
CONVERSION OE UNITS AND DIM ENSION FORMULAS
where
A’= i - ' <2'9)

When measuring the same quantities by means of the
units r2, a2 and b2, we correspondingly gel
C \ ^ A 2B2 (2 AO)
where
^ = - . “ O <2-«>
Hy dividing Eq. (2.8) by Eq. (2.10) and taking into acco
unt Eqs. (2.9) and (2.11), we get
Co _ ^2 ^2 (2.12)
Ci di bi
— —LVaM q<
lTTa (2.13)
at
JlL —LPbM<,b fTb (2.14)

h
£ l —[fa+Vb qbTra'rrb (2.15)

Cl
Tims the first theorem has been proved. It is quite obvious
ibat the remaining theorems can easily be proved in the
same way.
It is important to note the following circumstance. Since
Ilit' method of constructing a derived unit includes the
■•■Ination to unity (or to any other arbitrary constant number
not depending on the dimensions of the basic units) of the
lac. tor of proportionality in the defining relationship, this
means that we agree to consider this factor to have a zero
dimension or, as is generally said, to be a dimensionless
one. It is understood, in addition, that any constant nume
rical factor obtained in mathematical operations should
al so be considered as a dimensionless one.
Let us explain the above with examples.
4 6 UNITS OP PHYSICAL QUANTITIES AN D TI IE IR DIMENSIONS
1. The dimension of the area of a square

[Asq] -■=[I? - L 2M°M° (2.16)
or omitting here, as will be done in the following, the
symbols of the basic units having* a zero dimension,
[Aaq]~L* (2.17)
2. The dimension of the area of a circle
[Acir \ = \ ~ ] [ D ? ^ L ^ (2.18)
since the factor ji/4 is a constant one not depending on the
dimensions of the basic units, and therefore a dimensionless
one. For this reason the dimension of the area of any geo
metrical figure, regardless of its shape, will be
[A ]-Z /2 (2.19)
3. The dimension of velocity can be determined from
the formula for the velocity of uniform motion:
= ( 2 . 20)
4. The dimension of acceleration is determined from the

formula for uniformly accelerated motion:
[a] = = iT -t (2.21)
rl
For purposes of illustration let us use the last formula to
find how the unit of acceleration will change if we change
over from measuring length in metres and time in seconds
to kilometres and minutes, respectively. Here the unit
of length will increase 1 000 times, and that of time
60 times. According to formula (2.21), the unit of accelera
tion will change 1 000/602 = 10/36 times, i.e., the new unit
of acceleration will equal 0.278 of the old one.
5. The dimension of kinetic energy, determined by the
formula
£/<=— (2.22)
will obviously be equal (in the SI and cgs systems) to
[Eh] = L*MT^ (2.23)
CONVERSION OF UN ITS AN D DIMENSION FORMULAS 47
From the latter formula, in particular, it follows that if,

in measuring length, we change over from centimetres
to metres and in measuring mass from grams to kilograms,
and retain the second as the unit of time, the unit of kinetic
energy will increase (100)2 X 1 000 107 times.
6. Newton's second law, written in the form
Ft -- mv2~-mvl (2.24)
where the product Ft is called the impulse of the force,
and mv the quantity of motion or momentum *, determi
nes the dimension of force:
[F] = LMT~2 (2.25)
In the following, when investigating the units of derived
<|nantities, we shall always use the dimension formulas.
The dimension formula of a derived unit often determines
Im>lli its name and the symbols used to designate it. For
example, the unit of velocity “metre per second” is desig
nated m/s, the unit of area “square metre” m2, etc.
2.2. Conversion of Dimension When Using Different

Basic Units
I f we change the quantities whose units are taken as the
basic ones without changing the defining relationships,
i lien the dimension formulas will change correspondingly,
h»r example, when changing over from the SI or cgs system
in I lie mk(force)s system, in which the basic quantities
include force instead of mass. Here the transition from one
vs!cm to the other can be performed if in the dimension
Ini imilas we substitute for the dimension of the correspond-
■im>; basic unit its dimension expressed in the other system.
11 we designate the dimension of force in the mk(force)s
\ inn by the symbol F, then from formula (2.25) we can
"Main the dimension of mass in this system:
Af = [m l=Z"1/ T 2 (2.26)
1 The term “momentum” is at present mainly used in theoretical
in" lianics. The symbol p is sometimes used instead of mv. In the
ii" “i v of relativity, quantum mechanics, atomic and nuclear physics
11" ici m “impulse” is generally accepted, The term “quantity of
.... lion" is practically obsolete.
By inserting the dimension of M from this formula into the

dimension formulas of various quantities in the SI and cgs
systems, we shall obtain the dimensions of these quantities
in the mk(force)s system. Thus, for example, the corres
ponding dimension of kinetic energy is
lEh] = LF (2.27)
As an example of reverse conversion the dimension of pres
sure and mechanical stress can be given, which in the
ink(force)s system will be
[p] = L~*F (2.28)
and, correspondingly, in the SI and cgs systems
[p] = //W 2 ,' a (2.29)
2.3. Conversion of Dimensions with Different

Defining Relationships
When establishing a derived unit with the aid of a de
fining relationship, i.e., a mathematical formulation of
a definition or a law connecting the given quantity with
quantities taken as the basic ones or determined previously,
the factor of proportionality in the relationship is assumed
to equal unity or some other constant number.
From the viewpoint of construction of the dimension
formula, this means that we deprive it of a dimension rela
tive to the basic units or, which is the same, give it a zero
dimension. In other words, we agree to consider the factor
unchangeable upon any change in the basic units provided
that the defining relationship does not change. Should this
condition not be observed and another defining relationship
be used for determining the derived unit, then the factor
of proportionality may change correspondingly. For exam
ple, if we use the area of a circle instead of that of a square
for determining the unit of area, then, as we have seen
(in Sec. 1.4), the factor of proportionality in the formula
for the area of a square becomes equal to Ain instead of
unity, since the factor of proportionality is taken equal to
unity in the new defining relationship (the formula for
the area of a circle). The transition from square to round
CONVERSION OF UN IT S AND DIMENSION FORMULAS 49
units of area makes it necessary to correspondingly change

the factors of proportionality in all formulas relating to the
measurement of areas. Here, however, the dimension of area
remains the same, since as a matter of fact in this case

also we use the same theorem on the relationship between
the area of geometrical figures and their linear dimensions
and take the factor of proportionality in a particular expres
sion of this theorem (the formula for the area of a circle)
equal to unity.
The above example can be considered, by the way, not
as a transition from one defining relationship to another
one, but as the replacement of the factor of proportionality,
previously taken as unity, by a factor of 4/ji in the formula
for the area of a square.
Such a change of the defining relationship is, however,
possible that will make the factor of proportionality dimen
sional, i.e., dependent on the dimensions of the basic units.
This can be best illustrated by using the example of estab
lishing the unit of force as a derived unit in systems with
length, mass and lime as the basic quantities. In the usual
determination of the unit of force by means of Newton’s
second law, we obtain the following dimension of force
[see formula (2.25)]
\F\ = LMT-*
If we insert this dimension in the expression for the law
of universal gravitation [formula (1-10)1, then the following
dimension will be obtained for the gravitational constant:
iG l^Z rW -1^ 2 (2.30)
(Here and below we shall revert to the generally used sym
bol G for the gravitational constant.) The fact that the
gravitational constant has a dimension means that its
numerical value depends on the selection of the basic units.
To find this relationship it should be remembered that
a dimension formula shows how a derived unit changes
when the basic units are changed. Therefore, by conditio
nally introducing “the unit of the gravitational constant”,
we can say on the basis of formula (2.30) that this “unit”
k— KUO
50 UNITS OF PHYSICAL QUANTITIES AND TH E IR DIMENSIONS
changes in proportion to the cube of the unit of length,

inversely proportional to the unit of mass and inversely
proportional to the square of the unit of time. Since the
numerical value of a quantity upon a change in the units
measuring it changes in the reverse proportion [see for
mula (1.1)1, then, consequently, the numerical value of
the gravitational constant will be inversely proportional
to the cube of the unit of length and directly proportional
to the unit of mass and the square of the unit of time. Thus,
if the basic units are metre, kilogram and second, the gra
vitational constant is numerically equal to 6.67 X 10"11,
then upon changing over to the basic units, centimetre,
gram and second, its value will become 6.67 X 10~8.
If for determining the unit of force we use the law of
universal gravitation instead of Newton’s second law, then
we make the gravitational constant dimensionless, i.e.,
independent of the basic units, but equal to a constant
number, for example, unity. The dimension of force will be
equal to
\f] = L~2M 2 (2.31)
while the inertial constant, which was previously equal to
unity and had no dimension, now acquires the dimension
[Ci]=L-*MTz ( 2 . 32 )
The change in the dimension of force and the appearance
of a dimensional inertial constant with the simultaneous
disappearance of a dimensional gravitational constant will
lead, naturally, to a different mathematical expression of
the laws and definitions in the field of mechanics and to
a change in the dimension formulas. For example, the
dimension of work, which, as previously, is determined by
the product of the force, distance and the cosine of the
angle between their directions, will now be not
[W ]= L2MT~2
but
[W] - [/] \L] - L~2M 2•L = L~lM 2 (2.33)
The same dimension of work can be obtained in a different
way. If we consecutively deduce the relation betweei* the
CONVERSION OF UNITS AND DIMENSION FORMULAS 51
work and the change in the kinetic energy* in the new

system, we get
(2.34)
Inserting the dimensions of mass, velocity and the inertial
constant in the right-hand part, we get
[W] := L ~ m 2 (2.35)
Thus, when different systems differ from each other in
the selection of the defining relationships, it should be
remembered that the factors of proportionality, which
in one system are considered to be dimensionless (and
usually equal to unity), acquire a dimension in the other
system. Upon transition from one system to another the
dimensionless factor should be replaced by one having a
dimension, or vice versa, for determining the dimension.
If the number of basic units is reduced, as can be done,
for example, if we combine Newton’s second law and the
law of universal gravitation into one general law similar
to Kepler’s third law, then both the gravitational and the
inertial constants become equal to unity or another dimen
sionless number, and only the dimensions of length and
time will remain in the dimension formulas. The conver
sion of a dimension from systems with three basic units to
a system with two basic units can be performed if in the
relevant dimension formulas the dimension of mass is
replaced with its expression obtained from the formula
combining Newton’s second law and the law of universal
gravitation. If we write this formula as
(2.36)
and consider that C is dimensionless (for example, equal
lo unity), the dimension formula for the unit of mass
will be
\M] I?T~2 (2.37)
It this expression is inserted into any of the dimension
iormulas of force derived both from Newton’s second law
* We have used the term “kinetic energy” here for the expression
mi'1/2, although the actual expression for kinetic energy is Cimv2,l 2.
4*
52 u n its o f p h y s ic a l q u a n t it ie s and t h e i r Dim ensions
and the law of universal gravitation, the same dimensions

will naturally be obtained. Indeed
[F\ LMT~2- LAT~* (2.38)
IF] - Zr2A/a L4?1-4 (2.38a)
The same also concerns the dimensions of the units of the
other quantities relating to mechanics. Let us give some
of them for purposes of illustration. The dimension of work
and energy is
IW] = IJT~*
The dimension of the impulse and the momentum is
[ft] - 1mv\ - L*T~*
2.4. Determining the Relationship between Units

of Different Systems
It is the simplest to convert units of one system into
those of another when both systems are constructed on the
same defining relationships and the same basic quantities,
so that the basic units differ only in magnitude. It follows
from the above that since in this case the dimension formula
of a derived unit is the same in both cases, it is sufficient
to insert in this formula the ratios of the magnitudes of the
basic units, which should be given either by definition or
experimentally, for example by comparing the standards
of the relevant units. In addition to the examples given
above let us establish the relationship between two units
of force determined on the basis of Newton’s second law
with the following basic units: centimetre, gram, second,
and foot, pound and minute. The relations between the
basic units are as follows: 1 foot - 30.48 cm (comparison
of standards), 1 pound 453.0 grams (comparison of stan
dards), and 1 minute — 60 seconds (definition). On the
basis of the dimension formula
[F]=LMT~2
we determine the relation between the units of force
unit of system foot, pound, minute 30.48x453.6 _3 3 4
unit of system centimetre, gram, second ~~ (60)2
CONVERSION OF UNITS AND DIMENSION FORMULAS 53
Matters are more complicated when with the same defining

relationships, units of different quantities are used as the
basic ones, as, for example, in the SI and mkg(force)s
systems.
Since at least one of the quantities that is taken as a basic
one in one system is a derived one in the other system and
vice versa, the relationship between the corresponding
units should be established. This can obviously be done only
by experiment. When determining the relationships between
the units of the SI and rnkg(force)s systems, the free falling
of a body can be used as such an experiment. Here we shall
make use of the fact that in the mkg(force)s system the
unit of force is the weight (i.e., the force of attraction to
the Earth) of the prototype body—the kilogram-while
in the SI system the unit of mass is the mass of the same
prototype body. It is general knowledge that any free
falling body (in particular, the prototype kilogram) acquires
under the action of its weight an acceleration that at each
given point of the globe is the same for all bodies, but differs
at different points of the globe, increasing from the value
of 9.7805 m/s2 at the equator to 9.8322 m/s2 at a pole. Mea
surement of the acceleration of gravity at the place of storage
of the prototype kilogram (Sevres) gave the value
9.80665 m/s2. This value has been called the normal acce
leration and has been fixed as a constant value not to be
precised*. Whenever the accuracy of measurements or'
calculations allows an error of over 0.3%, at all points;
of the Earth’s surface the weight of one kilogram can be
laken equal to the established technical unit of force—the
kilogram (force).
Thus, the result of an experiment consisting in measuring
Ilie acceleration of free falling of a body (in our case the
prototype kilogram) can be formulated as follows: “a force
of 1 kgf imparts to a mass of 1 kg an acceleration of
9.(SI m/s2”.
Now let us establish the unit of force in the SI system and
llie unit of mass in the mk(force)s system, using in both
instances Newton’s second law. Obviously, in the SI system
* In the following we shall use only the approximate value

'i.Si m/s2 for the acceleration of gravity.
54 UN ITS OF PHY SICAL QUANTITIES AND TH E IR DIM ENSIONS
the unit of force is the force that imparls to a mass of 1 kg

an acceleration of 1 m/s2. This unit of force, as is known
(see Sec. 1.3) is called the newton (N). In the mk(force)s
system the unit of mass is a mass that under the action
of a force of 1 kgf acquires an acceleration of 1 m/s2. This
unit of mass lias not been given a special name established
by any State Standard. Sometimes it is called the “technical
unit of mass” (t.u.m.). USSR State Standard GOST 7664-01
requires this unit to be designated, according to its dimen
sion, i.e., kgf-s2/m. Some time ago professor M. P. Malikov
proposed calling this unit the inerta* (i). Although this
name has not only never been legalized anywhere, but has
not even come into use, we shall use it in the following
owing to its brevity and convenience of designation.
Let us now write the following three equations
1 k g f = l i-1 m/s2 (2.39)
1 k g f = l kg-9.81 m/s2 (2.40)
1 N = 1 kg-1 m/s2 (2.41)
The first equation expresses the definition of the technical
unit of mass, the inerta, the third the definition of the
newton, and the second the result of the experiment described
above. In the first two equations the same forces impart
different accelerations to different masses. Since the acce
lerations acquired are inversely proportional to the masses,
then, as a corollary, we get
1 i = 9.81 kg (2.42)
or, conversely,
1 k g - A r i=rj°-1 0 2 1 (2 .42a)
In the second and third equations different forces impart
different accelerations to the same masses. Here, since the
forces acting on the bodies are proportional to the accele
rations imparted to them, we have
1 kgf = 9.81 N (2.43)
* Other names proposed for this unit are the metric slug, the mug}
the par, and the TME.
CO NVERSION OP UNITS AND DIMENSION FORMULAS 55
or, conversely,
1 N = 9 ^ r k g f = = 0 -102 k g f ( 2 -4 3 a )
Mastering of these relationships will help our readers

lo avoid numerous errors that are usually made during
the initial period when solving problems in mechanics.
Since we now have the relationships between the units
of mass and force in the SI and mk(force)s systems, it will be
simple to establish the relationships between any derived
units of these two systems, using the dimension formulas.
It will be almost as simple to determine the relationships
between units of the cgs and mk(force)s systems, since the
former are very simply related to the units of the SI system.
Let us give two examples for purposes of illustration.
1. Find the relationship between the units of pressure
in the mk(force)s and cgs systems. Let us use dimension
formula (2.29) of the unit of pressure in the cgs system
Since the unit of length in the mk(force)s system (the metre)

is 100 times greater than that in the cgs system (the cen-
iimetre), the unit of mass in the mk(force)s system (the
iuerta) is 9.81 X 1 000 = 9 810 times greater than the
unit of mass in the cgs system (the gram), and the unit
of time (the second) is the same in both systems, then, accord
ing to the dimension formula, the unit of pressure in the
mk(force)s system will be 100'1 X 9 810 = 98.1 times
urea ter than that in the cgs system. The same result can be
oblained if we use dimension formula (2.28) of the unit
of pressure in the mk(force)s system
\p\ = L-*F
The ratio of the units of length here, as previously, is 100,

while the ratio of the units of force, as the reader himself
< in easily find, is 9.81 X 105. The ratio of the units of pres-
:ni*e will accordingly be 100"2 X 9.81 X 105 = 98.1, which
corresponds to the result obtained above.
2. Find the relationship between the units of power in
i lie mk(force)s and cgs systems. The dimension of the unit
5 6 UN ITS OF PHYSICAL QUANTITIES AND TH E IR DIMENSIONS
of power in the cgs system is

IP] - L2M T~s (2.44)
By using the known ratios between the units of length and
mass, we find the sought relationship
1002 x 9,810 = 9.81 x 107
In the same way when using the dimension formula of power
in the mk(force)s system
[PI = LFT-' (2.45)
the ratio of the units will be
100 x 9.81 x 105 = 9.81 x 107
Now let us consider the conversion of units in the most
complicated case when different defining relationships
are used to determine the derived unit in the two systems.
Here wc shall confine ourselves only to that case, which is
of the greatest interest, when the basic quantities in both
systems are the same.
Let us have a certain quantity A whose units av and a2
in two different systems (based on different laws) have the
dimensions
\A]{= and [A]2 —
the numbers A i and A2 that express the quantity A in
these units being in the following relationship
A i - CA2 (2.46)
Here C is a factor of proportionalitythat is now not an
abstract quantity, but one depending on the selection of the
basic units. The unit by means of which C is “measured”
obviously has the dimension
[C\ --- -Hr- - LK -nM n-nT'i-r* (2.47)
[-4]2
Since the numbers measuring a quantity in different units
are inversely proportional to these units, we can write
a{ ^ ~ a 2 (2.48)
CONVERSION OP UNITS AND DIMENSION FORMULAS 57
The ratio of the dimensions of the quantity A in the first

and second system gives the dimension of the factor C.
Thus, if we know the numerical value of this factor in one
system of units, it is possible to determine its numerical
value in any other system and thus find the relationship
between the corresponding units of the given quantity A.
Let us explain what has been said using the example of
force that we have considered. When measuring force with
the inertial unit, the law of universal gravitation has the
form
, /nr ^1^2
If the basic units in both systems are the same, then the
expression in the right-hand part represents the
same force of mutual attraction, but measured in gravi
tational units (the latter are sometimes called astronomical
units). Consequently, upon designating the number that
measures force in the inertial system by /*, and in the gra
vitational one by f g, we can write
ft = Gfe (2.49)
With a change in the basic units the numbers f t and f g
will also change, but not to the same extent. For this reason
the numerical value of the factor G will change. To deter
mine the nature of this change, let us use the dimension
formulas
[f)i^LM T~* and \f\g ^L~*M2
whence
[G] = 4 tt- = L m -n '-* (2.50)
1/Jg
As we have already seen (in Sec. 2.3), the numerical
value of the gravitational constant is inversely proportional
lo the cube of the unit of length and directly proportional
lo the unit ofmass and the square of the unit of time.
It should be remembered that with the basic units metre,
kilogram and second G = 6.07 X 10“u , and with the basic
units centimetre, gram and second G -- 0.07 X 10"8. Since
l lie relationship between the inertial and the gravitational
58 UNITS OF PHY SICAL QUANTIT IES AND T H E IR DIMENSIONS
units of force (let us designate them cp,- and (p^) is
then, when measuring the mass in kilograms and the dis

tance in metres, we get
<P/ = 6 . 6 7 x 1 0 - 1 1 ^ r" 1 ' 5 X 10,0 rpg ( 2 -5 1 a )
and when measuring the mass in grams and the distance

in centimetres, correspondingly,
<P*= 6.67xl0-» (p« ^ 1-5 x 1 0 7 (2.51b)

Introducing the designations of the units of force, we can
write instead of equations (2.51a) and (2.51b)
1 N ~ 1 .5 x 1010 kg2/cm2 units (2.51c)
1 d y n ~ 1 .5 x l0 7 g2/cm2 units (2.51d)
It is also not very difficult to determine the relationship
between units when the dimension of the basic units is
different in the two systems. This can be done in the sim
plest and most illustrative way if we first convert one of the
units into a system with the same basic quantities, but
with the dimensions of the basic units the same as in the
second system.
It goes without saying that all such conversions can be
accomplished only on condition that in the system having
a dimensional factor of proportionality the numerical value
of this factor is known either directly or can be obtained
by conversion from another system with the same defining
relationships.
2.5. Compilation of Conversion Tables

To avoid the conversion of one set of units into another
one in every calculation, it is good practice to compile
tables by means of which a quantity measured in one unit
can be expressed through any other unit of the same quan
tity. A special conversion table will be required for each
quantity.
CONVERSION OF U NITS A ND DIMENSION FORMULAS 59
The units that are to be converted are arranged at the

left-hand side of such a table, and those which they are
lo be converted to at the top of the table.
Let us take for example the units of length. The cor
responding table is given in Appendix 5 at the end of the
hook (Table 2). The number 39.4 at the intersection of the
lino “1 m = ” and the column “Inch” shows that one metre
contains 39.4 inches.
When compiling conversion tables, use is made either
of relationships based directly or indirectly on experiment
(as, for example, 1 kgf = 9.81 N), or on definition (for
example, 1 m = 100 cm), or established by the comparison
of standards or prototypes, or, finally, by calculations
similar to those given above and based on the use of dimen
sion formulas.
Such conversion tables, compiled for the most important
quantities considered and given at the end of the book, may
he useful in solving a great diversity of problems. These
Iables, in addition to the units of different systems, include
a number of the most popular non-system units.
2.6. On the So-called Meaning of Dimension

Formulas
The examples considered in the previous sections show
i hat the dimension formula of a unit of the same quantity
can have different forms depending on the defining relation
shi p used to establish the unit. We consider this tenet to
lx- quite important, since attempts are often encountered
in literature to find some “secret meaning” in dimension
formulas. Moreover, the widespread and abbreviated expres
sion “dimension of a quantity” used above is often under
stood literally as some unchangeable property characteristic
of the given physical quantity. The possibility of construct
ing different dimension formulas with a different selection
of the defining relationships clearly shows the erraneous
n il nre of such a view.
In this connection it will be appropriate to quote
M Planck: “From this we again see that the dimensions of
n physical quantity are not inherent in it, but constitute a
conventional property conditioned by the choice of the
£0 u n i t s op p h y s i c a l q u a n t i t i e s a n d t h e i r d im e n s io n s
.'system of measurement. If this circumstance had always

been properly appreciated, a great number of unfruitful
^controversies in physical literature, particularly concerning
that of the electromagnetic system of measurement, would
ihave been avoided'’*, and “the fact that when a definite
physical quantity is measured in two different systems of
units, it has not only different numerical values, but also
different dimensions, has often been interpreted as an
inconsistency that demands explanation, and has given
rise to the question of the 'real’ dimensions of a physical
quantity... it is clear that this question has no more sense
than inquiring into the ‘real1 name of an object”.**
Naturally, if we remain within the limits of a definite
system, then a definite dimension will be retained for the
unit of each quantify. In some of the simplest cases the
dimension formula will give a notion of how a derived unit
has been obtained from the basic ones.
This is natural, since, as we have already mentioned, the
principle underlying the construction of derived units
reflects the possibility of determining the value of a physical
quantity by indirect measurement. For example, the dimen
sion of velocity LT~l shows that to find the velocity the
distance and time should he measured, and the relevant
numerical values divided. It will be shown below that the
dimension of capacitance in the cgs system coincides with
that of length. This can he considered as a reflection of the
fact that the capacitance of insulated conductors having
the same shape is proportional to their linear dimensions.
Not many of such examples can be given, however, and
in the majority of cases the dimension formula does not
give a clear notion of the relation between a given physical
quantity and other quantities, in particular, those taken
as the basic ones.
Indeed, if we take as an example the dimension formula
of such a static quantity as pressure or mechanical stress,
which in the SI and cgs systems is
[p ] = i - w r 2
* Max Planck, / ntroduction to Theoretical Physics, vol, f, General

Mechanics* Sec. 28, London, 1933.
** H>jd, vol. 3, fileclrUity and Magnetism, vSec, 7i
CONVERSION OF UNITS AND DIMENSION FORMULAS ()1
il will hardly bo possible to find any physical moaning in

the unit of length and the square of the unit of time in
the denominator. And, of course, no concrete notions are
called forth by the dimension formulas of electrical units
in the cgs system, in which the symbols of the dimensions
of the basic units are quite frequently in fractional
powers.
The very limited significance of dimension formulas can
also be seen from the fact that the units of different quan
tities sometimes have the same dimension. This, of course,
should never be interpreted as to mean that these quantities
have a common physical nature. Moreover, we shall encoun
ter such quantities among those being considered whose
units are dimensionless in a certain system, i.e., do not
depend on the selection of the basic units.
A typical example of such quantities is an angle.
Although its units can be different (degrees, minutes, parts
of a circle, radians), none of them change upon a change
in the basic units.
Besides employing dimension formulas for the conversion
of units from one system into another and establishing
relationships between units, they are used for checking
the correctness of formulas obtained by theoretical deduc
tion. The constancy of a dimension formula within the
limits of a given system requires that the dimensions in
the left-hand and right-hand parts of any equation relating
different physical quantities (or, more exactly, the numbers
measuring these quantities) be the same. Otherwise when
going over from one set of units to another the equation
would be violated. For this reason, upon obtaining as a result
of reasoning or solution of a problem a formula expressing
the relationship between a quantity interesting us and
other quantities, the coincidence of the dimensions of the
left-hand and right-hand parts of the equation should be
checked. If these dimensions do not coincide, it can be
said that an error has been made and the equation is not
correct.
It should be understood that coincidence of the
dimensions is not at all a guarantee that the equation
obtained is correct,
62 UNITS OP PHYSICAL Q UANT ITIES AN D TH E IR DIMENSIONS
2.7. Brief Conclusions on Chapters One and Two

The previous sections set out the principles underlying
the construction of systems of basic and derived units and
dimension formulas, and also the methods of transferring
from one system to another.
For the reader’s convenience, the contents of these sec
tions is briefly summarized below in the form of the following
conclusions:
1. A measurement is a comparison of the given quantity
with another homogeneous quantity taken as a unit.
2. The condition for objective measurement and estab
lishment of units of measurement is the possibility of obtain
ing the absolute value of relative quantities.
3. The units of all quantities can in principle be selected
independently of one another. The existence of indirect
measurements together with direct ones, however, makes
it possible to relate the units of different quantities to one
another.
4. In constructing systems of mutually related units,
the units of several quantities are selected independently
of the others and of one another. Such units are convention
nally called fundamental or basic ones.
5. For all the remaining quantities there are established
the so-called derived units, that are related either to the
basic units or to one another with the aid of defining relation
ships that are mathematical expressions of physical laws
and the definitions of physical quantities.
6. The symbols of physical quantities in mathematical
expressions of physical laws and definitions do not represent
the quantities themselves, but are numerical values expres
sing these quantities in the units selected for measurement.
7. The relationship between a derived unit and the basic
ones is determined by the dimension formula (or, in brief,
the dimension), which is a monomial formed by the product
of the generalized designations of the basic units to diffe
rent powers.
8. A mathematical expression showing the relationship
between different physical quantities should have dimen
sional homogeneity (the dimension formulas of the left-
hand and right-hand parts should be the same). The equa-
CONVERSION OP UNITS AND DIMENSION FORMULAS 63
tion may have a dimensional factor of proportionality, i.e.,

one whose numerical value changes when the basic units
are changed.
9. A combination of basic and derived units forms a
system of units. The latter is constructed as follows:
(a) quantities are selected whose units are taken as the
basic ones (such quantities are conditionally called basic
ones);
(b) the dimensions of the basic units arc established;
(c) a defining relationship is selected for establishing
each derived unit;
(d) the factor of proportionality in the defining relation
ship is equated to unity (or to another constant value) and,
consequently, is assumed to be dimensionless.
10. The construction of a system is in principle quite
arbitrary, since the number of basic units and the ones
selected, and also their dimensions and the selection of the
defining relationships are all arbitrary.
11. The number of basic units is connected with the num
ber of dimensional factors in the mathematical expressions
of the physical laws. The greater the number of basic units,
the greater the number of such factors.
12. With a different selection of the defining relation
ships, the dimension formula of a unit of the same quantity
may be different. Consequently, the dimension is not a cer
tain unchangeable property of a given physical quantity,
but depends on the way the system of units has been con
structed.
13. The conversion of units from one system into another
is accomplished by means of the dimension formulas, for
wiiich purpose it is necessary to have the relation between
Ihe basic units, which is established according to the method
of constructing the systems, namely:
(a) if both systems have been constructed using the same
basic quantifies and the same defining relationships, then
(lie relationship between the basic units is determined
cither by comparing their standards or prototypes, or by
Ihe conditional definition of the relation existing between
Ihe units (for example, a unit of one of the systems is
defined as a multiple or a submultiple of a unit of the
oilier system);
04 UNITS OF P H Y SIC A L Q U A N T IT IE S AND T H E IR DIMENSIONS
(b) if the defining relationships in both systems are the

same, but the basic quantities differ, then it is necessary
to establish experimentally the relation between the units
of a quantity that is a basic one in one of the systems and
a derived one in the other;
(c) if the derived units are constructed with the aid of
different defining relationships, then it is first necessary
to establish the dimension of the factor of proportionality
in one of these relationships written in the system in which
it is not the defining one, determine experimentally its
value at some known values of the basic units and then,
by using the dimension formulas, calculate its value at the
dimensions of the basic units corresponding to the given
system.
14. While it is theoretically possible to construct a system
of units in an arbitrary way, practical considerations impose
certain limitations on the number of basic units, and on
the selection of the basic quantities and the defining rela
tionships. In particular, it is good practice to have a number
of basic quantities that is neither too small nor too great.
CHAPTER THREE
ANALYSIS OF DIMENSIONS
3.1. Determining Functional Relationships

by Comparing Dimensions
The application of dimension formulas is not exhausted
by the conversion of units and the checking of whether the
formulas are correct. If it is known in advance what physical
quantities participate in the process being investigated,
it is often possible to establish the nature of the relationship
connecting the given quantities by comparing the dimen
sions. In many branches of physics and related sciences—
heat engineering, fluid mechanics, etc.—such a method,
referred to as the analysis of dimensions, has come into
considerable favour. It is especially fruitful when direct
determination of a law being sought either encounters
considerable mathematical difficulties or requires a know
ledge of such details of a process that are unknown before
hand. In essence, analysis of dimensions is based on the
same requirement that the relationship between physical
quantities be independent of the selection of units which
is equivalent to the requirement of coincidence of the dimen
sions in both parts of equations. While in many instances
ibis does make it possible to rapidly establish the nature
nr the relationship being sought, the analysis of dimensions
is not at all an all-powerful method, and sometimes its
possibilities are found to be quite limited.
The objects of the present volume do not include a detailed
consideration of the methods and applications of the analysis
of dimensions, which special books are devoted to. We
shall limit ourselves only to a brief acquaintance with how
dimension formulas can be used to solve practical problems,
lor which purpose several of the simplest typical examples
will be considered below.
10 40
f5(j UNITS <>K PHYSICAL Ol ' ANTITIKS AND Til Kilt M MKNS! O NS
1. A weight having a mass of m is suspended from a spring

(Fig. 2). Upon elongation of the spring by h there appears
an elastic force equal (in absolute value) to / that tends to
return the spring to its initial position. Besides the force /,
no other forces act on the weight. Find the time t it takes
the weight to return to its initial
position.
To solve this problem, it is ne
cessary to represent time as a cer
tain function of the known quan
tities //I, h and /. Although this
function may in principle have
different forms, certain quite defi
nite considerations can be said
about it. Let us assume that this
function includes some trigonome
trical, exponential or other non-
algebraic functions. It is obvious
that only dimensionless quantities
can be the arguments of the latter.
It can be easily seen that in a sys
tem of units L, M and T no di
mensionless combination can be
formed from the quantities m, h and /, whose dimensions
are correspondingly M, L and LM T ~2, since T is included
only in the dimension of force, and for this reason
force cannot be included in such a combination,
while h and m, naturally, cannot give a dimensionless
combination. Thus the only possible kind of relationship
between t and h, m and / is an algebraic function. It seems
natural to seek this function in the form
t = X fphqmr (3.1)
where X is an unknown dimensionless factor of proportiona
lity, and p, q and r are unknown exponents. Let us equate
the dimension formulas of the left- and right-hand parts
of equation (3.1):
r = LpM pT-*pLqM r (3.2)
Equation (3.2) will be an invariant one with respect to
the dimension of the basic units (i.e., it will remain in
ANALYSIS nr DIMENSIONS 67
Force upon an increase or a reduction in the value of the

basic units) if the exponents of the basic units in the left-
and right-hand parts are equal. On the basis of this condi-
lion, we get the following equations for the exponents:
0 = p-\-q\ 0 = /H -r; and 1 = —2/> (3.3)
whence
—y ; 9 = and r - = \ (3-4)
Accordingly,
<= -y (3.5)
Naturally, the above analysis does not allow us to judge
of the value of the factor X . If the force / is proportio
nal to h (as is the case for elastic forces), then
f = Ceh (3.6)
where Ce is the coefficient of elasticity of the spring, and
we can write the equation
(= x / - (3.7)
so that the lime does not depend on the elongation h. Exact

solution of this problem based on the application of the
laws of mechanics leads to the same equation (3.7), but
with a definite factor X equal to jx/2*.
2. An ideal (non-viscous) liquid with a density of p is
poured into a cylindrical vessel with a cross-sectional area
of A x to a level at a height h from the bottom (Fig. 3). The
bottom of the vessel has an orifice with the cross-sectional
area A 2. Find the time t it will take the liquid to flow out.
Since the liquid flows out under the action of the force
of gravity, it is natural to assume that among the quantities
determining the process there should be the acceleration of
gravity. It is possible here in principle that the relationship
* It can be easily seen that for the force of gravity (the force is
proportional to the mass, / — mg) formula (3.5) transforms into the
formula for the duration of free falling Ar q/h/g, where X is a dimen
sionless factor whose numerical value, as is known, is equal to 1/2.
5*
68 UNITS of physical quantities and th eir dimensions
being sought contains a transcendental function including

the quantities h, A l and A 2 in the argument (p and g can
not be in this argument owing to the considerations given
above). Nevertheless we
shall here also try to
represent the time being
sought in the form of an
exponential monomial:
t ^ X p vf h rA\A[ (3.8)
where, as above, X is a
&S! dimensionless and non-dete-
Fig. S
rminable factor of propor
tionality, and p, q, r, k ,
and I are exponents to be
determined. Let us compile a dimension equation
T -- M vL~zpLqT~2QLrL2hL21 (3.9)
whence, by equating the exponents of the lef- and right-
hand parts, we get the following simultaneous equations
0 —- —3p + q f r -r 2 (k-\-1) ^
0 ■—p | (3.10)
i--2 q J
We have only three equations for determining five expo
nents. Two of them, however, are determined directly:
p—
~0 and (3.11)
This is already of certain interest, since it shows that
the duration of outflow does not depend on the density
of the liquid and is inversely proportional to the square
root of the acceleration of gravity. For determining the
remaining exponents it is essential to have either additional
data or make assumptions based on our ideas of how the
process goes on. Let us assume that the absolute velocity
of the liquid flowing through the orifice does not depend
on its cross section. Thus the duration of outflow should be
inversely proportional to A 2. At the same time the duration
of outflow from the same initial level h should be propor
tional to the total amount of liquid and, consequently, to
ANALYS IS OF DIMENSIONS 69
the section A v This gives the values of 1 and —1 for the

exponents k and I. With such an assumption we immediately
determine the exponent r = y , and the duration of outflow
can be expressed as
<3 - , 2 >
With respect to the factor X, an analysis of the dimension,

as in the previous example, does not make it possible to
determine it. Calculations show that this factor is equal
to V 2 .
The exponent r can also be found in another way. Since
the initial condition of the problem says nothing on the
shape of the orifice and that of the cross section of the vessel,
the unit of area can be related to the basic ones, not making
it dependent on the unit of length. In this case instead of
Eq. (3.9) we should write
T = M r)L-™LqT-™LrAkAl (3.13)
where A is the symbol for the dimension of area.
From this equation, in addition to the conditions that
j) 0 and q--= —y* we get
k = — I and r ~ — (3.14)
Thus, we have as the solution of the problem
<315>
where, in contrast to Eq. (3.12), however, q) {A2IAX) is an
unknown function of the ratio A J A ^
It is easy to see that in comparison with the original state
of affairs, when a special assumption was required, we were
able to achieve greater definity of the solution owing to]the
introduction of an additional basic unit.
3. A constant force F acts over a path of length h on
a body with a mass m. Find the speed the body acquires at
Ilie end of this path.
Similar to the previous examples, we write the velocity as
v ----- XFpm?hr (3.16)
70 UNITS OF PHY SICAL QUANTITIES AN D T H E IR DIMENSIONS
with the unknown factor X and the exponents p , q and r.

The dimension formula will be
LT~l = L pM l)T~21>M qL r (3.17)
Upon comparing the exponents, we easily get
P= q=-— \ \ and r = j (3.18)
whence
(3.19)
The solution of this problem on the basis of the law of
conservation of energy gives, as is known.
i.e., X = 1/2.
Let us approach the same problem in a somewhat diffe
rent way. Let us try to find the velocity by means of the
same analysis of dimensions, but in a system of units in
which the unit of force is determined not from Newton’s
second law, but from the law of universal gravitation.
In this system the dimension of force is
lf\=-.L~2M 2 (3.20)
The dimension equation based on Eq. (3.16) will be
L T '1= Lr2VM 2VM qL r (3.21)
We have arrived at an absurd contradictory result. In the
left-hand part the dimension of time has the exponent —1 ,
while in the right-hand part time is in general absent.
What is the reason for this contradiction? Upon considering
the essence of the problem, we see that the main law deter
mining the given process-acceleration under the action
of an external force—i.e., Newton’s second law, has dropped
out of consideration. This is important in the respect that
in the system of units which we have adopted, the expres
sion for Newton’s second law should contain an inertial
constant whose dimension is
[Ct[ = Ir*MT*
ANAL YS IS OF DIMENSIONS 7i
Upon introducing into the equation of dimensions the

dimension of the inertial constant, we obtain simultaneous
equations, but we shall have only three of them for deter
mining four exponents. Indeed, if instead of Eq. (3.16)
we write
y = Z /f,m?/irC{ (3.22)
we get the simultaneous equations
1 — —2/>-f r — 3k
0 = 2p -(- q -f- k I(3.23)
- 1 = 2k j
1
from which only k = — ^ can be directly determined, so
that in the relationship being sought one of the exponents
remains unknown, and it can be written, for example, as
follows:
v= (3.24)
The problem can be made completely definite if we intro
duce another basic unit, namely the unit of force. If we
designate its dimension, as previously, by /, the dimension
of the inertial constant will be
[C,<] = L“1Af“17,V (3.25)
We can now write the relationship being sought as
v = X fr>m'lhTC\ (3.26)
and obtain the following simultaneous equations for the
exponents:
1= r -&
0 = q —k
- 1 -2 k (3.27)
0 = ]o + k
whence it is quite simple to determine all the exponents:
1 1
p = r= Y and q = k r =— —
and, consequently,
v— (3.28)
It is easy to see that if, on the contrary, we reduce the
number of basic units, using for the solution of the problem
a system with two basic units—length and time—in which
the dimensions of force and mass are, respectively,
[/] = L4r ~4 (3.29)
[M] - z ? r - 2 (3.30)
then the problem will also become indefinite. Indeed, the
equation of dimensions will now be
LT~l L*vT-*vLmT-eJ1LT (3.31)
and we obtain only two equations for finding Ihe exponents:
(3.32)
The solution of the problem again requires additional

assumptions that are not quite obvious, notwithstanding
the simplicity of the problem itself. Equations (3.32) are
simultaneous ones and are satisfied if we substitute for
the exponents their values from Eq. (3.18).
3.2. The II-Theorem and the Method of Similarity

A consideration of the examples given above leads us
to the conclusion that the analysis of dimensions cannot
be a universal method making it possible to automatically
find the relationships between physical quantities parti
cipating in a process that are of interest to us. The use
of analysis of dimensions frequently requires a suitable
selection of the system of units, consideration of the dimen
sional factors that may enter the expressions for the laws
governing the given process, or the definitions of the physi
cal quantities. Additional assumptions are often required
that have to be chosen by intuition, and so on.
The examples also show that the smaller the number of
basic quantities and the greater the number of parameters
participating in a process (including the dimensional fac-
ANALYSIS OF DIMENSIONS 73
tors), the more incomplete will be the system of equations

that can be compiled for finding the exponents of the sym
bols of the quantities entering the relationship being sought.
It is also possible that the dimension equations will lead
to an unsolvable system of equations for the exponents
in the dimension formulas. As we have seen, this indicates
that a quantity essential for solving the problem was not
taken into consideration. A dimensional factor may also
be such a quantity.
The so-called Il-theorem, whose proof can be found in the
cited book by Bridgman and in a book by Sedov*, can
render appreciable assistance in analysing dimensions.
According to this theorem, if the functional relationship
between n physical quantities satisfies the condition of
invariance relative to the magnitude of the basic units, and
the number of basic units is /c, then n — k dimensionless
combinations of the quantities can be compiled. The smaller
this difference, the more definite will the solution of the
problem be. When n — k — 1 the problem becomes the
most definite and, as a rule, single-valued. By separating
the quantity whose relationship to the remaining ones we
want to determine from among the total number of quan
tities, we can express the relationship being sought in the
form of an explicit function.
Let us illustrate the above with examples, using for
this purpose the ones considered in the previous section.
In the first example (the returning of a weight pulled back
by a spring to its initial position) four quantities are related,
namely, the mass of the weight, the force of tension of the
spring, its elongation and the duration of returning to the
initial position. According to the Il-theorem, with three
basic quantities—length, mass and tim e-one dimension
less combination can be formed from four quantities.
Accordingly, the relationship between the.se quantities can
be written in the form of the function
(|) (fvhqmrtk) —const (3.33)
where the argument of the function is dimensionless, and
1be constant quantity forming the right-hand part also has
* L. I. Sedov, Melody podobiya i razmernosti v mekhanike (Methods
"f Similarity and Dimensions in Mechanics), Nauka, Moscow, 1967,
74 TJNTIS OF PHY SICAL QUANTITIES AND TH E IR DIMENSIONS
no dimension. In the argument all the exponents, while

retaining its dimensionless nature, can be changed by the
same number of times, as a result of which one of them can
be made to equal unity. It is most convenient to do this,
obviously, for the quantity being sought, in the given case
the time, so that, equating the exponent k to unity, we get
lfphqmrt] = i (3.34)
Equation (3.34) is equivalent to equation (3.2), the only
difference being that all the exponents have the reverse
signs.
Since the basic units have been selected, combined units
may be taken as such provided that their dimensions will
be independent. For this reason the number of quantities
whose dimensions are mutually independent can be taken
instead of the number of basic units in the formulation of the
II-theorem. This can be illustrated by returning to the
second example (on the outflow of a liquid from a cylindrical
vessel). In this example we sought the relationship between
the following quantities: the duration of outflow t, the
density of the liquid p, the acceleration of gravity g, the
height of the level fe, and the cross-sectional areas A x and A 2.
Among the dimensions of these quantities 71, L “37tf, LT~2,
L, L 2 and L 2 there are three independent ones, namely,
T, L~2M and L. Thus three dimensionless combinations
can be compiled from the quantities listed above:
, p°; and (3.35)
By separating the lime t from the combination h/gt2,
we can write it as follows:
* <3-36>
which is what we previously obtained.
Let us consider another example and find the velocity v
with which a ball sinks in a viscous liquid. The diameter
of the ball d, its density pl7 the density of the liquid p2
and its viscosity r] are given. Obviously, the acceleration
of gravity is also among the quantities determining the
process. Thus, to solve the problem we have six quan
tities with three basic units, which makes it possible to
ANALYSIS OE DIMENSIONS 75
compile three dimensionless combinations. As we have

already seen, the problem becomes the more definite, the
smaller the difference between the number of quantities
defining a phenomenon and the number of basic units. The
present problem can be made more definite if we introduce
at least one additional basic unit, preferably the unit
of force. The quantities included in the problem will have
the following dimensions;
[vl = LT~': [d\ = L\ [Pll - [p2] =--=L~*M;
[r]] ^ L ~ 2TF; and [g]~FM~1
The explanation of our writing the dimension F M _i for

acceleration instead of L T ~2 is that in the latter case we
would have to introduce another dimensional quantity—
the inertial constant. By writing the dimension of accele
ration as F M 'i, we retain the formula of Newton’s second
law / = ma.
Now we can compile only two dimensionless combinations.
One of them, similar to the examples previously considered,
will obviously be the ratio p2/pi- When compiling the
equations for the exponents of the remaining quantities,
we easily obtain a second combination including, in par
ticular, any of the densities, for example p1? namely,
i/rjp11d~2g~i. Hence for the velocity of sinking being sought
we get
(3.37)
The function cp (p2/pi) is not determined by the data of

the problem. Naturally, the problem would be still more
indefinite if we retained only three basic units. It is intere
sting to note that an almost identical problem on the velo
city with which an air bubble (whose density can be neg
lected) will rise to the surface of a liquid becomes quite
definite, since the number of quantities involved is less
by one. It is simple to show that here the dimensionless
combination has the form
d*p2g
76 UNITS OF PHYSICAL QUANTITIES AND T H E IR DIMENSIONS
whence the velocity with which the bubble rises in the

liquid is
v = X d ^ p 2g (3,38)
A comparison of equations (3.38) and (3.37) shows that
the function (p (p2/Pi) has the form
so that equation (3.37) becomes

v-X ^-fa-pJ (3.40)
Theoretical calculations give a value of X of 1/18.

It is easy to see that formula (3.40) describes all the cases
of motion of a ball in a viscous liquid, both when px > p2
and when pj < p2, up to px = 0 , since v can assume both
positive and negative values.
The examples given above show once more that when
employing analysis of dimensions it is necessary, together
with sufficiently obvious procedures, to use intuition not
only when determining the quantities of significance for
the given specific problem, but also when selecting the
basic units and even when writing down the dimensions.
Thus, in the last example it was not obvious that the dimen
sion of the acceleration of gravity should be written FM~l,
and not L7'~2*.
It can be noted that the 11-theorem in itself adds nothing
new to the method of employing the analysis of dimensions
described above, although it often does make it possible
to conduct the analysis in a more convenient way and give
its results in different forms depending on the parameters
we are interested in. Its main purpose, however, is that
it is convenient to introduce the so-called dimensionless
similarity criteria with its aid.
* It is not difficult to see that the reason for this is that in the
second case we would have to introduce another dimensional quan
tity—the inertial constant. By writing the dimension of acceleration
in the form FM~l , we retain the formula of Newton’s second law
/ — mn.
ANALYSIS U f 7 LUMfciNSiOiNS
In principle any of the dimensionless combinations of

quantities determining the phenomenon being investigated
may be such a criterion. If in such a combination the values
of the quantities forming it are so changed that the combi
nation itself does not change, its numerical value will
remain constant even when the dimension of the basic
units is changed. Consequently, when the remaining quan
tities are retained, the quantity being sought also remains
unchanged. Thus, for example, in the problem on the dura
tion of outflow of the liquid the time is a function of the
dimensional ratio hlg, the dimensionless ratio A J A l and,
it can also be said, the dimensionless quantity p°. The latter
simply means that the duration of outflow does not depend
on the density of the liquid. In the given instance the ratio
AJ A^ should be considered as the criterion of similarity.
Should the area of the vessel cross section A x and that of the
orifice A 2 be changed by the same number of limes, then with
a constant value of h (and, of course, of g), the time of
outflow will not change.
The introduction of criteria of similarity was found to be
especially convenient when sufficient information was not
available for complete description of a phenomenon, or
when the strict solution of a problem involved great mathe
matical difficulties.
The first criterion by means of which important theore
tical results were obtained relating to the flow of a real
(viscous) fluid was introduced by O. Reynolds and bears
his name. This criterion, or Reynolds number Re, is equal to
Re — ^ (3.41)
n
where v = velocity of fluid
D = diameter of pipe
p = density of fluid
r\ — viscosity.
'Hie latter, as will be shown below, has the dimension
L~iM T~i and, consequently, Re is actually a dimensionless
quantity. With a given value of Re the nature of the flow
of different liquids in different pipes with different velo
cities has been found to be the same; the distribution of
pressures, velocities, etc. is identical. It has been established
experimentally that when the value of Reynolds number

Re reaches 2200 (the so-called critical Reynolds number),
the regular lam inary flow of a liquid becomes ch ao tic-
turbulent.
The introduction of similarity criteria has been quite
helpful in solving a variety of problems in aero- and hydro
dynamics, heat transfer, etc. Of especial significance is
the fact that the method of similarity can be used to study
various phenomena on models. Thus, for instance, Rey
nolds number (which is applicable not only to the flow
of fluids in pipes, but also to the flow of a fluid around bodies
submerged in it) makes it possible to investigate the resi
stance of bodies in a stream of fluid if the bodies are replaced
by geometrically similar models of smaller dimensions and
the velocity of flow is increased correspondingly.
The similarity criteria in the way they are formed are
dimensionless only with a certain selection of the defining
relationships, and should the latter be changed, the dimen
sions of the units entering the expression of the given cri
terion will also change, and it will acquire a definite dimen
sion. It can be shown, for instance, that when the inertial
unit of force is replaced by the gravitational one, Reynolds
number acquires the dimension
[Re] = Ljr r (3.42)
It can be easily seen that the criterion can again be made
dimensionless if we introduce into it the inertial constant
whose dimension, as we know, is equal to L~3M T2.
In conclusion it should be noted that the compiling
of dimensionless combi nations is also useful when a problem
is solved without any great difficulty in the usual way.
By so transforming the solution that the quantity being
determined is represented as a function of a number of
quantities of which at least a part can be collected into
dimensionless combinations, it is possible to get an expres
sion convenient for analysis and generalizations.
CHAPTER FOUR
UNITS OF GEOMETRICAL
AND MECHANICAL QUANTITIES
4.1. Introduction
For constructing the units of geometrical quantities, only
the unit of length is required of all the basic units, the
metre in the SI and mk(force)s systems and the centimetre
in the cgs system. In kinematics a second basic unit, the
unit of time—the second, the same in all the systems, is
added to the unit of length. Finally, in dealing with dyna
mics a third basic unit is introduced, the unit of mass
kilogram or gram in the SI and cgs systems, respectively,
and the unit of force the kilogram (force) in the mk(force)s
system. All these units were given previously, and we shall
not consider them here.
In the following sections of this chapter all the most
important geometrical and mechanical units will be con
sidered—their formation, definition, determination and
dimension formulas in the SI and cgs systems (i.e., with
respect to the units L, M and T). The dimension formulas
in the mk(force)s system (L, F and T) are given in the sum
mary table (Appendix 5, Table 1) of geometrical and mecha
nical units relating to the SI, cgs and mk(force)s systems.
For each quantity the table gives its name, symbol, the
formula used to determine it, the dimension formulas in
the SI, cgs and mk(force)s systems, and the symbols of
the corresponding units in all three systems. The basic
units of each system are given in the table in bold-face
lype.
Some of the most widespread non-system units of the
rorresponding quantities will also be given in the following
sections,
4.2. Geometrical Units

In addition to the basic system units metre and centi
metre, a number of decimal multiples and submultiples
of these units are used, of which the following are in the
greatest favour:
Kilometre: 1 km = 1000 m — 105 cm
Decimetre: 1 dm — 0.1 m — 10 cm
Millimetre: 1 mm = 10-3 m = 0.1 cm
Micron: 1 [x -- 10"6 m = 10~4 cm - - 10'3 mm
Nanometre: 1 run = 10“9 m = 10~7 cm = 10~6 mm ~ 10~3 jx
Angstrom: 1 A = 10"10 m 10~8 cm -- 10-7 mm = 10~4 jx
Two of these units require additional remarks. The micron
is generally designated as written above, i.e. jx. In con
nection with the introduction of the International System
it was proposed to call it micrometre and designate it jum.
The nanometre, equal to 10~9 m, was previously called
millimicron and designated mja.
In X-ray spectroscopy and X-ray diffraction analysis,
a unit of length called the X-unit and designated XU is
used. This unit was first introduced as 10"3 A (or 10“H cm),
owing to which it was the same as a milliangstrom. Care
ful comparison, however, showed a certain discrepancy
between the X-unit, which is determined with high accu
racy in X-ray spectroscopy, and the milliangstrom. Since
for a number of years all wavelengths and lattice constants
in X-ray diffraction analysis were measured in X-units,
and the latter have been used in numerous tables, it was
found expedient to retain the X-unit as an independent
unit of length
1 XU = 1.00206 x 10~3A
The number of units of length of a non-metric origin
is exceedingly great and, probably, cannot be counted,
since in each country at various times different units were
introduced, sometimes not related to one another in any
way, and the same name could be given to units of different
sizes. Below only some of these units will be given, the
selection being determined by the fact that either they are
used to a more or less considerable extent, or are often
mentioned in literature.
UN ITS OF GEOMETRICAL AND MECHANICAL QUANTITIES 81
In engineering, mainly in machine-building, wide use

up to the present time is made of the inch
1#= 2.54 cm = 0.254 m
In navigation the international nautical mile is used,
which is equal to one angular minute (see below) of a meri
dian, i.e., 1852 m. One tenth of a nautical mile (185.2 m)
is called a cable or cable-length.
In astronomy the following specialûnits of length are
used:
— the parsec (pc) is the distance from which half the
diameter of the Earth’s orbit is seen at an angle of one
angular second (see below). One parsec is equal to 3.084 X
X 1013 km. In addition to the parsec its multiples are
frequently used—the kiloparsec (kpc) and the megaparsec
(Mpc);
— the astronomical unit of length (AU) is the mean dis
tance from the Earth to the Sun, equal to 1.496 X 108 km
(a more accurate value is 1.495993 X 108 km);
— th e light year (e n c o u n te re d m a in ly in p o p u la r science
lite r a tu r e ) is th e d is ta n c e covered b y lig h t in one y e a r,
eq u al to 9.4605 X 1042 k m .
In English-speaking countries the following main units
of length are in use:
inch: 1" = 2.54 cm
foot: 1 ft = 1 2 " = 0.3048 m
yard: 1 yd = 3 ft = 0.9144 m
mile: 1 mile = 5,280 ft = 1.609 km
mil: 1 mil = 0,001" = 2.54
Area. In all systems the unit of area is the area of a square

whose side is equal to a unit of length. From the formula
A = l* (4.1)
we obtain the dimension
[A] = L* (4.2)
The system units of area are the square metre in the SI
and mk(force)s systems, and the square centimetre in the
cgs system:
1 m2 =■ 104 cm2
82 UNITS OF PHY SICAL QUANTITIES AN D TH E IR DIMENSIONS
Formula (4.1) serves as the basis for constructing non

system units of area, among which the ones in greatest use
are 1 km 2 = 106 m2, 1 dm2 = 1 0 '2 m2, 1 mm2 = 10~2 cm2,
and 1 in 2 = 6.4516 cm2.
A unit of area of 100 m2 is called the are, and 100 ares
equal one hectare (ha), a generally used unit for measuring
land area:
1 ha = 102 ares —104 m2 = 10~2 km 2
In English-speaking countries some units of area are
1 in 2 - 6.4516 cm2, 1 ft 2 - 0.09290 m2, 1 yd2 = 0.8361 m,
I acre = 4,046.86 m2, and 1 mile2 = 2.590 km2.
Volume. The unit of volume in all systems is the volume
of a cube with, an edge equal to a unit of length
V= P (4.3)
Correspondingly the dimension is
[F] = £ 3 (4.4)
In the SI and mk(force)s systems the unit of volume
is the cubic metre (m3), aiid in the cgs system the cubic
centimetre (cm3 or cc):
1 cm3 - - 10~6 in3
Among other units are
1 dm3= 10~3 m3— 103 cm3
and
1 in3 = 16.384 cm3
The litre (1), which is frequently called “a unit of capa
city’’, was previously defined as the volume occupied by
one kilogram of water at 4°C, equal to 1.000028 dm3.
In 1964 the litre was equated to one cubic decimetre:
I I = 1 dm3.
Angle. In all systems of units an angle is defined as the
ratio of the length of an arc to its radius. According to this
definition a unit of angle is an angle the length of whose
nrc is equal to a unit of length with the radius also equal
to the unit of length. Since according to this definition
the angle
(4.5)
UNITS OF GEOMETRICAL AND MECHANICAL QUANTITIES 83
where I is the length of the arc and r the radius, then it is

not difficult to see that angle is a quantity with a zero
dimension with respect to all the basic quantities, in other
words, its unit does not depend on the dimension of the
basic units. This universal unit of angle is named the radian
(rad).
The circumstance that the unit of angle in all three
systems has no dimension is often absolutely erroneously
interpreted in the sense that angle is an abstract quantity.
Actually angle is a full-fledged geometrical quantity. It
can be directly measured with the aid of an arbitrary angu
lar measure—a unit of angle, which is sometimes even
taken as a basic unit with its own dimension (designated Si).
The absence of a dimension of angle in the SI, cgs and
mk(force)s systems only means that with the defining rela
tionship (4.5) accepted in these systems the unit of angle
is the same regardless of the magnitude of the basic units.
This also makes it possible to easily introduce independent
non-system units of angle, namely, the revolution (rev)
equal to 2n radians and the degree (deg or °) forming l/360th
of a revolution; the degree is divided into 60 minutes
1° = 60'
and a minute is divided into 60 seconds
1' —60"
In addition, a right angle is used (designated ID or 1 L)
3T 1
equal to 90°, or rad, or rev. A right angle is divided
into 100 parts, each of which is called a gon (1*):
l e -= 0.01D -- 0.9° = 0.0157 rad
One gon consists of 100 metric minutes (1°) and 104
metric seconds (lcc):
l g= , 102c = 104cc
It follows from the above that
1 rad = 57°17'45" = 57.296°
1° = 0.017453 rad
6*
UNITS OF PHYSICAL QUANTITIES AND THEIK DIMENSIONS
Figure 4 shows angles of 1° (AOB), 1 rad (AOC) and

(ADD). For purposes of illustration it may be indicated
that a length 1 mm long is seen at
an angle of 1' from a distance of
3.44 m, and at an angle of 1" from
a distance of 206 m.
Solid angle. Before determining
a unit of solid angle, let us con
sider in greater detail the concept
of solid angle itself, since it is
frequently not completely under
stood. Let us take a sphere on
which a certain closed line is drawn
(Fig. 5), If all the points on this
line are connected to the centre
of the sphere, a cone* is formed enclosing a certain
part of space. The cone will be the wider or, in other words,
*'ig- *
its flare or divergence will be the greater, the larger the
part of the surface of the sphere that is enclosed by the
* A cone in the broad meaning of the term denotes any figure
formed by the motion of a straight line with one of its ends fixed
and any point on it moving along a closed line.
UNITS OF GEOMETRIC All AND MECHANICAL QUANTITIES
line. If we now construct from the same centre a number

of spheres having different radii, then the cone that we
obtained will cut sections out of them that are similar
to the one used to construct the cone. The areas of these
sections, as is obvious from simple
geometrical considerations, will be
proportional to the squares of the ra
dii of the spheres which they were cut
out from. For this reason the ratio
of the area of each of them to the
square of the corresponding radius
will remain constant regardless of
the radius of the sphere and will
be the greater, the larger is the
flare of the cone. This ratio of the
area cut out by a cone on a sphere
to the square of its radius is taken
in all three systems [SI, cgs and mk(force)s] as a measure
of solid angle. Thus, the solid angle £2 is determined
hv the formula
q = 4 (4.6)
A complete sphere forms a solid angle equal to £2 —
4JI/-2 .
r2
Since when three mutually perpendicular planes inter
sect a sphere through its centre, it is divided into eight
light angles (Fig. 6 ), the magnitude of each right angle
will be 4or = -77Z , the same as that of a right angle on a plane.
It follows from the definition that a solid angle, like an
angle on a plane, is a quantity having no dimension. For
IIris reason the unit of solid angle accepted in all systems
is the steradian (sr), defined as the solid angle subtended
a I the centre of a sphere by an area on its surface nume
rically equal to the square of the radius.
In astronomy a unit of solid angle is used called the
s<{uare degree (Q 0)—a solid angle whose cone is a tetrahed
ral pyramid with an angle between its edges equal to 1°.
I : -]0 = 3.046 x 10~4 sr = 2.424 X 10~5 solid angle of a com
plete sphere.
8 6 UNITS OF PHYSICAL QUANTITIES AND T H E IR DIMENSION S
The remark on the possibility of introducing an inde

pendent non-system unit of angle on a plane also relates
to the measurement of solid angle, with the only difference
that in the latter instance only the right angle is used in
practice as such a unit (except for the astronomical unit—
the square degree).
In connection with the above the following remark should
be made. In the SI system, the units of angle and solid
angle—the radian and steradian—have been placed into
a separate group of “supplementary units”. It seems to us
that such a separation is absolutely unsubstantiated and
may lead to misunderstanding, in particular it will make
us consider these units to be outside of any system what
soever. Actually, however, as noted above, the circum
stance that the radian and steradian have no dimension
with respect to the basic units of a system does not at all
mean that they are non-system units. Equations (4.5) and
(4.6) giving definitions of plane and solid angles are typical
defining relationships in which the factor of proportionality,
as usual, is assumed to equal unity and to be deprived of
a dimension. Thus it should be considered that the units
of plane and solid angles—the radian and steradian—are
full-fledged derived units, with the only distinguishing
feature that these units are the same in all systems.
If it is considered that dimensionless units should be
placed in a special group of “supplementary units” instead
of being included among the basic or derived ones, then
this group should also include such quantities relating
to the theory of oscillations (see below) as phase, quality,
and, naturally, any dimensionless combinations of quanti
ties, in particular the criteria of similarity mentioned above.
Curvature. Any curved line has a certain curvature at
each point. An element of the curve adjoining the given,
point can be considered as part of a circle with a certain
radius r (Fig. 7). The quantity that is a reciprocal of r,
P= 4 (4-7)
serves as a measure of curvature of a curve at a given point,

while the radius r itself is called the radius of curvature.
Thus, the unit of curvature is defined as the curvature at
UNITS OF GEOMETRICAL AND MECHANICAL QUANTIT IES 87
such a point at which the radius of curvature of the given

curve is equal to a unit of length.
Curvature of a surface. When we have to deal with a sur
face, the concept of curvature becomes more complicated.
If at any point of a surface planes N x and N 2 are drawn
perpendicular to a tangent (Fig. 8 ), then the intersection
of the surface with these planes
gives two curved lines A 1M B1 and
A 2M B2, which may be characteri
zed by the corresponding radii of
curvature rx = OxM and r 2 0 2M,
and the curvature = \irx and
p2 = l/r2.
It is proved in differential geo
metry that no matter how these
two intersecting planes are drawn,
the sum
fP = Pi + P* = - " l ~ (4-8)
will remain constant. This sum is called the mean curva
ture at the given point of the surface. Sometimes half of
A i
Fig, 8
lliis value is called the mean curvature:

(4.8a)
8 8 UNITS OF PHY SICAL QUANTITIES AND T H E IR DIMENSIONS
The mean curvature of a sphere will obviously be

(4.9)
P“ T
(4.9a)
P '= T
where r is the radius of the sphere.
In addition to the mean curvature, a surface is some
times characterized by the Gaussian curvature, determined
by the expression
1
(4.10)
rir2
where rt and r 2 are the same as in formula (4.8).
For a sphere
K=± (4.11)
The dimension both of the curvature of a curved line
and of the mean curvature of a surface is
[p] = [p'] = L - i (4.12)
The dimension of the Gaussian curvature is
[#] = (4.13)
The unit of curvature of a curve in the SI and mk(force)s
systems is the inverse metre—the curvature of a curve whose
radius of curvature at the given point is equal to one metre.
In the cgs system the unit of curvature is correspondingly
the inverse centimetre. The units of mean curvature of a sur
face are also the inverse metre and inverse centimetres and
p equals unity for a sphere with a radius of two metres [in
the SI and mk(force)s systems], two centimetres (in the cgs
system), or for a cylinder with a radius of one metre or one
centimetre. Correspondingly p' equals unity for a sphere
with a radius of one metre or one centimetre or for a cylinder
with a radius of 0.5 metre or 0.5 centimetre. The units
of Gaussian curvature (1/m2) and (1/cm2) are the Gaussian
curvatures of spheres with radii of one metre or one cen
timetre. Obviously, 1 m "1 = 10“2 cm "1 and 1 n r 2 =
= 10"4 pm~4.
UN ITS OF GEOMETRICAL AN D MECHANICAL QUANTITIES 89
Moments of plane figures. In strength of materials wide

use is made of special geometrical characteristics of plane
figures that are, for example, sections of various structural
members. These characteristics also have the correspond
ing units.
The statical moment relative to an axis is a quantity deter
mined as
(4.14)
A
where dA is an element of area, and r is the distance from

this element to the axis relative to which the moment is
being determined (Fig. 9). Integ
ration is carried out over the entire
area of the figure. The dimension
of the statical moment is
[SZ] = L3 (4.15)
and its units are m3 and cm3.
The dimension and designations
of the units of the statical moment
coincide with those of the units of
volume, although there is nothing
in common between these quan
tities. This can serve as an illust
rative example of the fact that
the coincidence of dimensions does [not at all mean [the
coincidence of the physical (or in the given case geometrical)
essence of the quantities.
In accordance with formula (4.14), the statical moment
»)f a rectangle with sides a and b relative to side b is
(4.16)
I'or this reason the statical moment of a rectangle with

sides of 1 m and 2 m (or correspondingly 1 cm or 2 cm)
relative to the side with a length of 2 m or, respectively,
'1 cm may be taken as the unit of statical moment.
According to the dimension of the statical moment
1 m3 = 10° cm3
90 UNITS OF PHVSICAL QUANTITIES AND TH E IR DIMENSIONS
The axial (<equatorial) moment of inertia is determined

(see Fig. 9) as
/ z- j r 2<M (4.17)
A
The dimension of the axial moment of inertia is

(/z] = £ 4 (4.18)
and its units are m4 and cm4:
1 m4 = 108 cm4
For a rectangle with sides a and b the axial moment of

inertia relative to side b is
T a * h
Jl = — (4.18a)
and correspondingly the unit may be the moment of inertia

of a rectangle with sides of one and three metres (centi
metres) relative to the second of
these sides.
The polar moment of inertia is
computed in the same way as its
axial counterpart, the only diffe
rence being that the distance is
taken not to ail axis, but to a
certain definite point (Fig. 10):
Fig. 10 dA (4.19)
I '12
The dimension and units of the polar and axial moments
of inertia coincide. From formula (4.19) it is easy to find
the polar moment of inertia of a circle:
nr 4
/o = ~2~ (4.20)
For this reason the polar moment of inertia of a circle

whose radius is \ / 2/n = 0.89 metre or centimetre can be
taken as the unit of the polar moment of inertia.
U NITS OF GEOMETRICAL AND MECHANICAL QUANT ITIES 91
4.3. Kinematic Units

Time. Since kinematics considers processes of motion,
a unit of time is required. In all the systems such a unit
is the second, defined above as one of the basic units. Greater
non-system units of time are the minute, 1 min == 60 s
and the hour, 1 h — 60 min = 3 600 s. Fractions of the
unit of time are constructed according to the decimal prin
ciple, i.e., millisecond (ms), microsecond ([is), nanosecond
(ns).
Velocity. The unit of velocity is found from the formula
for uniform rectilinear motion
..i (4-21)
According to this formula, the unit of velocity is the velo
city of such uniform rectilinear motion at which a point
travels a unit of length in a unit of lime. The formula of
uniform motion also determines the dimension of velocity
[v]^LT~l (4.22)
In the SI and mk(force)s systems the unit of velocity is the
metre per second (m/s), and in the cgs system—the centimetre
per second (cin/s).
It should be noted that 1 m/s is sometimes called a mesy
though this name has not been legalized. Of the non-system
units of velocity the most widely used in everyday life
is the kilometre per hour, 1 km/h = 0.278 m/s.
In navigation the unit of velocity is the knot, equal to one
nautical mile per hour or 1.852 km/h.
Acceleration. The unit of acceleration is established on
i lie basis of the formula for uniformly accelerated motion
(4.23)
where v1 = initial velocity
v2 = final velocity
t = time
a — acceleration.
Acceleration can be defined as the increase in velocity in
a unit of time. Hence the unit of acceleration is taken as
i lie acceleration of such uniformly accelerated rectilinear
92 UNITS OP PHYSICAL QUANTITIES AND T H E IR DIMENSIONS
motion at which the increase of velocity in a unit of time

is equal to a unit of velocity. The dimension of acceleration
is found from formula (4.23)
[a] = LT~z (4.24)
The unit of acceleration in the SI and mk(force)s systems
is the metre per second per second (m/s2)—the acceleration
of such uniformly accelerated motion at which the velocity
grows by 1 m/s every second. In the cgs system the unit
of acceleration is correspondingly the centimetre per second
per second (cm/s2). This unit is sometimes (mainly in geo
physics when measuring the acceleration due to gravity)
called the gal (in honour of Galileo). The relationship
between the units of acceleration is
1 m/s2 = 102 cm/s2
The unit of acceleration equal to the normal acceleration
due to gravity, 9.81 cm/s2, is widely used in aviation and
astronautics. This unit is designated g. Acceleration mea
sured in these units is often called overload, since it shows
how many times the weight of a body moving with the
given acceleration is greater than the weight of the same
body at rest or moving uniformly near the surface of the
Earth.
Angular velocity. In uniform rotation the angular velocity
is equal to the ratio of the angle of rotation of a body to the
time during which this rotation took place:
= (4.25)
The dimension of angular velocity is

[co] = T~1 (4.26)
The unit of angular velocity is the angular velocity of
uniform rotation at which a body turns through one radian
in a unit of time (rad/s).
Angular acceleration. Angular acceleration is defined
as the increase in the angular velocity in a unit of time
0)2—0)!
UNITS OF GEOMETRICAL AN D MECHANICAL QUANTIT IES 93
The unit of angular acceleration is the angular accelera

tion of such uniformly accelerated rotation at which the
angular velocity increases in a unit of time by a unit of
angular velocity (by one radian per second).
Since in all the systems the unit of time is the second,
then the units of angular velocity and angular acceleration
will be the same in all of them. The dimension of angular
velocity is determined by formula (4.26), and of angular
acceleration by the formula
[a] = r-* (4.28)
For the non-system units of angle, the corresponding units
of angular velocity and angular acceleration are respecti
vely 1 rev/s, l°/s, 17s, 17s and 1 rev/s2, l°/s2, 17s2, 17s2.
The relationships between these units are the same as
between the relevant units of angle.
As an example we shall note that the minute hand of
a timepiece moves with the angular velocity
0.17s = 67s
Period. Any periodical process consists of a number of
cycles. By a cycle is meant a complete set of repeating
values of a periodically changing quantity. The interval
of time required to complete one cycle is called a period.
The dimension of period is
[t ] = T (4.29)
and its unit is the second (s).
Frequency. The number of cycles completed in a unit of
(ime is called the frequency (v). Obviously,
(4-30)
The dimension of frequency is
M - 7 11 (4.31)
The unit of frequency in all the systems is the hertz (Hz),
a frequency equal to one cycle per second. The term “cycle
per second” (cps or c/s) is still encountered in technical
lilerature instead of hertz. In other words, the hertz is the
frequency of such a periodic process that repeats every
94 UNITS OF PHYSICAL QUANTITIES AND TH E IR DIMENSION S
second. In radio engineering multiple units are used—

kilohertz (kHz), megahertz (MHz), etc. A unit called the
fresnel and equal to 1012 Hz was introduced about 1930,
but it has never been popular.
In uniform rotation the following relationship can be
established between frequency and angular velocity:
c o ^ 2 ftv=---— (4.32)
The concept of angular velocity is also very useful, howe

ver, when applied to other periodical processes (for example,
to rectilinear oscillations). Here the angular velocity or,
as it is also called, the angular frequency is determined
directly from equation (4.32).
Phase. The instantaneous stale of an oscillatory process
is characterized by phase, which is the argument of the
periodical function describing the process. For instance,
in harmonic oscillatory motion the deviation x from the
position of equilibrium is described by the equation
x ~ A sin cl) (4.33)
Here A is the amplitude, i.e., the maximum deviation
from the position of equilibrium, and is the phase. With
an angular frequency to, at the moment t from the beginning
of registration of the oscillations
O —o)/+cp (4.34)
where (jp is the initial phase, i.e., the phase at the initial
moment. It is obvious that phase is a dimensionless quan
tity.
In electrical engineering, phase and phase difference are
sometimes measured by the unit “electrical degree’’, which
is the interval of time corresponding to I/360th of the period
of an alternating current. Since in the USSR the frequency
of alternating current in all electrical mains is 50 Hz,
then an interval of time equal to 55.6 ps corresponds to an
“electrical degree”. In countries where a frequency of 60 Hz
is employed, the corresponding figure will be 46.3 ps.
Volumetric flow rate. The volume of a fluid flowing through
a cross section in a unit of time is called the volumetric
tJNil'S OP GEOMETRICAL AND MECHANICAL QUANTITIES 95
flow rate (Qv). Its dimension is

[QV\ = L*T-' (4.35)
and its units are m3/s and cm3/s.
Volumetric flow rate density (q0) is the flow rate per unit
of cross-sectional area:
<sh
(4.36)
ii
Its dimension
lq,]=LT~l (4.37)
coincides with that of velocity. This should be expected,
since the volumetric flow rate density is actually the linear
velocity of a flow.
Velocity gradient. When a fluid flows in such a way that
its different layers move with different velocities, a spe
cial quantity called the velocity gradient is introduced:
[grad v] = ~ (4.38)
which is the increment of velocity per unit of distance

between the layers in the direction of the change in velo
city. The dimension of the velocity gradient is
[grad y ] [ - ^ - ] = T~x (4.39)
while its units m/(s»m) and cm/(s-cm), are the same in all
the systems and can be written as s"1.
The velocity gradient is also applicable to the curvilinear
motion of a solid body. For example, for a uniformly rotat
ing disk, the velocity of its points grows from the centre
lo the periphery. If the angular velocity of the disk is co,
then at a distance r from the centre the linear velocity v
is equal to cor. Hence
gradz; =- - ^ = 00 (4.40)
Thus, the velocity gradient is here equal to the angular

velocity of rotation of the disk and, consequently, is the
same for all points of the disk.
4.4. Static and Dynamic Units

Mass. Previously we have already defined the basic units
of mass in the SI system (kilogram) and the cgs system
(gram), and also the technical unit of mass (or as we have
agreed to call it in this book, the inerta), which is a derived
unit in the mk(force)s system. We should only remember,
however, that the latter has the dimension
[ m ] = L - ' F T * (4.41)
and according to the relevant USSR State Standard is de
signated kgf -s2/m.
Among the units of mass obtained from the basic ones
according to the decimal principle, the most widely used
are the ton, equal to 103 kg, which was previously a basic
unit in the metre-ton-second system, the centner, or quintal
(q) equal to 100 kg, the milligram (mg) equal to 10‘3 g,
and the microgram (fig) equal to 10~6 g. In German and
English technical literature, the name gamma (symbol y)
is sometimes used instead of microgram. The mass of pre
cious stones is generally measured using a special unit
named the carat equal to 2 X 10“4 kg or 0.2 g.
It is sometimes useful to have a unit of mass containing
a definite number of molecules. Such a unit of mass will
naturally differ for different substances. Such a unit is the
kilogram-molecule or kilomole—the amount of a substance
containing the same number of kilograms as there are units
in the molecular weight of the substance. A unit 1 000 times
smaller is called the gram-molecule or simply mole. A kilo-
mole of any substance contains a constant number of mole
cules (Avogadro’s number), equal, according to numerous
investigations, to (6.0249 ± 0.0002) X 102(i; a mole cor
respondingly contains (6.0249 ± 0.0002) X 1023 molecules.
In rounded numbers a mole of hydrogen contains 2 grams,
of oxygen 32 grams, of water 18 grams, etc.
Force. It should be remembered that the derived units
of force in the SI and cgs systems, determined from New
ton’s second law, have the dimension
[/] —LMT~2 (4.42)
and are respectively called the newton (N) and the dyne
(dyn). The newton is defined as the force imparting to a mass
UNITS OP GEOMETRICAL AND MECHANICAL QUANTITIES 97
of 1 kg an acceleration of 1 m/s2, and the dyne as the force

imparting to a mass of 1 g an acceleration of 1 cra/s2. It can
therefore be written that
1 N = 1 kg-m/s3
1 d y n = l g-cm/s 2
and
1 N = 105 dyn
The unit of force of the abolished metre-ton-second system
is sometimes encountered in literature. This unit of force,
the sthene (sn), is defined as the force imparting to a mass
of 1 ton an acceleration of 1 m/s2;
1 sn = 1 t •m/s2
Obviously, 1 sn = 103 N.
The unit of force in the mk(force)s system—kgf*, which
is the basic one in this system, is determined by a prototype
1 kgf = 9.81 N = 9.81 X 106 dyn
The ton (force), equal to 103 kgf, and the gram (force),
equal to 10 3 kgf, are also sometimes used in practice.
Impulse. The impulse of a force is measured by the product
of the force and the duration of its action, f t . The unit of
impulse is the impulse of a force equal to unity and acting
during a unit of time. Accordingly, the units of impulse
in the different systems are:
S I : 1 N*s = 1 kg-m/s
cgs : 1 dyiX'S -- 1 g«cm/s
mk(force)s : 1 kgf-s
Momentum (impulse). Momentum is defined as the product
of the mass of a body and its velocity (mv). The unit of
momentum of a body is the momentum of a body with
a unit mass moving with a unit velocity.
We have previously mentioned (see Sec. 2.1.) that in
modern literature on physics the term “impulse of a body”
* According to the standard definition, a kilogram (force)—
Kid - is the force that imparts to a mass of 1 kg the normal accele-
iMiioii of gravity, i.o., 9.80665 m/s2.
i H)/jO
98 U N IT S OP PHYSICAL Q UANTITIES AND T H E IR DIMENSIONS
is often used instead of the term “momentum”. From the

formula
ft — mv2—mvi (4.43)
(the impulse is equal to the change in the momentum) it
follows that the dimension of impulse and momentum should
coincide. Indeed,
[ft] = LMT~*T LMT~l (4.44)
[mu] = M LT-1=~- LM T-1 (4.45)
The units of momentum and of impulse are the same, in
the SI system—kg-m/s, in the cgs system—g-cm/s; in the
mk(force)s system the unit can be written as i -m/s, which,
of course, is equal to kgf -s
1 kg-m/s =■-■=105 g*cm/s
1 kgf*s = 9.81 kg»m/s
Pressure. With a uniformly distributed load, pressure
is determined by the force acting per unit of area,
p=4- (4>46)
The unit of pressure is such a uniformly distributed
pressure when a unit of force acts on a unit of surface.
The dimension of pressure is
[p]= --L "W ra (4.47)
In the SI system the unit of pressure is the newton per
square metre (N/m2). It has been proposed to call this unit
the pascal (Pa). In the cgs system the unit is dyne per square
centimetre (dyn/cm2). The dimension of pressure establishes
the relationship between the units N/m2 and dyn/cm2:
1 N/m2= 1 0 dyn/cm2
In the mk(force)s system the unit of pressure is kgf/m2,
which is equal, obviously, to 9.81 N/m2 = 98.1 dyn/cm2.
A pressure of 1 kgf/m2 is, with a high degree of accuracy,
equal to the pressure of a column of water 1 mm high.
Indeed, a layer of water with an area of 1 m2 and a thickness
of 1 mm occupies a volume equal to 1 dm3, consequently
its weight with a high accuracy is equal to 1 kgf. For this
reason in engineering the unit of pressure kgf/m2 is often
U N ITS OF GEOMETRICAL AND MECHANICAL Q UANTITIES 99
called a millimetre of water column (mm of water, or mm

H 20). This is especially convenient when water pressure
gauges are used (for example, when measuring the velocity
of a gas in a pipe).
The unit of pressure in the cgs system dyn/cm2 in litera
ture on physics was previously called the bar. In meteoro
logy this name is used to denote a unit 106 times greater,
and equal to 105 N/m2. The bar is given this value in
GOST 7664-55 covering non-system mechanical units.
It is not recommended to use this name to denote dyn/cm2.
In addition to the units mentioned above, a number of
non-system units are widely used in physics and engineering.
One of them that is in great favour is the standard atmo
sphere—the pressure of air balanced by a column of mer
cury 76 cm high with a density of the mercury of 13.595 g/cm3
at normal acceleration of gravity. Such a column applies
a pressure equal to its weight to each square centimetre.
The exact value of a standard atmosphere is
1 atm = 76 cm x 13.595 g/cm3 x 980.665 cm/s2=
= 1.01325 X 106 dyn/cm2=-1.01325 X 105 N/m2
Since this pressure is approximately equal to 1.033 kgf/cm2,
instead of it use is often made of the technical atmosphe
re (at), exactly equal to 1 kgf/cm2. Obviously, 1 at =
= 104 kgf/m2 = 9.81 X 104 N/m2. In the metre-ton-second
system the unit of pressure was called the pieza (pz) and was
equal to a pressure of 1 sthene per square metre. Hence
1 pz — 103 N/m2 = 104 dyn/cm2= 0.01 bar
(here a bar is equal to 106 dyn/cm2).
The pressure is quite frequently measured directly in
millimetres of mercury column (mm Ilg), also called torrs
(after Torricelli). The latter name is seldom encountered
in literature on the subject.
Obviously, 1 mm Hg (1 torr) = 10~3 m X 13.595 X
103kg/m3 X 9.80665 m/s2 = 133.3 N/m2 = 1 333 dyn/cm2.
The system units of pressure and their multiples and sub-
multiples are also used to measure mechanical stresses.
Pressure gradient. The flow of fluids in channels and pipes
is determined by the pressure difference per unit of length
7*
of the stream. For a constant cross section of the stream

this quantity can be written as
Pi—Pi (4.48)
U—L
where I is the distance along the stream measured from its
origin. When the cross section of the stream is not uniform,
the expression — —should be substituted for expression
(4.48).
The quantity
^ - = gvadp (4.49)
is called the pressure gradient.
It is easy to see that the dimension of the pressure gra
dient is
[grad p ] = L -2MT-* (4.50)
where grad p is measured in units of pressure related to a
unit of length
N N kg dyn dyn g kgf _ kgf
m 2- m ni3 m 2«s2 ’ c m 2«cm cm3 c m 2’S2 ’ m 2*m m3
and in the non-system units
atm torr
cm
etc.
The dimension of the pressure gradient determines the
relationship between the units:
1 N/m3 —0.1 dyn/cm3; 1 kgf/m3 —9.81 N/m3
Work and energy. Work under the action of a constant
force is defined as the product of the force, the distance,
and the cosine of the angle between their directions
W=f l cos( f J) (4.51)
The unit of work is the work done by a unit of force
over a path equal to a unit of length when the force and
the path coincide in direction. The dimension of work is
determined from its formula
IW] = [f] [1i - M r 2 (4.52)
U NITS OF GEOMETRICAL AND MECHANICAL Q UANTITIES 101
In the SI system the unit of work is the joule (J) —

the work performed by one newton over a distance of 1 metre:
1 J = 1 N«m
In the cgs system the unit of work —the erg —is the
work performed by one dyne over a distance of 1 cm:
1 erg — 1 dyn-cm
In the mk(force)s system the unit of work—the kilo gram-
metre—is the work done by 1 kilogram force over a distance
of 1 metre (1 kgf-m). The relationship between these units
of work is obtained from formula (4.52)
1 kgf-m -9 .8 1 J; 1 J = 107 erg
Before the introduction of the International System of
Units, special heat units of work—units of the quantity of
heat—the calorie (cal) and kilocalorie (kcal) were used in
all thermal calculations. The first of these units can be
approximately defined as the quantity of heat required to
heat 1 gram of water by 1°C; 1 kcal = 1 000 cal. These
units will be considered in greater detail together with the
other thermal units. In connection with the introduction
of the International System of Units, it is recommended
l o use the general units of work—the joule and its multiples
and subinultiples—instead of the calorie and kilocalorie.
The following relationship has been established for the
conversion of calories to joules:
1 cal-4.1868 J
(the coefficient 4.1868 is sometimes called the mechanical
equivalent of heat).
When measuring the work performed in various processes
involving a change in the state of a gas, use is sometimes
made of a unit that is determined according to the work
of expansion of the gas at constant pressure
W = pAV (4.53)
where p is the pressure of the gas and AF the change in
ils volume. If p = 1 atm and AF = 11, then the correspon
ding work can be called a litre-atmosphere (l*atm):
1 1-atm = 10“3 m3 x 1.01325 X 105 N/m2- 1.01325 X 102 J
102 U N ITS OF PHYSICAL Q UANTITIES AND T H E IR D IM ENSIO N S
Both potential energy Ep, also called the energy of rest,,

and kinetic energy Ek, or the energy of motion, are mea
sured in the same units as work. The sum of both kinds
of energy gives the total energy of a system:
E = Ek + Ep (4.54)
Energy density. Sometimes energy is stored in a certain
volume. For instance, a compressed gas has a certain stock
of energy uniformly distributed through its volume. The
energy per unit of volume is called the energy density e:
e= - (4.55)
Its dimension is
[e] = L-lMT-s (4.56)
and its units are J/m3, erg/cm3, kgf-m/m3, and kgf/m2.
Since the dimension of energy density coincides with
that of pressure, then the relationship between the units
will also be the same:
1 kgf/m2—9.81 J/m2; 1 J/m3 = 10 erg/cm3
Power is the rate of performing work. The power of a uni
formly working system, for example, a machine, mechanism
or the like, is equal to the work performed in a unit of time
The unit of power is the power of such a uniformly wor

king system that performs a unit of work in a unit of
time. The dimension of power is
[P] = L*MT~* (4.58)
The unit of power in the SI system is the watt, or joule
per second (W — J/s), in the cgs system the erg per second,
and in the mk(force)s system the kilogram-metre per second
(kgf-m/s). Since the unit of time in all these systems is the
second, then the relationship between the units of power
remains the same as between those of work.
Larger and smaller decimal multiple and submultiple
units of power, the kilowatt (kW), megawatt (MW), milliwatt
(mW) and microwatt (jTW), are in great favour. The hecto
watt (hW) is used less frequently. The watt and its decimal-
UNITS Ofr GEOMETRICAL AND MECHANICAL Q UANTITIES 103
derivatives are almost exclusively used to form units of

energy for measuring electrical energy. These units are the
watt-hour (W-h), hectowatt-hour (hW-h), kilowatt-hour
(kW-h), and megawatt-hour (MW-h). They denote the work
done at the corresponding power during one hour. It follows
from this definition that 1 W-h = 3 600 J, 1 hW-h =
- 3.6 x 105 J = 360 kJ, 1 kW-h = 3.6 x 106 J = 3.6 MJf
and 1 MW-h = 3 600 MJ.
The thermal units of work related to a unit of time give
the corresponding units of power. Of these units the ones
in greatest use are the calorie per hour (cal/h) and the kilo
calorie per hour (kcal/h). The relationship between the
kilocalorie per hour and the watt is
1 kcal/h = 1.163 W
Up to the present time quite widespread use is made of
a unit of power with the absurd name horsepower (hp), both
in the metric system, and in the system of weights and
measures still in use in some English-speaking countries.
These “horsepowers” differ slightly in value. The metric
horsepower is equal to 75 kgf-m/s or 736 W, while the
IBritish (or U.S.) horsepower is equal to 550 ft-lb/s or
745.7 W.
Coefficient of friction. When a body moves along a surface,
a braking force appears—the force of friction. Its magnitude
ilr depends on the nature of the contacting surfaces and is
proportional to the normal force fn of the pressure forcing
Ilie body against the surface:
ffr = Cfrf n (4.59)
The coefficient CiT is called the coefficient of friction
and, according to its definition, is a dimensionless quantity,
Ilie same in all systems.
Coefficient of resistance. If a body moves in a viscous
medium (a gas or liquid), a force of resistance appears that
depends on the velocity. At relatively low velocities this
force is proportional to the velocity:
f = rv (4.60)
The coefficient r, called the coefficient of resistance, depends
on (he properties of the medium, and the dimensions and
shape of the body. Its dimension is

[ r j^ M T -1 (4.61)
and its units are N -s/m = kg/s, dyn*s/cm = g/s, and
kgf’s/m. The relationships between these units are the
same as between the units of mass, namely, 1 N -s/m =
= 103 dyn*s/cm, 1 kgf-s/m = 9.81 m-s/rn.
Flexibility. If an external force is applied to an elastic
system, the latter is deformed. If Hooke’s law is observed,
then the linear deformation is proportional to the applied
force
Ax — kf (4.62)
The coefficient k is called the flexibility of the system.
Formula (4.62) determines the dimension of flexibility:
{k] = M~lT* (4.63)
and its units m/N, cm/dyn and m/kgf.
According to equation (4.63), 1 m/N = 10“3 cm/dyn,
1 m/kgf - 1/9.81 m/N - 0.102 m/N.
Moment of force. The moment of a force relative to a cer
tain point is measured by the product of the force and its
arm, i.e., the distance between the direction of the force
and the point relative to which the moment of the force is
taken. For this reason the unit of moment of a force can be
taken equal to the moment of a force equal to unity with
an arm equal to a unit of length. From the formula for the
moment of a force
M = fl (4.64)
where M is the moment of the force / and I is the arm,
it follows that the dimension of the moment of a force is
[MJ = L2iMT~2 (4.65) j
We see that the dimension of the moment of a force coin
cides with that of work and energy. It should be noted,,
however, that these quantities are of an absolutely diffe
rent nature. While work and energy do not have a direction,
and are scalar quantities, the moment of a force has a direc
tion, i.e., is a vector quantity.
The units of moment of a force are formed in the same;
way as the units of work, but the names erg, joule, etc. are-
U N IT S OF GEOMETRICAL AND MECHANICAL Q U A N T ITIES 105
not used with respect to it. Thus, for moment of a force we

have the units N dyn^cm, and kgf*m.
Moment of inertia of a body (dynamic moment). In mecha
nics, especially when considering the rotary motion of
a body, it is very convenient to use a special quantity—
the moment of inertia of a body—which
is calculated relative to a certain axis.
For purposes of illustration let us first
find the moment of inertia of a material
point relative to an axis. It is equal to
I = mr2 (4.66) IX
where m is the mass of the material point,
and r the distance from it to the axis . 4 . ---------------
relative to which the moment of inertia
is being determined.
For a system of rigidly connected
material points or for a solid body the Fig. 11
moment of inertia can be determined as the
sum of the products of the masses of the separate material
points which the system consists of or which system can be
divided into and the squares of the corresponding radii—
Ihe distances to the axis of rotation (Fig. 11):
for a system of points I = 'Lmr2 (4.67)
for a solid body / = j r*dm (4.68)
v
In the process of studying problems connected with the
rotation of bodies, the essence and role of the moment of
inertia are ascertained. It is found that all the formulas
describing the rotary motion of a solid body have a form
similar to the corresponding formulas for translational
motion if the linear quantities in the latter (velocity,
acceleration) are replaced by the relevant angular quan
tities (angular velocity, angular acceleration), and the
mass by the moment of inertia relative to the axis of rota-
iion. The unit of moment of inertia follows from its defi
nition, and is the moment of inertia of a material point
liaving a mass equal to unity, with a distance to the axis
<<|ual to a unit of length. The dimension of moment of
inertia is accordingly
[I] = L2M (4.69)
and its units are kg-in2, g-cni2, and i-m2 (1 i -m2 =
= 1 kgf -m -s2). The relationships between them are 1 kg *m2=
= 107 g*cm2, and 1 kgf-m-s2 = 9.81 kg-m2.
Impulse of moment of force. The impulse of the moment
of a force relative to a certain axis is the product of the
moment of the force relative to this axis and the duration
of action of the force
M t = fit (4.70)
The formula defining the impulse of the moment of a
force gives for its dimension
[ M tj^ L U lT -1 (4.71)
Moment of momentum. The moment of momentum (also
called angular momentum) of a material point rotating
about an axis is the product of the momentum of this point
and the distance to the axis of rotation.
X = pr = mur (4.72)
Since the linear velocity in rotation can be expressed
through the angular velocity by the formula
v — (or
then the moment of momentum can be written as
# = /© (4.73)
Thus, the moment of momentum is equal to the product
of the moment of inertia of a rotating point relative to the
axis of its rotation and the angular velocity. It follows
from formulas (4.72) and (4.73) that the dimension of the
moment of momentum, similar to that of the impulse of
the moment of a force, is
[X] = L2MT~l (4.74)
The equality of the dimensions of the impulse of the moment
of a force and the moment of momentum, naturally, follows
from the law that “the impulse of the moment of a force
relative to the axis of rotation is equal to the change in
Units of geometrical and mechanical quantities 107
the moment of momentum”

M t = X 2- X , (4.75)
The units of the impulse of the moment of a force and the
moment of momentum also coincide. They can be defined
as the impulse of the moment of a force equal to unity during
a unit of time, or as the moment of momentum of a body
having a moment of inertia (dynamic) equal to unity, and
rotating with an angular velocity equal to unity. These
units are:
N «m •s = kg • m • m/s —kg •m2/s
dyn •cm •s = g • cm •cm/s = g . cm2/s
kgf •m . s = i •m • m/s = i •m2/s
The relationships between them are the same as between
the corresponding units of work or those of the moment of
inertia, since the unit of time in all the systems is the same.
Action. A quantity called action and whose dimension is
the product of energy and time plays an appreciable part
in analytical and quantum mechanics and in a number of
other branches of physics. Without going into its physical
essence, it should be noted that its dimension coincides
with that of the moment of momentum or the impulse of
the moment of a force and is accordingly measured in the
same units.
Mass flow. In investigating the flow of liquids and gases,
in addition to the volumetric flow rate considered above,
use is made of a quantity called the mass flow (<?m). Its
dimension is
[Qm] = M T-' (4.76)
and its units are kg/s, g/s, and i/s (kgf-s/m). The relation
ships between them are obviously the same as between the
units of mass.
Mass flow velocity. The mass flow velocity or mass flow
density qm is determined similar to the volumetric flow
rale density as the mass flow related to a unit of cross-
sectional area of the flow. The dimension of qm is accordingly
[qm] = L~*MT-1 (4.77)
ami its units are kg/(s-m2), g/(s*cm2) and i/(s*m2).
Dynamic characteristics of oscillatory motion. Together

with the kinematic quantities—frequency, period, phase
and amplitude—an oscillating system is characterized
by a number of dynamic quantities, including kinetic and
potential energy and their units, considered above. Of

great significance are the quantities characterizing the
properties of an actual oscillating system. Since every such
system has attenuation, or damping, its motion in the
absence of an external force (Fig. 12) can be written as
x — A0e"6/ sin (co£ <p) (4.78)
where x deviation from the position of equilibrium
A 0 — initial amplitude
e = base of natural logarithms
co = angular velocity
<p == initial phase
6 — damping factor.
The dimension of 6
[S l-T 7' 1 (4.79)
determines its unit s '1. Attenuation isalso usually charac
terized by the logarithmic damping decrement x, equal to
x = 8t (4.80)
where t is the period of oscillations. It can be seen from
definition (4.80) that x is a dimensionless quantity. It
UNITS OF GEOMETRICAL AND MECHANICAL Q U A N T ITIES 1 q9
can be easily shown from formulas (4.78) and (4.80) that the
logarithmic decrement is equal to the natural logarithm
of the ratio of two consecutive amplitudes. Another impor
tant characteristic of an oscillating system is its quality Q,
determined by the formula
Q= (4.81)
where Etot = total energy of the system in resonance
Ei — loss of energy during one period.
It is obvious that Q is a dimensionless quantity. It can be
shown that
i
(4.82)
26 y~km
where k is the flexibility of the system [sec formula (4.62)].
4.5. Units of Mechanical and Molecular Properties

of a Substance
Both in scientific research and in numerous spheres of
everyday life we utilize the most diverse properties of the
materials that we have to deal with. These properties, as
a rule, are characterized hy definite quantities and can be
measured quantitatively in some way or other. Different
materials have different mechanical strength, different elasti
city, etc.
In addition, any substance also has a number of molecular
characteristics, connected either with its structure or with
the processes in which its molecular nature appears. In
this section we shall mainly consider the mats of the mecha
nical and molecular properties of a substance which we have
to deal with in physics and related subjects.
Density. Density is defined as the ratio of the mass of
a homogeneous body to its volume
P= f (4-83)
The unit of density is the density of such a homogeneous
substance whose amount in a unit of volume is equal to the
unit of mass. The unit of density in the SI system is kg/m3,
in the cgs system g/cm3, and in the mk(force)s system i/m3
or kgf*s2/m4. Frequently, especially when dealing with

a gas, the density is measured in g/1. The value of the density
measured in g/1 coincides with that expressed in kg/m3.
The dimension of density
[pj = Z rW (4.84)
allows the relationship between its various units to be
established quite easily *.
Specific volume. The reciprocal quantity of density is
called the specific volume
_F
m (4.85)
P
All the units of specific volume are reciprocals of the
corresponding units of density. In the same way its dimen
sion is
(4.85a)
Specific weight is the ratio of the weight of a body to
its volume
Since the weight of a body and its mass are related by

the formula
f^ -m g (4.86)
where g is the acceleration of gravity, then the relationship
between the density p and the specific weight y will be
V ~ Pg (4.87)
The unit of specific weight is the specific weight of such
a homogeneous substance, a unit of volume of which is
attracted to the Earth with a force equal to a unit of force.
Thus we obtain the units N/m3, dyn/cm3 and kgf/m3. For
example, the specific weight of water in the cgs system is
981 dyn/cm3. Specific weight is often measured in gf/cm3
and kgill. Both these values coincide and are practically
equal to the specific weight of water at 4°C. The numerical
* When measuring the density of a liquid with an areometer

various conditional scales were previously used (see Appendix 2).
U N IT S OF GEOMETRICAL AND MECHANICAL Q UANTITIES 1H
value of specific weight expressed in gf/crn3 or kgf/1 coincides

with that of density expressed in g/cm3 or in kg/1.
Molecular weight (molecular mass, relative molecular mass)
is the ratio of the mass of a molecule of a given substance to
the so-called atomic unit of mass. Previously, in chemistry,
this unit was taken equal to one-sixteenth of the mass of an
atom of oxygen (the mean value of the mass of the
three isotopes of oxygen with account taken of their
relative contents in per cent). In 1901 the atomic unit of
mass was taken equal to one-twelfth of the mass of the
isotope of carbon with a mass number of 12 (i.e., containing
six protons and six neutrons in its nucleus). This unit
will be considered in greater detail in Sec. 9.2.
As mentioned above (Sec. 4.4), the mass of a substance
containing the same number of kilograms or grains as there
are units in the molecular weight of the given substance
is called the kilomole or mole.
Molecular volume is the volume occupied by one kilomole
or mole of a substance. Molecular volume in standard con
ditions, i.e., at 0°C and a pressure of 1 atm, is called stan
dard molecular volume. It is equal to 22.42 m3 for a kilo
mole and 22.420 cm3 (22.42 1) for a mole.
The number of kilomoles (or moles) contained in a given
mass of a substance can be determined as the ratio of this
mass to that of one kilomole or mole. If we define the latter
as the number of kilograms per kilomole or grams per mole
and designate it in the same way as molecular weight, we
can find the number of kilomoles or moles from the expres
sion
r, m
(4.88)
Coefficients of extension and shear and moduli of elasticity
and shear. If a solid specimen is subjected to linear (uni
axial) tension or compression, it deforms (becomes stretched
or compressed), its deformation or strain within certain
limits following Hooke’s law
AZ= a - | (4.89)
where AI ~ deformation or strain
I = initial length
/ = load
A = cross-sectional area.
The coefficient a is called the coefficient of extension of
a material. It is the deformation of a specimen of unit
length and unit cross-sectional area when a unit deforming
force is applied to it. The dimension of this coefficient is
[a] - LM~lT2 (4.90)
In the SI system its unit is m2/N, in the cgs system cm2/dyn,
and in the mk(force)s system m2/kgf. In strength of mate
rials the reciprocal quantity is generally used:
(4.91)
which is called the modulus of elasticity, or Young’s modu

lus. Its units are reciprocals of those of the coefficient of
extension—in the SI system N/m2, in the cgs system dyn/cm2,
and in the mk(force)s system kgf/m2. In engineering Young’s
modulus is often measured in kgf/cm2 or kgf/mm2. It is the
load that would have to be applied to a specimen with
a cross-sectional area equal to 1 m2, 1 cm2 or 1 mm2 to
double its length (if Hooke’s law would be constantly
observed during the process and the specimen did not fail).
Similar to the coefficient of extension and Young’s modu
lus, there can be defined the coefficient and modulus of shear.
The relationships between the units of Young’s modulus
or the modulus of shear are the same as between the units
of pressure.
Coefficient of bulk compression. If a specimen is subjected
to uniform or bulk (triaxial) compression under a certain
pressure p, its volume will decrease by AF, determined
from the formula
AV = kpV (4.92)
where k is the coefficient of bulk compression. The unit
and dimension of this coefficient obviously coincide with
those of the coefficient of extension. In contrast to the
latter, the concept of the coefficient of bulk compression is
applicable not only to solid bodies, but also to fluids (liquids
and gases). In this case it is usually Called the compressi
bility factor. With a view to the greater compressibility
UNITS OF GEOMETRICAL AND MECHANICAL QUANTITIES H3
of gases, it is more convenient to write the compressibility

factor as
1 dV
k= -
V dp (4.93)
The minus sign shows that a growth in pressure corresponds
to a reduction in volume.
Hardness. The resistance of bodies to destruction or the
formation of permanent set when sufficiently great deforming
forces act on their surface is characterized by hardness.
Since with a different nature of the action on the surface
of a body it behaves in different ways, it is difficult to
indicate a sufficiently objective and single-valued charac
teristic of hardness. Upon destruction of a solid body an
attempt can be made to appraise hardness by the work of
destruction related to a unit of area of the newly formed
surface, taking into consideration the fact that the surface of
the body increases upon destruction. With such a definition,
hardness should be measured by the same units as the coef
ficient of surface tension (see below), determined by the free
energy per unit of surface. It should be noted, however,
that the actual work of destruction is considerably greater
than the increase in the free energy of the surface, since
the predominant part of the work done is dissipated as heat.
The fact is also of importance that with different methods
of treatment the work actually done may vary quite con
siderably. This is why different methods of conditionally
appraising the hardness of materials have come into favour
in engineering.
In mineralogy hardness scales are used in which numbers
in growing order designate materials so arranged that each
following one is capable of leaving a scratch on the surface
of the previous one. Talc and diamond are at the extremes
of these scales. The arrangement of minerals in the Mohs
and Breithaupt scales is given in Table 47.
Methods of determining hardness are used in engineering
Ibat are based on measuring the dimensions of the inden
tations obtained when steel balls, diamond cones or prisms
are pressed into the surface of the material being tested
(Ibinell, Rockwell or Vickers hardness). For purposes of
iIlustration let us consider the method used to determine
flu* Brinell hardness of a material. For this purpose a har-
h in/iU
dened steel ball is pressed into the surface of the material

by a definite load, and the diameter d of the indentation
formed is measured. If the diameter of the ball is D , and
the load P, then the measure of hardness is the quantity
B H N , determined by the formula
______ 2P______ (4.94)
BHN =
siD (D—
Here P is measured in kgf, D and d in mm. Hence BH N
is measured in kgf/mm2.
Impact strength (ductility). In addition to hardness, the
resistance of a material to destruction is characterized
by the impact strength, also called ductility, which is mea
sured by the work done for impact destruction of a specimen
related to a unit of its cross-sectional area at the place of
fracture. The units of impact strength are J/m2, erg/cm2
and kgf-m/m2. The relationships between them are
1 J/m2 = 103 erg/cm2, and 1 kgf-m/m2 = 9.8 J/m2.
Viscosity (coefficient of internal friction). If laminary
(striated) flow of separate layers takes place in a fluid, then
a force directed tangentially to the surface of these layers
appears in them. The existence of viscosity also leads to
the appearance of a force acting on every body moving
in a fluid or around which a fluid flows. This force, known
as the force of internal friction, is expressed by the formula
f= -*iw A (4-95)
where dv/dl ~ velocity gradient
A — area acted upon by the force /
r| = viscosity.
The minus sign in the formula shows that the force is
directed toward the layer moving with a higher velocity.
Both the definition and the dimension of the unit of vis
cosity ensue from formula (4.95). Viscosity is measured by;
the force acting on a unit of surface area of one of the inte-j
racting layers from the other layer if the distance between
the layers is equal to a unit of length and the layers move
with respect to each other with a unit of velocity. The
dimension of viscosity is
[r\] ^ L - 'M T '1 (4.96;
UNITS OP GEOMETRICAL AND MECHANICAL Q UANTITIES H5
In the SI system the unit of viscosity has no special

name and is designated N -s/m2. In the cgs system its unit
is the poise (P), defined as the viscosity of such a fluid in
which at a velocity gradient of 1 cm/s per cm, a force of
friction equal to one dyne acts on each square centimetre
of a flowing layer. From the dimension formula of visco
sity or its designation in the SI system it is easily found
that 1 N *s/m2 = 10 P. It can similarly be shown that the
unit of viscosity in the mk(force)s system is 9.81 N -s/m2
or 98.1 P.
The viscosity of water at 20.5°C is quite accurately equal
to 0.01 P, i.e., 1 centipoise (cP). The viscosity of a fluid
grows with a reduction in temperature. In particular, the
viscosity of water at 0°C is 1.79 cP. The ratio of the vis
cosity of a fluid to that of water at the same temperature
is called the relative viscosity. The ratio of the viscosity
of a fluid to that of water at 0°G is called the specific vis
cosity. Both the relative and the specific viscosities are
dimensionless quantities.
A great diversity of various instruments generally known
as viscosimeters have been proposed for practical deter
mination of viscosity. Some of them allow the viscosity to
he determined in any of the units given above. Viscosi
meters are also used, however, in which the value of vis
cosity is given in conditional units. Among the latter con
siderable use is made in the USSR (especially for measuring
Ilie viscosity of petroleum products) of the Engler visco
simeters, in which the duration of outflow of 200 grams of
liquid is directly measured. This time serves as a measure
<>T viscosity in so-called Engler seconds. The ratio of this
lime to the time of outflow of the same volume of water
a l a temperature of +20°C gives the viscosity of the liquid
iu Engler degrees (°E). The relationship between Engler
degrees and poises is given by the approximate formula
t) = (0.0731°£- P (4.97)
where r\ = viscosity in poises

°E = viscosity in Engler degrees
p = density of liquid in g/cm3.
8*
116 UNITS OF PHYSICAL QUANTITIES AND T H E tR DIM ENSIONS
Fluidity. A quantity that is the reciprocal of viscosity

is called fluidity:
<P— - (4-98)
The dimension of fluidity is
[cp] ~LM~'T (4.99)
The unit of fluidity in the SI system is measured in m2/N -s,
and in the cgs system—in reciprocal poises. Sometimes this
unit is called and designated rhe.
Kinematic viscosity. Besides the viscosity considered above,
which is frequently called the dynamic viscosity, wide
use is made in hydrodynamics of kinematic viscosity, defined
as the ratio of the dynamic viscosity to the density of
a fluid:
(4.100)
The dimension of kinematic viscosity
[v] = L^T'1 (4.101)
coincides with that of the diffusion coefficient (see below).
In accordance with its dimension, the unit of viscosity in
the SI system is designated m2/s. The unit is the same in
the mk(force)s system. The unit of viscosity in the cgs
system, cm2/s, equal to 10~4 m2/s, is called the stokes (st).
The coefficient of surface tension of a liquid is determined
by the force acting on each unit of length of the boundary
of the liquid film. This coefficient can also be defined as
the free energy* of a unit of surface area of the liquid film.
The dimension formula of the coefficient of surface tension
follows from both these definitions:
toi w ~ MT % (4.102)
The unit of the coefficient of surface tension is defined as
the coefficient of surface tension of such a liquid film on.
each unit of length of whose boundary there acts a unit]
--------- |
* The free energy is approximately defined as that part of thej
energy of a system that can he converted into work. A stricter defini-j
tion is given in the course of thermodynamics. ]
U N ITS OF GEOMETRICAL AND MECHANICAL QUANTITIES H 7
force, or each unit of whose surface area has a free energy

equal to unity.
The definition of the coefficient of surface tension and
its unit as the free energy of a unit of surface area makes
it possible to extend the concept of the coefficient of surface
tension to solid bodies, since the molecules in the surface
layer of a body have a higher potential energy than those
inside it.
The unit of the coefficient of surface tension in the SI
system is N/m or J/m2, in the cgs system dyn/cm or erg/cm2,
and in the mk(force)s system kgf/m or kgf-m/m2. Tables
sometimes give this coefficient in milligrams force pei
millimetre (mgf/mm). It is simple to find that 1 mgf/mm =
= 9.81 dyn/cm.
The dimension “particle”. In molecular and atomic phy
sics, together with macroscopic quantities (density, vis
cosity, etc.) we also have to deal with quantities characte
rizing the properties of separate particles—molecules, atoms,
electrons, ions, etc. Such quantities as the energy of mole
cules, the mass of an atom and the charge of an electron,
should be expressed by units of energy, mass, or quantity of
electricity per “piece'’, i.e., per particle. Although the
dimension “particle” is usually not introduced into the
designations of the relevant units, it is present in latent
form in the measurement of a number of quantities. Obvio
usly, the first of such quantities is simply the total number
of particles in a certain volume or mass.
Concentration. By relating the number of particles to
a unit of volume, we obtain a quantity called concentra
tion. Its dimension is
[n] = L"3 (4.103)
and its units are m “3 and cm~3. Introducing the dimension
“particle” mentioned above, we can write
1 particle/m3 — 10~6 particle/cm3
In chemistry concentration is measured not by the number
of particles, but by the number of kilomoles or moles per
unit of volume. If the concentration in kilomoles per cubic
metre or in moles per cubic centimetre is known, the con
centration of particles can be determined if we multiply
118 UNIST OF PHYSICAL Q UANTITIES AND T H E IR DIM ENSIONS
the former concentrations by Avogadro’s number, expressing

in the first instance the number of particles per kilomoie,
and in the second the number of particles per mole. Most
frequently concentration is measured in moles per litre.
A concentration of 1 mole/1 is called the normal concen
tration.
Diffusion coefficient. When its density or concentration
is not uniform, a liquid, gas or dissolved substance will
diffuse in a direction opposite to the density gradient or
concentration, the amount of substance Am diffusing during
the time At being determined by the formula
Am = - D - ^ - A A t (4.104)
where ^ = density gradient
A — surface through which diffusion is taking place
D = diffusion coefficient.
The equation can also be written as
AN — — D ~ A A t (4.105)-
where AN = number of molecules that have diffused
~ = gradient of concentration.
Both formulas are identical, since the first of them can
be obtained from the second by multiplying both sides by
the mass of a molecule. Each of these formulas can serve
for determining the diffusion coefficient. It is measured by
the mass or number of molecules that diffuse in a unit of
time through a unit of surface area with a density or con
centration gradient equal to unity. Either of the formu
las (4.104) and (4.105) gives the same dimension of the
diffusion coefficient
|D |= M M lW = 'C' 7'-‘ <4-106>

or
i e | = T 5 w W “ t!7T" <4 J0 7 >
which, as indicated above, coincides with that of kinematic
viscosity. In the SI and mk(force)s systems the unit of
the diffusion coefficient is m2/s, and in the cgs system cm2/s.
UNITS OP GEOMETRICAL AND MECHANICAL QUANTITIES HQ
Distribution functions. The statistical nature of molecular

processes appears in that the quantities characterizing
the behaviour of molecules and other atomic particles are
not the same for all the particles in a given system, but
have the most diverse values distributed according to
a definite law. Shown in Fig. 13 as an example is the maxwell
velocity distribution of molecules. The hatched area under
the curve between the values
of the molecule velocities v1
and v2 shows the number of
molecules whose velocities are
greater than ux and smaller
than v2. The number of mole
cules in the small interval bet
ween the velocities v and v +
du is
dN = F (v) dv (4.108)
where F (u), being the ordinate
of the curve, is called the
function of the velocity distribution of the molecules. By
definition,
= (4.109)
The dimension of the distribution function by velocities is
IF (v)] - (particle) L~lT (4.110)
The functions of distribution by any other statistical
characteristic—energy, length of free path, etc.,—can be
determined in the same way as the function of distribution
by velocities.
It can easily be seen that all these distribution functions
have a dimension that is the ratio of the dimension “par
ticle” to the dimension of the quantity, the distribution
according to which is characterized by the given function.
The distribution function is often related to the total
number of particles. A distribution function determined in
Ihis way is called “normalyzed per unit”. Designating this
function / (*;), we can write
f(v) = ~ F { v ) (4.111)
where N is Hie total number of particles. Obviously

oo
^ f (v) dv —1 (4.112)
o
It is easy to see that the dimension of the normalized
distribution function is simply the reciprocal of the dimen
sion of the quantity, the distribution according to which is
determined by the given function.
CHAPTER FIVE
THERMAL UNITS
5.1. Temperature
The basic quantity in the science of heat is temperature.
The concept of temperature is known to everybody from
childhood. Moreover, it is “familiar” to every living creature,
and even to every plant. Nevertheless or, perhaps, just
for this reason it is very complicated to give a definition
of temperature. In elementary textbooks temperature is
sometimes defined as “the degree of heating of a body”,
sometimes as “the cause of feeling heat and cold”. These
definitions, while being illustrative to a certain extent,
do not give a quantitative characteristic of temperature.
Such a requirement can he met by strict definitions relating
temperature to other thermodynamic functions. The latter,
however, have another drawback—they are not so illu
strative and require preliminary acquaintance with more
complicated and abstract conceptions. For this reason, with
a view to the objects of the present book, we shall proceed
as follows: we shall assume that the reader is qualitatively
acquainted with the concept of temperature and consider
the problem of how to measure temperature. It does not
have to be proved that everybody understands the terms
“cold”, “warm” and “hot”, and also knows how to measure
temperature with a conventional liquid thermometer.
It is easy to see, however, that with such measurements
we cannot answer the question of how many times one tem
perature is greater or smaller than another one. According to
the centigrade (one-hundred degree) scale used in everyday
life we may have both positive and negative temperatures,
so that the ratio between two temperatures may be either
positive or negative, and may even be equal to infinity.
The “absolute temperature scale” (designated °K) intro

duced by W. T. Kelvin is quite widely known. In this scale
all temperatures are positive, and the drawback mentioned
above seems to disappear. The question nevertheless remains
as to the extent to which the temperature measured accord
ing to the absolute scale is indeed “absolute”, and what is
the criterion that 600°K is twice as great as 300°K, or that
the interval from 1 000°K to 1 500°K is five times greater
than the interval from 400°K to 500°K. The matter is that
although we do have the ability of feeling temperature
(thermal feeling) and qualitatively comparring temperatures
within a feasible range, we have at our disposal no methods
for the direct measurement of temperature. To have an
indirect method, we must relate temperature to other
quantities whose measurement is accessible.
First of all we should turn to such properties of bodies
surrounding us that according to our observations change
with a change in temperature. It is natural here to make
use of the expansion of bodies when heated. This gave birth
to thermometers measuring the temperature according
to the change in the volume of a liquid. Upon more tho
rough investigation it was found that an appreciable scat
ter of the results of measurements was hidden in this method,
which can readily be illustrated. Assume that several ther
mometers filled with different liquids have been made. Let
us mark the same “basic” or “fixed” points on them, for
example, the melting points of two substances. Let us divide
the scale between these points on all the thermometers into
the same number of equal graduations. If all the thermo
meters are now placed in a medium having an intermediate
temperature, the readings of different thermometers will
be different. The thermometer that we decided to fill with
water would behave especially queerly. At a temperature
somewhat higher than the melting point of ice its column
would be not higher, but lower than this point.
Thus, a different law of the change in the volume of diffe
rent liquids with temperature (up to a change in the sign
of the law) seems to deprive us of the possibility of finding
a single-valued method of measuring temperature. Matters
noticeably improved when Gay-Lussac discovered that all
gases expand practically the same with a rise in temperature.
THERMAL UNITS 123
Mendeleev and Clapeyron succeeded in combining Gay-

Lussac’s law (an experimental one) with the experimental
law of Boyle and Mariotte into a general law expressing the
relationship between the volume of a gas and its pressure
and temperature. Assuming that the volume of a gas at
constant pressure or, more generally, the product of the
volume of a given mass of gas and its pressure, is a linear
characteristic of temperature, the combined law could be
written as follows:
pV = C(i + at) (5.1)
where p — pressure of a gas
V = its volume
t = temperature read from any initial point
a = constant depending on the selection of the
initial tempera-
ture point and the
scale of measu
ring
C = factor depending
on the mass of
the given gas, the
units of pressure
and volume and
the scale of meas
uring the tempe
rature.
Formula (5.1) can be depicted graphically by a straight
line (Fig. 14) intersecting the axis of ordinates. It was found
expedient to extrapolate the straight line to its intersection
with the axis of abscissas and select the point of intersection
as the beginning of the temperature scale. Thus the concept
of “absolute temperature” was introduced. With respect to
the scale for measuring this temperature, it could naturally
be quite arbitrary. It was so selected as to divide the inter
val between the melting point of ice and the boiling point
of water into 100 parts called degrees. With such a scale
the point of intersection of the straight line with the axis
of abscissas in Fig. 14 will be about 273 degrees to the left
of the origin of coordinates. This point, as is known, was
called “absolute zero”. Formula (5.1) was transformed accor-
dingly, and it can now be written

pV = - ^ R T (5.2)
where T = absolute temperature
in — mass of a gas
M — mass of a kilomole or mole
R = so-called universal gas constant, whose nume
rical value depends on the selection of the units
of the quantities in the formula.
In this form, together with Boyle-Mariotte’s and Gay-Lus
sac’s laws, equation (5.2) also includes Avogadro’s law.
This equation can in essence be interpreted as the definition
of temperature as a quantity proportional to the product
of the pressure and the volume of one mole of a gas.
Equation (5.2) makes it possible to measure the concen
tration of a gas by the so-called “reduced pressure’ . If the
equation is rewritten as
m p
~MV~ ~ ~RT
(5.2a)
then the left-hand part will denote the number of moles
or kilomoles per unit of volume, i.e., the molar concentra
tion. The concentration will obviously be the same when
at a temperature of T0 = 273°K the pressure of the gas
will be
Po = - y T 0 (5.2b)
The pressure p 0 is called the reduced pressure, and it
singularly determines the molar concentration and, con
sequently, the concentration of the molecules of a gas at
a pressure p and temperature T. The corresponding rela
tionship can easily be found if it is taken into consideration
that 1 kilomole of a gas in standard conditions occupies a
volume of 22.42 m3. Thus, at a reduced pressure equal to
one atmosphere, the molar concentration of a gas will be
equal to 0.044616 kmole/m3.
Knowing that one kilomole contains 6.023 X 102G mole
cules, we find that such a concentration corresponds to
2.687 X 1025 molecules/m3. The values of the molar concen
tration and the concentration of molecules at the reduced
pressure expressed in various units are given in Table 18.
THERMAL UNITS 125
The development of the kinetic theory of ideal (perfect)

gases made it possible to deduce equation (5.2) with a num
ber of simplifying assumptions, including the one that
the absolute temperature is proportional to the mean kinetic
energy of translational motion of molecules. This relation
ship can be expressed by Boltzmann’s formula
(5.3)
where K is a universal constant (not depending on the gas).
In the generally accepted form, equation (5.3) is writ
ten as
(5.3a)
The constant k in this equation is called Boltzmann’s cons
tant.
Equation (5.3) makes it possible to give a definite physi
cal meaning to temperature as a quantity proportional
to the mean kinetic energy of the molecules. Such a defini
tion of temperature, however, does not exhaust the possible
relationships of temperature with other physical quanti
ties. Let us consider some of them.
Suppose we have an enclosed envelope or shell isolated
from the surrounding space and kept at a constant tempe
rature, with an ideal vacuum inside. Notwithstanding the
latter circumstance, it will not be absolutely “empty”.
The space within the shell will be filled with electromagnetic
radiation whose radiant energy density er, according to
the Stefan-Boltzmann law, is proportional to the fourth
power of the absolute temperature of the shell
er = o f 4 (5.4)
where a is a constant depending on the selection of the units.
The radiation inside the space is distributed by wavelength
as shown in Fig. 15 for three different temperatures. As
established by Wien, the wavelength of the maximum energy
in this distribution is inversely proportional to the absolute
temperature
(5.5)
126 UNITS OF PHYSICAL QUANTITIES AND TH EIR DIMENSIONS
where b is a constant also depending on the selection of the

units.
The two formulas (5.4) and (5.5) can used for measuring
and determining the temperature to the same extent as
formulas (5.2) and (5.3). Such a determination of tempera
ture from the formulas of radiation is even more general,
since it can be used both for
a space filled with a substance
and for a vacuum. For this
reason the widespread defini
tion of temperature as a
quantity proportional to the
mean kinetic energy of trans
lational motion of molecules
should be considered as a
particular one, namely, as
the definition of the tempe
rature of a body consisting
of molecules, atoms and
electrons. Quantum mecha
nics, however, limits even
this definition, making it unsuitable for low temperatures.
At the same time formula (5.4) is true for any conditions.
According to the second law of thermodynamics, no
heat machine, even the most ideal one, working without
friction and heat losses to the surrounding medium, can
have an efficiency (the efficiency r) is the ratio of the useful
work to the entire energy received by a system) equal to
unity, since part of the heat must pass without fail from
the heat source (heater) to the heat sink (cooler). If the
temperature of the source is 7\, and that of the sink T2j
then the maximum efficiency of a machine (which, of course,
cannot be achieved practically) is equal to
(5.6)
If we define absolute zero as the temperature which a
sink should have* for the ideal efficiency to equal unity,
* It should l)t* noted that absolute zero cannot be reached in prin

ciple, but can be approached very closely.
THERMAL UNITS 121
we can use formula (5.6), only theoretically, of course, for

constructing a temperature scale.
It is proved in thermodynamics that all the formulas given
above determine the same temperature, which for this rea
son has been called the thermodynamic temperature. Any
of the coefficients /?, k, o and b in formulas (5.2), (5.3a),
(5.4) and (5.5) could, if desired, be assumed equal to unity,
which would give different dimensions of temperature, viz.,
L?MT~2, L -1/ 4 M 1/471“1/2, and L “*. Moreover, it would even
be possible to change the definition of temperature itself,
making it proportional not to the mean kinetic energy of
translational motion of the molecules, but to the density
of the radiation energy in an enclosed shell. Correspondingly
all the formulas including temperature would change. For
example, the product of the volume and the pressure of a
gas, and the mean kinetic energy of the translational motion
of molecules would be proportional to the fourth root of
the temperature defined in this way. Naturally, such a step
would lead to very appreciable inconveniences. It would
also be inconvenient to replace temperature with a quantity
proportional to it, for example, the product pV for one kilo-
mole or mole of ideal gas, or the kinetic energy of one mole
cule, etc.
The exceedingly important place occupied by tempera
ture in modern physics, since in a macroscopic system
(i.e., in a system containing a large number of molecules
and other particles) it determines most of its properties and
the phenomena taking place in it (density, electrical con
ductivity, rate of chemicalTeaction, phase transformations,
etc.), makes it expedient to relate temperature to the group
of quantities having their own dimensions of their units,
and accordingly it is desirable to include the unit of tempe
rature among the basic ones. The designation of the dimen
sion of temperature is 0.
According to the International system of units, the abso
lute temperature is defined as the thermodynamic tempera
ture, a degree of this temperature being so established that
the triple point of water has a temperature of exactly
273.16°K. By triple point is meant such a point at which
all three phases of water, namely, ice, liquid water and
saturated vapour are in equilibrium. While equilibrium
between two phases (water-vapour, ice-water, ice-vapour)

is possible at different temperatures, the equilibrium of
all three phases is possible only at a quite definite tempera
ture, known as the triple point. According to the conventio
nal scale of temperatures, the triple point of water is quite
accurately equal to +0.01°C, so that the zero point of the
conventional one-hundred degree scale corresponding to
the melting point of ice at a pressure of 1 atm is equal
to 273.15°K.
5.2. Temperature Scales

The absolute thermodynamic temperature scale (Kelvin
scale) is used in scientific research to establish the relation
ships between temperature and other physical quantities.
In everyday life, in engineering and even in laboratory
practice use is made of the so-called centigrade or Celsius
scale, named after Anders Celsius (it should be noted that
the scale actually proposed by Celsius was inverted with
respect to the one that bears his name, i.e., it had zero
for the boiling point of water and 100 for the melting point
of ice). Temperature measured by means of the Celsius
scale is designated °C. For temperature intervals measured
in degrees Celsius or Kelvin, use is made of the symbol deg,
which also enters the designations of combined names of
derived units.
In some countries the Reaumer scale is still in use (°R).
In this scale the interval between the melting point of ice
and the boiling point of water at a pressure of 1 atm is
divided into 80 parts. In the Fahrenheit scale (°F) used
in Great Britain and the USA, a temperature of 32°F is
assigned to the melting point of ice, and of 212°F to the
boiling point of water, so that this temperature interval
is divided into 180 parts*.
We can now easily establish the relationship between
the different temperature scales. Indeed, if we designate
* A scale in which the magnitude of a degree is the same as in the

Fahrenheit scale, but where the temperature is counted from absolute
zero, is called the Rankine scale. In this scale a temperature of 459.67°
corresponds to 0°F, 491.67° to the freezing point of water and 671.67°
to the boiling point of water.
THERMAL UNITS 129
the temperalure interval between the melting point of ice

and the boiling point of water by 0, then we obtain for one
degree Kelvin or absolute (°K), Celsius (CC), Reaumur (°R)
and Fahrenheit (°F) the following values:
° K - ° C - 100 ‘ °R :- — * and °F —180
80 ’ (5.7)
and consequently, any other interval At will be expressed
by the values
A^°K = A2°C = 100
A«°R = ^ 8 0 (r.,8)
A«°F = ^ 180
whence
A«°K At°C A/°R Az°F
(5.9)
100 100 80 180
or
A/°K A<°C At°l\ AZ°F
5 9
(5.9a)
5 4
We stress the fact that the symbols A/°K, A/°C, A/°R
and A/°F represent numbers measuring the same tempera-
hire interval in different degrees—temperature interval
'mils. These numbers can be represented as the difference
between the extreme temperatures of a selecled interval
measured according to the respective scale, in other words
A*°K - *°K - f0°K; A*°C = fC - /0°C;
A ^ ° R - / R - / 0°R; and M °F = toF - t 0°V
laking /0°C —0 and, consequently, /0°C ■---=273°K *; /0°R-~
(fR; t0°F --- 32°F, we get
(t — 273)°K _ t°C _ *°R __ (I — 32)°F
(5 .1 0 )
5 ” 5 “ 4 9
The last expression makes it very simple to convert tempo-
mlures from one scale to another.
Hero 0(/0 is taken approximately equal to 273°K.

!I)40
130 UNITS OP PHYSICAL QUANT ITIES AND T H E IR DI MEN SIONS
5.3. Fixed Temperature Points

The thermodynamic temperature scale defines tempera
ture as a measured physical quantity and establishes its
unit. The latter is taken as a basic one and is defined as
follows: “the degree Kelvin is the unit of temperature accord
ing to the thermodynamic scale, in which the temperature
of the triple point of water has the value of 273.16°K (preci
sely)”. The word “precisely’* denotes that this point is fixed
as an unchangeable one.
in praclicc direct measurements in the thermodynamic
scale arc too complicated, and it is desirable to have the
possibility of comparing various instruments serving to
measure temperatures within comparatively narrow tempe
rature intervals while retaining a sufficiently high accu
racy. For this purpose use could be made of a gas thermome
ter, preferably a hydrogen or helium one, since these gases
in comparison with others obey the laws of ideal gases to
the highest degree. The use of a gas thermometer in practical
conditions, however, is highly inconvenient, and for this
reason several permanent fixed points were selected whose
reproduction in laboratory conditions is not difficult. One
of these points is given by the definition of the thermodyna
mic scale itself—the triple point of water, to which a con
stant temperature of 273.16°K is assigned. The remaining
points have been established on the basis of measurements
made as carefully as possible. All of them are temperatures
of phase conversions at standard pressure (1 atm). These
points arc as follows:
fo ilin g p o in t of o x y g e n .................................................................... - - - l N 2 . 9 7 f C
boiling point of w a t e r ........................................... 10i)L0
F reezin g p o i n t o f z i n c .................................................................. 4 1 9 . 5 0 , r>oC
Boiling point of s u l p h u r ....................................... 444.6°C
Freezing point of silver ........................................... 900.8°C
Freezing point of gold ............................................... 10G3°C j
5 A . Other Thermal Units

Quantity of heat. When speaking of the units of the quan-.
tity of heat, it should be noted first of all that the quantity
of heat is in essence a measure of work, and not of energy,
as is often thought. Indeed, if we subject a gas close to ai}
THERM AL UNITS 131
ideal one to isothermal expansion, then we shall have to

impart to it a certain amount of heat, which will not he
used to increase its “thermal” energy, hut will he completely
spent for performing external work. We have put the word
“thermal” in quotation marks, since no specific thermal
energy as a special form of energy actually exists.
Sometimes, in our opinion unsuccessfully, the term
“thermal energy” is used with respect to the kinetic energy
of the molecules of a substance. Heat received by a body,
however, may be converted to a certain extent info internal
energy of a system even at a constant temperature, if chan
ges in the internal structure of the system occur, for instance,
if a phase conversion takes place. The best example here
is the melting of bodies that requires the addition of heat
at a constant temperature. This heat is sometimes called
“latent heat”.
The first law of thermodynamics makes it possible to
establish a measure of the quantity of heat according to
the well known relationship
+ (5.11)
where AQ ----- quantity of heal supplied to a syslem
AU ----- change in the internal energy of the system
ATT--- work done by the system to overcome external
forces.
The quantity A U may include various kinds of energy,
both an increase in the kinetic energy of various kinds of
motion of the molecules (translational, rotational, oscil
lating) and a change in the energy of bond between separate
molecules. It may also include the energy of dissociation,
ionization, etc. Equation (5.11) shows that the quantity
of heat can be measured in the same units as any kind of
energy and any kind of work, including mechanical work.
Hence, the dimension of the amount of heat
[Q] = LHIT~2 (5.12)
is the same as the dimension of work, and the units for
measuring the quantity of heat should be the same as those
for measuring work. Accordingly, in the SI system the
quantity of heat is measured by the unit of work and
energy—the joule.
9*
132 UNITS OF PHYSICAL QUANT ITIES AND T H E IR DIMENSION S
Special units of the quantity of heat mentioned above

(Sec. 4.4)—the calorie and kilocalorie—however, are also
in great favour. These units were established in connection
with calorimetric measurements of the quantity of heat with
the use of a heat exchange process. Since the main substance
used in comparing the quantity of heat was water, then
the unit, the calorie, was correspondingly defined as the
quantity of heat necessary for heating one gram of water
by 1°G. The greater units established are the kilocalorie,
equal to 1 000 calories, and the therm, equal to 10** calo
ries. Accurate measurements, however, have shown that
these quantities of heat are not constant and depend on the
temperature interval used for heating. For this reason the
mean calorie was introduced, which is defined as one-hun-
dredth of the quantity of heat that must be imparted to one
gram of water to heat it from its melting point to its boil
ing point. This calorie corresponds to heating water from
14.5 to 15.5°G.
When the equivalence of heat and work was established,
special experiments were conducted to find the relation
ship between the units of the quantity of heat and work.
These experiments determined the so-called “mechanical
equivalent of heat”—a relationship according to which one
kilocalorie is equal to 427 kgf-m.
With a view to the fact that a noticeable discrepancy
exists between the values of a calorie or kilocalorie deter
mined in different ways (calorimetrical, Lhermochemical),
which led to the necessity of introducing corrections in
accurate calculations, it was decided to do without deter
mining the units of the quantity of heat by some kind of
thermal measurements and establish a constant relationship
between these units and the units of work, which was taken
as follows:
1 c a l- 4.1868 J
It is assumed that measurement of the quantity of heat

by means of the calorie and its multiples and submultiples
is to be retained as a temporary measure, and in the future
it will be replaced with measurements using the unit of
work of the SI system—the joule.
THERMAL UNITS 133
In conclusion it should be noted that in refrigeration

engineering use is made of the concept “quantity of cold”,
which is in essence the quantity of heat that can he extracted
by a refrigerating installation from the surrounding medium.
The unit of the “quantity of cold” is the jrigorie, numerically
equal to one kilocalorie, but according to its definition
having the reverse sign. It may be said that one frigorie
is equal to minus one kilocalorie.
Temperature gradient. Similar to the pressure and velocity
gradients considered previously, if is also possible to intro
duce the temperature gradient
dT
grad T dl (5.13)
which with uniform distribution of the temperature can
be written as
T2~Ti
Lo—/ !
Its dimension is
lg r a d 7 V ^ M lm O (5 .M )
and its units are deg/m and dcg/cm if the temperature is

measured according to the thermodynamic or one-hundred
degree scale.
The heat flow (heat flux) is determined as the quantity
of heat flowing in a unit of time in the direction of the
temperature drop:
(5.15)
Its dimension
[(D ]==[^]^ZA W Y -3 (5.16)
coincides with that of power.
Depending on the unit used to measure the quantity of
beat, the heat flow is measured in watts, kilowatts, mega
watts, etc., or calories and kilocalories per second, minute
or hour. The relationship between these units is given in
Table 15.
The surface density of heat flow (specific heat flow) is the
ratio of the heat flow to the cross-sectional area of the flow,
134 UNITS OF PHYSICAL QUANTITIES AND TH E IR DIM ENSIONS
i.e., the flow per unit of cross-sectional area perpendicular

to the direction of flow. According to the definition
dO
? = dA (5.17)
and its dimension is
\q] = MT~* (5.18)
Its units, accordingly, are equal to the units of heat flow
related to one square metre or one square centimetre.
Entropy. Thermodynamics divides processes into rever
sible and irreversible ones. The former include isothermal
and adiabatic changes in the state of an ideal gas. Ideal rever
sible processes, however, cannot be carried out in practice.
All processes accompanied by friction, heat exchange,
diffusion, etc. cannot be completely conducted in the reverse
direction. Statistical physics connects this irreversible
nature with a transition of the system from a less probable
to a more probable distribution of the elements forming the
system. We can consider as an example the process of mixing
two gases that were initially separated in a certain vessel
by a partition after the latter is removed. Another example
is the levelling out of the temperatures of several bodies
in contact that originally had different temperatures.
A quantitative measure has been established that allows
us to judge of the degree of irreversibility of a process.
This quantity is called entropy and designated S. If a system
is transferred from a state that we shall designate by the
subscript “1” to a state designated by the subscript “2”,
then according to the definition of entropy, its change in
this process will be
AS =■- j (5.19)
1
It is proved in thermodynamics that
dS = ^ h (5.20)
is a to ta l differential, so that the integral of dS around
a closed con lour is equal to zero. This means that entropy
is a function of state. For a particular non-iso la ted system
'.THERMAL UNITS 135
the change in entropy may have any positive or negative

value, and in particular be equal to zero. As follows from
the second law of thermodynamics, however, in a closed
system
A S>0 (5.21)
The quantity AS characterizes the degree of irreversibility
of the processes taking place in the given system.
Equation (5.19) determines the dimension of entropy
[5j = LW 2rT“20'1 (5.22)
and its units are J/deg, erg/deg, kgf.m/deg, cal/dcg, etc.
5.5. Units of Thermal Properties of Substances

Heat capacity. Ileat capacity is measured by the quantity
of heat that must be imparted to a body to heat it by one
degree. There are distinguished the specific heat capacity,
or simply the specific heat (the quantity of heat required
to heal one gram or kilogram), and the molecular, or mo
lar, heat (the quantity of heat necessary for heating one
mole or kilomole). The specific heat is found by the formula
C$P m dT (5.23)
where m -- mass of the body
Q — quantity of heat
T = temperature.
The dimension of specific heat is
-“ i W (5-24)
The relationship between the specific and molar heals is
determined by the simple relationship
C/noz—’c$pAI (5.25)
Frequently use is made of the concept volumetric specific
heat, which is the quantity of heat required for healing
a unit of volume of a given substance by one degree. It
is determined by the formula
Cuoi £spp (5.2G)
where p is the density of the substance.
UNITS OF PHYSICAL QUANTITIES AND T H E IR DIMENSIONS
The dimension of volumetric specific heat is

L 'W r ^ 1 (5.27)
The units of specific heal: arc J/(kg.deg), crg/(g.deg),
cal/(g.dcg), and kcal/(kg.deg).
Obviously 1 J/(kg.deg) 104 erg/(g-dcg); 1 cal/(g-deg) —
= 1 kcal/(kg-dcg) ----- 4.19 J/(kg-deg).
The units of molar heat are J7(kmole -deg), erg/(moie -deg),
cal/(mol-deg) and kcal/(kmole-deg).
The relationship between these units is the same as between
the corresponding units of specific heat.
The units of volumetric specific heat are J/(m3-deg),
erg/(cm3 -deg), cal/(cm3 -deg), kcai/(l -deg).
The relationship between them is 1 ,T/(m3-deg) --
— 10 erg/(cm:l-deg); 1 cal/(cm3-deg) 1 kcal/(I-deg) —
— 4.10 J I(kg -deg).
Ileal of transformation. When a substance passes over
from one state of aggregation to another it is necessary to
spend a certain quantity of heat at a constant temperature,
called the heat of transformation (or transition). The
latter, as heat capacity, can be related to a unit of mass,
a mole or kilomole, or to a unit of volume. The correspond
ing dimensions differ from those of specific heat in the
absence of the symbol of the dimension of temperature.
In the same way the units of the heat of transformation
differ from those of specific heat in the absence in their
denominator of Ihe unit of temperature interval-degree.
Heating value. Any fuel is characterized by its heating
value, i.e., the quantity of heat that a certain amount of
it can liberate upon combustion. The heating value, the
same as the specific heat and the heat of transformation,
can lie related to a unit of mass, mole or kilomole, and to
a unit of volume. The volumetric heating value is used exclu
sively for combustible gases, and is generally related to the
volume of the gas taken in standard conditions (i' -- 0°C
and p 1 atm). The units of heating value are the same as
those of the heat of transformation.
Thermal conductivity. When there is a difference of tem
peratures in a medium, a heat flow will set in from the
layer with the higher temperature to the one with a lower
temperature. For a stationary univariate case this heat
THEKMAL UNITS 137
flow can be expressed by flic formula
f “ - '- 4 U (S-28)
where ^7at -- heat flow
~ temperature gradient
A cross-sectional area of flow
X — thermal conductivity of medium.
The unit of the thermal conductivity should he taken
equal to the thermal conductivity of such a medium in which
a heat flow equal to a unit of quantity of heat in a unit
of time sets in through a unit of surface perpendicular to
the direction of the flow at a temperature gradient equal
to a unit of temperature over a unit of length. This defini
tion and formula (5.28) give the dimension and units of
the thermal conductivity:
[X] - :/>A/7,-30-1 (5.29)
The units of thermal conductivity are \\7(m*dcg),
erg/(cm -s -deg), cai/(cm -s -deg), and kcal/(m -h -deg).
The relationships between them are
1 W/(m*deg) = l()r>erg/(cm «s*deg)
1 kcal/(m-h*deg) —— cal (cm -s •deg)
Heat transfer coefficient. When there is a temperature dif

ference AT at the boundary between two bodies, a heat
flow sets in through this boundary that is determined by the
form ula
— = atsT A (5.30)
The coefficient a is called the heat transfer coefficient. It
depends on the conditions at the boundary, in particular,
;it the boundary between a solid body and a fluid, and on
1lie rate of flow of the latter. The heat transfer coefficient
ran be defined as the heat flow through a unit of boundary
.iroa at a temperature difference of one degree. Its dimen
sion is
(5.31)
The units of the heat transfer coefficient arc W/(m2-deg),

erg/(cm2 -s -deg), cal/(cm2 -s -deg) and kcal/(m2 -h -deg).
The relationships between them are:
1 W/(m2-dcg) —103 erg/(cm2*s-deg)
1
1 kcal/(m2*h*deg) -- 104 erg/(cnr-s*deg)
Thermal diffusivity. Since the concept of thermal diffu-

sivity is not always sufficiently well known, we shall deal
with it in somewhat greater
detail. Let us consider a homo
geneous bar whose sides are
perfectly thermally isolated,
i.e., do not exchange heat with
the surrounding medium. Let
all the points of the bar origi
nally have the same tempera
ture If wc now bring one
of the ends of the bar into
contact with a medium having
Fig. 16 the temperature 1\ (for pur
poses of dctcrminacy let us
assume that T1p> T0) then a heat flow will set in along
the bar, and the temperature of all the points along it
will begin to rise (Fig. 16). The curves £0, . . ., too
correspond to different consecutive moments of time.
Part of the heat passing through the bar will be spent for
increasing the temperature of its different points, and a
temperature gradient will appear along it. It is this process
of the establishment of a temperature gradient that is
called thermal diffusion. The process of thermal diffusion
is obviously not stationary, since with a stationary heat
flow througli the bar the temperature gradient at all its
points should be constant, not changing with time. The
rate of change of the temperature at each point of the bar
in the example described above (called a linear or univariate
case) is determined by the equation
THERMAL UNITS 139
since
d2T d grad T
di
i.e., tlie derivative of the gradient along the axis is the chan
ge in Hie gradient per unit of length of the bar. The factor a
is known as the thermal diffusivity and, as shown by theory,
is related to the specific heat csv , the thermal coductivity %
and the density p as follows:
X X
a (5.33)
cspP cvol
Formula (5.32) defines the thermal Tdiffusivity as the
increase in temperature in a unit of time if the change in
temperature gradient per unit of length is unity. A simpler
definition is given by formula (5.33), according to which
the thermal diffusivity is equal to the increase in tempera
ture of a unit of volume of a given substance if it receives
a quantity of heat numerically equal to its thermal condu
ctivity.
The dimension of the Ihernial diffusivity
[XI _ /.Af7’-30-i
[a] (5.34)
[Cap] IP] ~
coincides with that of the diffusion coefficient [see formu
la (4.10(5)1. This coincidence is not accidental. For a gas
even the numerical values of the two quantities are quite
close. This can be understood if we take into account that
the kinetic theory of gases gives the following approximate
relationship between the thermal diffusivity and coefficient
of diffusion:
X = Dpcv (5.35)
where cv is the specific heat of a gas determined at con
stant volume.
From formula (5.35) we get
£> = —
P CD
i.e., in essence this is the same as formula (5.33). A stricter

theory gives a factor of proportionality in formula (5.35)
that differs somewhat from unity.
140 UNITS OF PHY SICAL QUANTITIES AND THEIR DLYtEYHONS
Temperature coefficients. Most of the physical properties

of a substance depend on its temperature. If at a certain
temperature rf 0 the property we are interested in has the
value A a, then at a different temperature T this property
will have the value A , which can he expressed in the form
of the series
A -= A{) (1 4 a{t 4 a2t2Jr a As 4- *• •) (5.36)
where t ™ T — 7’0.
The coefficients a±, a 2, « 3, etc. may have most diverse
values, bolli positive and negative, and depend on the
selection of the initial temperature T0.
The absolute values of these coefficients often comply
with the condition
l > | a i | > | a 2| > | a 3| > . . .
Frequently the coefficients a->, a 3, etc. may be conside
red to be so small that
A = A0(l beij) (5.37)
In particular, for the volume of a gas this gives Gay-Lus
sac’s equation and, if T0 = 273 K 0°C, then, as is known,
= 1/273. Since tlie product a xt is an abstract quantity,
then a1 is measured in the units deg-1.
The other coefficients, naturally, are measured in the
units deg"“, dcg“;h etc.
Coefficients of the van dev Waals equation. The van der
Waals equation of the state of a real gas has the form
(p ± 7 * )(V -b )= %RT (5.38)
here p =- pressure of the gas

V — volume occupied by the gas (volume of the vessel)
m mass
T -- absolute temperature
M — molecular weight
R universal gas constant (see Sec. 5.1).
The quantities a and fo, which are constant for a given
mass of the given gas, have been introduced to take account
of the forces of cohesion between the molecules and the
THERMAL UNITS 141
volume of the molecules themselves. The quantity

= (5-39)
owing to the forces of molecular cohesion, has the dimension
of pressure, and for this reason it is often (though unsuccess
fully) called the internal pressure.
The units used to measure pressure and volume also
determine the unit used to measure a. Since
V= —
P
and when p = const pt = const, then

a = V2pi —— m2 (5.40)
i.e., a is proportional to the square of the mass.
If this constant for a kilomole or mole is designated a0,
then we can write
a "-=a° ( i r ) 2 (5-41)
The constant b, proportional to the total volume of all
the molecules, should be proportional to the mass of
a gas, i.e.,
6 - ' ' • ( ! ) <5 - « >
where is the value of the constant for one kilomole or

mole. The dimension of a from formula (5.40) is
fa] = U*MT~2 (5.43)
The dimension of b is equal, naturally, to that of volume:
[6] = L3 (5.44)
The dimensions ofa, a0, b and b0 determine their units. In
practice a is frequently measured in atm/12 and b in litres.
CHAPTER SIX
ACOUSTIC UNITS
6.1. Objective Characteristics

of Mechanical Wave Processes
In media having elasticity, mechanical deformations
propagate with a velocity depending on the elastic proper
ties and density of the medium. If the deformation is perio
dic, then waves propagate in the medium, their length being
related to the frequency of oscillations v and the velocity
of propagation c by the equation
As previously indicated, the frequency of oscillations is

measured in hertz (Hz), and the wavelength in units of
length—metres, centimetres, etc.
Oscillations whose frequency ranges from 1(5 Hz to lb-
20 kHz are perceived by man’s organs of hearing and are
called acoustic, or sound, oscillations. Oscillations with lower
frequencies are called injrasonic, and with higher ones —
ultrasonic.
The characteristics of oscillations connected with Ihe
features of their psycliophysiological perception are described
in the next section. Here we shall deal with the quanti
ties that have an objective nature and arc determined by
the corresponding general mechanical quantities that we
already know. Although the SI system of units is recommen
ded for use in all fields of science and engineering, in acou
stics the cgs system is still in the greatest favour. The
mk(force)s system is practically not used. Below are listed
the most important quantities and their units in the SI
and cgs systems.
Sound pressure. The appearance of sound oscillations
in a gas or liquid is accompanied by oscillations of the
ACOUSTIC UNITS 143
pressure of (he medium. Thus the pressure at the given

point at each given moment of time can he represented as
the sum of Die pressure in the non-excited medium, i.e.,
in the absence of oscillations, and a variable additional
pressure called the acoustic, or sound pressure. During one
period of oscillations the sound pressure changes its value
and sign between the positive and negative amplitude
values.
Sound pressure, as any other one, is measured in N/m2
or dyn/cm2. The latter was formerly called the bar in acou
stics. But since the bar is now
used lo denote a pressure of 10f;
dyu/cjn2, the use of this name
to do note a unit of 1 dyn/cm2,
has been discontinued, and the
latter is now called a microbar
(pb).
Volumetric velocity. In a sound
wave the particles of the me
dium oscillate with a velocity
depending on the amplitude of
oscillations, the frequency, and t'ig. 17
the phase. Assume we have
a flat longitudinal wave (sound waves are longitudinal
ones) propagating along the x-axis (Fig. 17). Let Die parti
cles of the medium in plane M have the velocity v at the
given moment. Let us place plane N at a small distance
\x from plane M. During the time At --- Axlv all the par
ticles contained between M and N will pass through N.
If we select an area A on plane A, then during the time At
Die volume = vAtA will pass through it, and during
a unit of time the volume vA. This quantity is called the
volumetric velocity. It is easily seen that ils dimension and
units are the same as those of the volumetric flow rate,
».e., m3/s arid cm3/s.
Sound energy. Any volume of a medium in which waves
are propagating has an energy consisting of the kinetic
energy of the oscillating particles and the potential energy
<>f elastic deformation.
Sound energy, as any other kind of energy, is measured
m joules and ergs.
1 44 U NITS OF PHYSICAL QUANTITIES AN D T H E IR DIMENSION S ^
Density of sound energy. Sound energy related to a unit

of volume of a medium is called the density of sound energy
and is measured accordingly in J/m3 and erg/cm3.
Flow of sound energy. Waves spreading in a medium carry
along with them a flow of energy. The energy conveyed in
a unit of time through a given area perpendicular to the
direction of propagation measures the magnitude of this
flow. Obviously, the dimension and units of the flow of
sound energy coincide with those of power, W and erg/s.
Sound intensity is the density of the flow of sound energy,
i.c., the flow of energy related to a unit of surface perpendi
cular to the direction of the flow. The dimension of sound
intensity is
[1\^MT~* (0.2)
The corresponding units arc W/m2 and crg7(cm2-s). The
relationship between them is 1 W/m2 — 103 crg/(cm2 *s).
Acoustic resistance. The amplitude of oscillalions and,
accordingly, the velocity of the oscillating points depend
on the mechanical stress appearing in the medium, while
for waves in a fluid on sound pressure. The instantaneous
value of the velocity is determined by the relationship
where /; is the sound pressure and p the density of the

medium.
If the left-hand and right-hand parts of equation (0.3)
are multiplied by the area of the flow (for example, by the
cross-sectional area of a pipe), then we can write
P
vA = P cl A
(6.4)
The quantity at the left is the volumetric velocity of oscil

lations. The ratio of the pressure to the volumetric velocity
is called the acoustic resistance, since the appearance of
formula (6.4) is the same as that of Ohm’s law if the sound
pressure is considered to be similar to the difference of
potentials, and the volumetric velocity to the current
intensity.
ACOUSTIC UNITS 145
According lo the definition of acoustic resistance, its

dimension is
llia\--L-*M T-1 (6.5)
The units of acoustic resistance are N -s/mr> and dyn-s/cm5.
The relationship between them is 1 N -s/m5 = 10~r’ dyn -s/cm5
The name acoustic ohm is quite widely used for the unit
dyn -s/cm5, although it is not recommended by the relevant
USSR State Standard.
In the general case the variable sound pressure and the
variable volumetric velocity may not coincide in phase,
and in such instances, similar Lo the impedance when dealing
with alternating current, there is introduced the concept
of complex acoustic resistance or acoustic impedance.
The acoustic resistance of a unit of surface area is called
the specific acoustic resistance or the acoustic resistivity and
is a characteristic of the given medium. Jt follows from
formula (0/i) that the acoustic resistivity is equal to the
product of the density of the medium and the velocity of
propagation of oscillations
r) = (>c (0.6)
The dimension <>f acoustic resistivity is
[q] -- L~2A/7,"1 (0.7)
The resistivities of some media are given in Table 58.
Mechanical resistance. In addition to the acoustic resist
ance, it also becomes necessary in acoustics to deal with the
so-called mechanical resistance, defined as the ratio between
the periodic force and the velocity of oscillations. Accord
ing to the definition
Hmech ~ ~ ~ (6.8)
Us dimension is
[Jlrneck] = M T ^ (0.9)
The units of mechanical resistance arc N -s/m and dyn-s/cm.
The latter unit is sometimes called (lie mechanical ohm.
Formula (0.9) determines the relationship between the
units: 1 N *s/m = 103 dyn-s/cm.
in— 1 0 4 0
14G UNITS OF PHYSICAL QUANTITIES AND THE III DIMENSIONS
Levels oj sound intensity and sound pressure. For characte

rizing the quantities determining the perception of sound,
of importance are not so much the absolute values of the
sound intensity and sound pressure as their ratios to certain
threshold values. For this reason there have been introduced
the concepts of the relative levels of intensity and sound
pressure. If the intensities of two sound waves are / 2 and
In then Hie logarithm of the ratio I J I i is called the diffe
rence between the levels of these intensities
£i = logio7^- (6.1<>)
The unit of level difference is the bel (B), defined as the
difference between the levels of two intensities whose ratio
is equal to ten and, accordingly, the common logarithm
of the ratio is unity. A tenth fraction of a bel, correspond
ing to the logarithm of a ratio equal to 0.1, is called the
decibel (dI3)*. With a level difference of 1 dB the ratio
A = io »-1=1.259 (6.11)
The difference in intensity levels measured in decibels is
determined by the formula
L (dB) =- 10 log,0—- (6.12)
In the same way as the difference in intensity levels, the
difference in levels of sound e n e r g y flow (sound power)
can be measured.
The following relationship exists between sound inten
sity and sound pressure:
I K (6.13)
pc
Consequently
logio 7 7 = 2 lo8io -7- (ti- U )
The method of measuring the difference in sound pressure
levels is so established as to ensure that this difference will
* Other names have been suggested in recent years for the decibel,
such as logit, decilit, decilog, decomlog and decilu (translator’s note).
ACOUSTIC UNITS 147
coincide with the difference in the intensity levels of the

same oscillations. Accordingly, the difference in sound pres
sure levels measured in decibels can he found from the
formula
L p -- 20 logJ0~ (G. 15)
Together with measuring differences of levels in bcls
and decibels, they are also measured in nepers (Np). A
difference in the levels of intensities of one neper corresponds
to a ratio of the intensities equal to the base of natural
logarithms. It follows from this definition that
1 B = 2.303 Np (6.16)
The level of sound intensity and sound pressure is frequently
related to a conditional threshold corresponding to a sound
pressure of 2 X 10~5 N/m2 or 2 X 10~4 dyn/cm2*.
6.2. Subjective Characteristics of Sound

The subjective perception of sound is characterized by
a number of quantities that can be compared to a certain
extent with some of the objective quantities considered
above.
Pitch of sound. The main qualitative characteristic of
a sound is determined by its frequency. We perceive different
sounds as having equal intervals in pitch if the ratios of
their frequencies are equal. Thus we can introduce the concept
of musical interval, determined by the ratio of the frequencies
of the sounds forming it. For example, an interval between
sounds with frequencies of 200 and 500 kHz is equal to an
interval between sounds with frequencies of 100 and 250 Hz.
A number of units constructed according to the logarith
mic principle are used to measure musical intervals. The
basic one is the octave, which is the interval between sounds
with a frequency ratio of two. An octave is divided into
1 000 millioctaves or 1 200 cents. Another unit of musical
interval is the savart (Sav) which is defined as an interval
for which the common logarithm of the ratio of the frequen
cies of the sounds forming it is equal to 0.001. The magni-
* Logarithmic units are described in greater detail in Appendix 1.
10*
tude of an interval measured in savarLs is expressed by the

formula
fra--.. lOO()log10^ (0.17)
The relationships between musical intervals and the ratios
of the frequencies of sounds forming them are given in
Table 22.
A series of tones with an interval of one octave between
the first and last ones is called a musical scale. To get harmo
nic musical sounds, the separate
intermediate steps of a scale—
its tones—must have frequen
cies whose ratios form conse
cutive small integers. A scale
whose tones satisfy this condition
is called a just, or natural, scale.
For transition from one tona
lity or key to another, however,
it is essential that it will be
possible to form a new .scale
with the same ratios between
the frequencies of the conse
cutive steps as in the basic scale, beginning from any
tone. It is absolutely impossible to simultaneously meet
both requirements in conventional musical instruments with
the note system of recording music in general use. For this
reason there was introduced a tempered scale in which an
interval of one octave is divided into 12 semitones (half
tones) with equal intervals between them. In accordance
with the above, the interval between adjacent semitones
is equal to 100 cents.
Table 23 gives the musical intervals forming the natural
and tempered scales. Figure 18 shows part of a piano keyboard
including one octave and gives the names of the respective
keys and the notes corresponding to them. (In some coun
tries the letter B is used instead of Ii to denote the seventh
tone of the scale).
Timbre of sound. Different sounds even of the same pitch
may differ from one another in their quality or timbre.
The latter depends on the presence and relative intensity
of additional oscillations, usually of higher frequencies^
ACOUSTIC UNITS 149
than the fundamental one determining the pilch of the

sound. There are no quantitative parameters that could
serve as a single-valued characteristic of timbre. In ana
lysing musical tones, the relative intensity of the separate
components is measured. In oilier words it can be said that
timbre is determined by the kind of function of the distri
bution of the intensity of a sound by frequencies.
Loudness or volume of sound. Although the perception
of a sound depends on its intensity, this relation, however,
dyn
is not a simple or single-valued one. First of all it should be

noted that the sensitivity of the human car to sounds of
different frequencies is different. The lowest curve in
Fig. 19 shows the so-called threshold of audibility—the mini
mum intensity of sounds of different frequency that the
normal ear is capable of hearing. A logarithmic scale has
been used in this figure along both axes—of abscissas and
ordinates. The left-hand scale of ordinates indicates the
intensities in erg/s-cm2 and the levels of the intensities
in decibels, the zero level being taken as the level of a sound
of the minimum audible intensity at a frequency of
150 UNITS OP PHYSICAL QUANTITIES AN D TH E IR DIMENSIONS
1 000 Hz. The right-hand scale of ordinates gives the cor

responding sound pressures in dyn/cm2.
The top curve corresponds to the appearance of mechanical
perception that passes over into a feeling of pain. Upon an
increase in the intensity of the sound of a given frequency,
the feeling of loudness of the sound grows. The curves given
in Fig. 19 are so constructed that the same loudness of
sounds of different pitch heard corresponds to each curve.
Thus, different levels of intensity correspond to sounds of
equal loudness, but differing in frequency. The curves have
been drawn in such a way that at a frequency of 1 000 Hz
they are spaced 10 decibels apart. At other frequencies the
difference in levels of adjacent curves is different.
Sounds are assumed to have equal intervals of loudness
if the differences in the levels of sounds having the same
loudness, but with a frequency of 1 000 Hz, arc equal to
10 decibels. Since different intervals of the level of inten-
sity correspond to equal intervals of the loudness level,
a special unit, the phon, lias been introduced for characte
rizing the level of loudness. The phon is defined as the diffe
rence in the loudness levels of two sounds of a given fre
quency, for which equally loud sounds with a frequency
of 1 000 Hz differ in intensity by 10 decibels. By faking
the level corresponding to the threshold of audibility as
the zero one, we can directly measure the level of loudness
of a sound in phons as the difference between the level of
loudness of the given sound and the zero level.
All the units given above, constructed on a logarithmic
basis, are, naturally, dimensionless ones.
6.3. Some Quantities Connected with the Acoustics

of Buildings
When a sound wave impinges on a surface, part of the
sound energy is reflected and part is absorbed. Correspond
ingly, the acoustic reflection and absorption factors are
introduced. The acoustic reflection factor p is the ratio of
the quantity of sound energy reflected from a large flat
surface of uniform material to the quantity of energy inci
dent on the surface during the same interval of time. The
acoustic absorption factor a is equal to the difference between
ACOUSTIC UNITS 151
unity and the acoustic reflection factor

l--p
a -- - (6.18)
The acoustic penetrance of a partition, d, is Hie ratio of the
quantity of energy passing through a partition to the quan
tity of energy incident on it during the same time.
All three quantities (p, a, and d) are dimensionless ones.
The aeons!ie penetrance of a partition is determined by
superposition of the processes of absorption in the substance
which the partition is made of, multiple reflection from
its front and rear surfaces, and partial passage through
these surfaces. Account also has to he taken of the phonomc-
non of interference of waves that are superposed on one an
other in different phases. Pure absorption is observed if the
thickness of the layer is so great that the intensity of the
waves reflected from the rear wall can be neglected. If a
plane wave is incident on the layer and its intensity after
entering the layer is 70> then at a certain distance x from
the boundary of the layer the intensity will be
/ = (6.19)
The factor 6 is called the linear absorption coefficient. Its
dimension is
[ 6 J -/T 1 (6.20)
and its units are m -1 and cm-1.
To characterize the absorption ability of individual
bodies the concept of the total absorbing power of a body
is introduced. It is determined by the product of the area
of the body and its absorption factor. It is measured by the
area of a perfect absorbing body having the same absorp
tion as the given one. The unit of total absorbing power
is the square metre of open window, since an opening in a wall
practically reflects no sound.
Reverberation. When a sound is produced in a hall, the
generated waves are repeatedly reflected from the walls,
floor, ceiling and all the articles filling it. During each
reflection part of the sound energy is absorbed, so that after
the generation of oscillations is stopped the density of the
sound energy at all the points gradually attenuates. If at
the moment when the generation of sound is stopped the
152 UNITS OF PHYSICAL QUANTITIES AND TI1K1H J)JMENSlONS
density of the sound energy is Uq, then after (lie time i it

becomes equal to
u := ui)e~t/x (0.21)
The process of the production of a sound with its fol
lowing decay is called reverberation. The characteristic time
constant r, as shown by W. Sabine, is equal to
4V
( 0 . 22 )
T cZaA
where V is the volume of the hall, and ^)ocA (lie sum of the
total absorbing powers of all the bodies in the hall, includ
ing the walls, floor, ceiling, furniture, people, etc.
The time t is that during which the density of the sound
energy decreases to 1/c-th of the initial density. In practice
a different quantity T is used, called the standard reverbe
ration time, defined as the time during which Ihe density
of sound energy will decrease by 00 dll, i.c., 10(i times.
Since
cZaAT
10-« -c /iV
we get
V
r - 2.M X 0 t )D. ((•>.2:5)
The reverberation time determines the acoustic proper
ties of a room or hall. If tins time is too short, the sounds
will be muffled and “dull”. If it is too long, the sounds will
be superposed on one another and speech will become unin
telligible. The optimal reverberation times depend on the
designation of the premises and range from several tenths
of a second to one or three seconds.
CIIAPTEU SEVEN
ELECTRICAL
AND MAGNETIC UNITS
7.1. Introduction
The systems of electrical and magnetic units have gone
through a complicated and, to a certain extent, contra
dictory period of formation, owing to the features of the
development of our knowledge of electrical and magnetic
phenomena. Up to the discovery by 11. C. Oersted in 1820
of the magnetic action of an electric current, electrical and
magnetic phenomena were studied independently, though
by the same scientists (W. Gilbert, G. Coulomb, etc.).
An appreciable part in the history of development of our
knowledge of magnetic phenomena was played by the cir
cumstance that man lirst became acquainted with them back
in ancient times upon the discovery of the magnetic pro
perties of iron.
When the time of quantitative investigation of electrical
and magnetic phenomena arrived, then, owing lo the exter
nal similarity between the interaction of permanent mag
nets and of electric charges, the same terminology was
introduced for describing these interactions. This termino
logy has been retained up to the present time, although it
does not correspond to our modern notions. Quite a few
scientists, basing their investigations on the above-men
tioned similarity, attempted to find the common nature of
electrical and magnetic phenomena, but with no success.
Wthough after the discovery made by Oersted and sub
sequent investigations it became clear that electrostatic
end electromagnetic phenomena arc quite different in
nature, the description of these phenomena in courses on
physics up to comparatively recent times was given in the
following sequence: first electrostatics was studied, then
magnetostatics, i.e., the science of interaction of permanent
154 UNITS OF PHY SICAL QUANTITIES AND TH E IR DIMENSIONS
magnets and their fields, next the laws of direct current,

and only at the end of the course was the magnetic action
of an electric current discussed. Magnetostatics also served
as the basis for constructing the units of magnetic quanti
ties, from which there were later formed the units of quan
tities characteristic of the magnetic action of electric cur
rent, electromagnetic induction, etc. Such a sequence of
studying the material created difficulties for understanding
the essence of phenomena and led to confusion in mastering
the fundamentals of the subject.
At present most courses in physics use different methods
of studying electromagnetism, in which a magnetic action
of a current is taken as the basic magnetic phenomenon.
This basis has also been used to introduce the unit of cur
rent intensity in the SI system—the ampere—which within
the limits of this system is conditionally considered as a
basic unit.
In this direction also, however, there is still no generally
accepted method of setting out the course, and it will
hardly be possible to indicate such a method. For this rea
son in the following section we shall consider different
kinds of electrical and magnetic interactions and show how
they can serve as the basis for constructing different systems
of units.
7.2. Possible Ways of Constructing Systems

of Electrical and Magnetic Units
Depending on the interactions and their form used to
define the physical quantities serving to describe electrical
and magnetic phenomena, a group of defining relationships
is established by means of which the relevant derived units
are introduced. In the mathematical expressions given below,
that describe the quail Lila live aspect of the interactions,
we shall, as previously, designate the factor of proportio
nality in the general form by the vSymbol C, regardless of
its specific numerical value, and only when necessary will
we supply it with a subscript.
Electrostatic interactions. Two electrically charged bodies
are mutually attracted or repulsed with a force depending
on the signs, magnitudes and distribution of the charges
ELECTRICAL AND MAGNETIC UNITS 155
on these bodies, their mutual arrangement, and the nature

of the medium in which interaction occurs. In the general
case, if charges of different signs are present on one or both
bodies, a torque may appear in addition to the resultant
force. The interaction will be the simplest if the bodies are
small in comparison with the distance between them, and
the charges may accordingly be considered as point ones.
Here, assuming that the interaction takes place In a vacuum,
the force of interaction can be written as the formula of
Coulomb’s law
(7.1)
Assuming in formula (7.1) that / ~ 1, r — 1, and Qx
Q2 and, as usual, that C - 1 and has a zero dimension,
we obtain the derived unit of charge (quantity of electricity),
by means of which we can construct the units of all Ihe quan
tities describing the properties of an electric held, condu-
ctors and dielectrics. The first of these quantities—the
vector characteristic of an electric field —Ihe electric field
intensity E is measured by the force acting on a positive charge
equal to unity, placed in the given field. By using this
definition and writing it without the factor of proportiona
lity in the form
E -^-L (7.2)
we can define the unit of electric Held intensity as the inten

sity of such a field in which a unit of charge is acted upon
by a force equal to unity. Further, by using the correspond
ing definitions, we can establish the units of other quanti
ties such as potential, capacitance, and polarizability.
The adopted unit of charge also makes it possible to establish
the unit of current intensity by the formula
/ = - (7-3)
according to which the intensity of an unchanging current
is defined as the amount of electricity flowing through a
cross section of a conductor in a unit of time. Strictly
speaking, a factor of proportionality should also be used
in formula (7.3), since an electric current is a new phenome-
156 UNITS OF PHYSICAL QUANT IT IES AND TIIE IR DIMENSIONS
non and its registration and measurement may not be con

nected with the measurement of an electric charge. Since
in all systems, however, formula (7.3) is considered as a
definition of current intensity, we have omitted this factor.
It should be noted in passing that Abraham proposed
a system of units in which the right-hand part of formu
la (7.3) contained a dimensional factor differing from unity.
By using Ohm’s law, we can further determine the unit
of resistance and thus construct an electrostatic system of
units. By taking as the basic units the unit of length—centi
metre, of mass—gram, and of time—second, we shall obtain
a system that lias been called the electrostatic system and is
designated cgse (or esu for “electrostatic units”). Here the
letter e is not a symbol of an additional basic unit, but only
serves to show that the system is based on electrostatic
interactions.
Interaction of permanent magnets. When investigating the
interaction of magnets, Coulomb established that if long
straight magnets are so arranged that the distance r between
their poles is much smaller than their length, then the force
of interaction between the poles is inversely proportional
to r2. Upon comparing this with the law of interaction of
electric charges also discovered by him, he introduced
the concept of magnetic charge, quantity of magnetism or
magnetic mass m, which should be similar to an electric
charge. By writing Coulomb’s law for the interaction of
magnets in a form similar to formula (7.1)
/ (7.4)
it was possible to establish a unit of magnetic mass similar

to the way used to establish the electrostatic unit of the
quantity of electricity. By analogy with the electric held
intensity, there was introduced also Ilie concept of the
vector quantity of the magnetic held intensity //, determi
ned by the force which a unit magnetic pole is subjected
to in the given held, j,eM such a pole whose “magnetic
mass” is equable unity
H / (7-5)
m
We indicated above that the similarity between electro

static and magnetic interactions has a purely superficial
nature that does not correspond to the essence of the pheno
mena. This, in particular, was revealed in the fact that,
as we shall see below, the coincidence of the names “field
intensity” for the vectors E and II does not conform with
the part that these vectors play.
In the same way as the unit of electric field intensity
was introduced in electrostatics, the unit of magnetic field
intensity was introduced in magnetostatics as the intensity
of such a field in which a pole whose magnetic mass is equal
to unity is subjected to a force equal to unity.
Although the concept itself of magnetic mass was found
to be completely fictitious, it was possible with its aid to
establish the definitions of all the quantities describing
a magnetic field and the magnetic properties of a substance,
and construct a system of magnetic units on the basis of
these definitions.This system, in which the basic units are
also the centimetre, gram and second, has been called the
electromagnetic system and is designated cgsm (or emu for
“electromagnetic units”). Here the letter m, as the letter e
in the designation of the cgse system, serves only to show
that the system is based on magnetic interactions. The sys
tem has been given the name “electromagnetic’ since it
includes the units of quantities characteristic of the magne
tic properties of a current, and the units of all the electrical
quantities have also been constructed on its basis.
Electromagnetic interactions. Various experiments that
are different in appearance, but have a common nature,
can be conducted to show the magnetic field of an electric
current. The first experiment of this kind was the deviation
of a magnetic needle under the action of an electric current
observed by Oersted. By determining experimentally what
magnetic field of a permanent magnet causes the same
deviation of the needle, Biot and Savart laid (he founda
tion for establishing the law determining the magnetic field
of a current. The difficulty involved in obtaining a gene
ral law on the basis of such experiments consisted in that
whereas the interaction of electric charges could be studied
on very small charged bodies that behaved as point ones,
il was impossible in principle to create a “point current”,
158 UNITS OF PHYSICAL QUANTITIES AN D TH E IR DIMENSIONS
since any current must flow along a closed circuit. This

difficulty was evaded by Laplace, who proposed a formula
representing the magnetic field of any closed circuit as
the geometrical sum of the fields created by separate ele
ments into which the given circuit can be imaginarily di
vided. Laplace’s formula (expressing the Biot, Savart and
Laplace law) for an element of a current can be written as
(7.6)
where dl = length of an element of the circuit

r -- distance between this element and the point
at which the field intensity is being determined
a -- angle between r and dl (Fig. 20).
The direction of the vector dH, coinciding with the dire
ction of the force acting on the north pole of a magnetic need
le, is determined by one of the mnemonic rules, for instance,
the right-hand screw rule.
To determine the magnetic field of a closed circuit having
an arbitrary shape, formula (7.6) should be integrated
around the entire circuit
Formula (7.6) gives several

possibilities for selecting units.
If we equate C to unity and
measure current intensity in
electrostatic units, we will have
to introduce a newr unit for
magnetic field intensity and
name it the electrostatic unit
of magnetic field. All the remaining units characteriz
ing a magnetic field will change accordingly. Another pos
sibility is to measure?magnetic field intensity using an
electromagnetic unit, also with C equal to unity, and esta
blish a new electromagnetic unit of current intensity.
Finally, we can retain both the electrostatic unit of cur
rent intensity and the electromagnetic unit of magnetic
field intensity and determine experimentally or theoretical
ly the numerical value (and, accordingly, the dimension)
of the factor C> This way will lead us to the construction of

the so-called symmetrical or Gaussian system of units,
which at present is in the greatest favour among physicists.
It should he stressed here that all the systems listed above
are cgs ones, since they are constructed on the same basic
units —the centimetre, gram and second. In contrast to
the cgse and cgsm system, the Gaussian system is designated
cgs.
Attention must also be drawn to the following very im
portant circumstance. If we accept the cgse system for
both electrical and magnetic quantities, then a dimensional
factor of proportionality will appear in the equations of
magnetism. It will be found immediately if we consider
Coulomb’s magnetic law (7.4). Since the units of force and
distance are established, while the unit of magnetic field
intensity determined by formula (7.6) gives us, according
to formula (7.5), a unit of magnetic mass different from
that introduced above, then formula (7.4) can be retained
only if the factor C differs from unity and, as can be easily
seen, has a definite dimension. Upon establishing the
electromagnetic unit for current intensity and, according to
formula (7.3), for electric charge, we shall have to introduce
into formula (7.1) a dimensional factor differing from unity.
The numerical values and dimensions of the factors will be
considered below.
It is quite obvious that the methods listed above do not
exhaust all the possibilities of constructing systems of
electrical and magnetic units, even if we remain within
the limits of the same three basic units (cm, g, and s).
Factors could be introduced into any of the formulas (7.2),
(7.3) and (7.5). It is possible to construct such a system of
units, for example in which, with a view to certain practical
considerations, we shall establish a prototype of one of the
electrical or magnetic units and use not three, but four
basic units.
The electromagnetic phenomenon that is the opposite
to the action of a current on a magnetic needle—the action
of the field of a permanent magnet on an electric c u rre n t-
gives the same possibilities for constructing systems of
electrical and magnetic units as the ones listed above. It
is possible either to determine the electromagnetic unit
160 UNITS OF PHYSICAL QUANTITIES AND TH EIR DIMENSIONS
of current intensity if the unit of magnetic field intensity

has been established, or determine the electrostatic unit
of magnetic field intensity if the unit of current intensity
has been established. in both instances we shall use Ampe
re’s formula for the force acting on an element of current
in a magnetic field:
/ \
d/ = C///dZsin(ff, dl) (7.7)
If we take, for instance, a straight conductor arranged
in a magnetic held perpendicular to the direction of the
lines of force of the held, and, if we assume that C = 1,
then a unit of length of this conductor will be acted upon
by a force equal to unity if: (a) the held intensity is equal
to unity in the cgsm system and the current intensity is
equal to an electromagnetic unit of current, or (b) the
current intensity is equal to unity in the cgse system and
the held intensity is equal to an electrostatic unit of inten
sity. If the current intensity is measured in cgse units, and
the magnetic held intensity in cgsm units, then we shall
have to introduce a dimensional coefficient C differing
from unity.
A somewhat different way of establishing the units will
appear if we abandon the use of the interaction of magnets
or a magnet and a current and turn to the interaction of two
currents. There are sufficient physical grounds to adopt this
way as the basic one. The interaction of currents can he
related with full right to the fundamental phenomena of
nature, such as universal gravita tion and Ihe interaction
of electric charges. At the same time the magnetic proper
ties of iron ami oilier ferromagnetic substances are inherent
only in these substances and reflect the features of their
structure. Ferromagnetism belongs to the most complicated
phenomena and its explanation became possible only on the
basis of consideration of the interaction of electrons from
the viewpoint of quantum mechanics.
If we lake the interaction of currents for constructing
systems of electrical and magnetic units, we can write this
interaction in different ways depending on the configuration
and mutual arrangement of the currents. We can, for exam
ple, consider the interaction of very long straight conductors
of small plane circuits at a distance that is great in compa

rison with their linear dimensions, etc.
We shall use the expression for the mechanical moment
acting on a small plane circuit in the field of any arbitrary
circuit. This expression can be obtained from Ampere’s
formula if we consider the forces acting on separate ele
ments of a closed circuit arbitrarily arranged in a homoge
neous magnetic field. For a field created by an arbitrary
circuit to be considered homogeneous, we select a small
“trial’* circuit, and for the moment acting on it we can write
(7.8)
where I 2 = current of small circuit
A = its area
/, =- current in arbitrary circuit that is a source
of the field which is an external one with respect
to the small circuit
<p - angle between direction of a normal to the
area of the small circuit in the given position
and in the position in which the moment is
maximum.
Up to now we did not consider the question of the part
played by the medium and included the parameter characte
rizing the properties of the medium in the factor C, or assumed
that all the interactions occurred in a vacuum. Let us now
write Coulomb’s law and formula (7.8) in such a way that
the properties of the medium in which interaction takes
place will be shown in explicit form. To distinguish the
factors of proportionality, we shall supply them with
numerical subscripts
(7.9)
(7.10)
The characteristics of the medium er and pir are called, as is

known, the relative permittivity (dielectric constant) and
the relative magnetic permeability.
Since the concepts of the electric field intensity E , electric
displacement (induction) D, magnetic induction B , and
I 1— 1 0 4 0
162 UNITS OF PHY SICAL QUANTITIES AND TH E IR DIMENSIONS
magnetic field intensity II are introduced for describing

electrical and magnetic phenomena, the following groups
of equations can be substituted for equations (7.9) and (7.8).
Electrostatic interactions Electromagnetic interactions
l-=C,EQ, (7.11) M =- C^BI2A cos <() (7.12)
D =c>%- (7.13) = n C I\ dl sin a (7.14)

E-= C A (7.13) B == Cs\irH (7.16)
er
The sequence in which the above equations have been
written is not accidental. Equations (7.11) and (7.12) relate
to the mechanical action (force or moment) resisted by a
charge or circuit in the given specific conditions with
account taken of the influence of the medium. Equations
(7.13) and (7.14) characterize the field of the charge Qx and
current / x without account taken of the influence of the
medium. Finally, the last two equations (7.15) and (7.16)
relate the characteristics E and B of a field determining the
mechanical action to the properties of the medium and
through the quantities D and II to the charges and currents
that arc the sources of the field.
Thus, a certain analogy can be established between the
following pairs of quantities:
E and B
D and II
er and i/\xr
It shows that the characteristics of a magnetic held hu. ..
been named unsuccessfully. The origin of these names is
connected with the fact that they were introduced during
the development of the science dealing with the properties
of permanent magnets. Coulomb’s magnetic law, with ac
count taken of the influence of the medium, was written as
(7.17)
7 ^
The external contradiction between formulas (7.10) and
(7.17) is explained by the circumstance that in magnetosta
tics it was assumed that the “magnetic masses7' do not depend
on the properties of the medium. An analysis of this ques

tion has shown that the “magnetic masses” of poles so change
with a change in the medium that if we denote the “magnetic
mass” in a vacuum by m0, then in a medium with a magnetic
permeability of pr the “magnetic mass” will be md\ir, so
that instead of formula (7.17) we can write
mot mQ2
(7.18)
The group of equations (7.11) to (7.16) makes it possible
to construct systems of units of electrical and magnetic
quantities in a great variety of ways if we add equation
(7.3) relating the current intensity to the charge. As has
been noted above, we omit the factor of proportionality that
could have been used in this equation, since in all systems
it is taken equal to unity.
From among all the diverse possibilities of constructing
systems that are given by the group of equations (7.11) to
(7.16) and (7.3) we shall consider only those combinations
of factors that are realizable in practice. It must first be
noted that we cannot simultaneously manipulate with all
the factors, since at least one of them is determined as the
result of an experiment. T
The relative permittivity and relative magnetic permeabi
lity, according to their definitions, are so selected that in
a vacuum er = pr — 1 , and they are dimensionless quali
fies. In addition, in all systems C3 = 1. As a result five
factors remain, four of which we can deal with at our discre
tion. We could, if desired, arbitrarily establish the fifth
factor too, but for this purpose, as will be seen below, the
number of basic units must be reduced.
Let us first consider the alternatives of constructing
systems with the basic units ern, g and s. In the cgs and
cgse systems, Ch ---- CG 1 . Here, as shown by experiment,
Ihe product of the factors
C\C6CH- C2- - (7.19)
where c is the velocity of light in a vacuum. It can be seen
that c has the dimension of velocity by comparing the
dimensions of the quantities in the formulas given above.
11*
164 UN ITS OF PHY SICAL QUANTITIES AND TH E IR DIMENSIONS
In the cgse system the factors are selected as follows:

C4= Cq= 1 and C8 —-i-
The factor C8 in the cgse system is frequently designated p,0.
In the symmetrical cgs system
C4^ C q- — and C8--~ 1
and in the cgsm system
C4^ C 6-: C8- 1
This selection of factors determines the unit of current
intensity and the units of B and II. The unit of charge is
also established correspondingly. By combining formu
las (7.11), (7.13) and (7.15) and reverting to experiment, we
find that
C3C5C7 —Cx—c1 (7.20)
As mentioned above, C3 = 1. In addition, the factor C5
is also taken equal to unity. Thus C7 = c2. This factor is
generally designated l/e 0.
All three systems, cgs, cgse and cgsm, can be combined
into a single one if we assume that c = 1 , for which purpose
it is obviously necessary to reduce the number of basic
units. Since the factor c is the velocity of light, this will
be possible if we make one of the units, time or length, a
derived one instead of a basic one. When discussing the
question of the number of basic units in a system (Sec. 1.4)
we indicated the arbitrary nature of this number and noted
that beside the possibility of converting the unit of mass
from a basic to a derived one (by simultaneously equating
to unity the inertial and gravitational constants), a further
reduction of the number of basic units is possible by equat
ing to unity the velocity of light in a vacuum. Here we have
seen directly that this is possible.
The electromagnetic theory of light developed by Maxwell
made it possible to calculate the velocity of light on the
basis of the general equations of the electromagnetic field
(Maxwell’s equations). Depending on the system used to
write these equations (cgs, cgse or cgsm), the velocity of light
K1YKCTR1 CAL AND MAUNRTIC UNITS in.
will correspondingly be equal to c\ i /J /p 0 ov l/c 0, which,

of course, gives the same value. Should we cousin id a system
using the coefficients c0 and p 0 , but in which neither of them
is equal to unity in a vacuum (the Sf system is such a system
of units), then the velocity of light will be equal to 'l/Kpoô-
Of all the (liree systems listed above, in the following we
shall consider in detail only the cgs (symmetrical) system,
turning to the cgse and cgsm systems when this will be
essential in particular cases.
He re we shall only note Unit the units of electrical quan
tities (charge, field intensity, potential, current intensity,
capacitance, resistance, etc.) of the cgs system coincide with
the relevant units of the cgse system, while the units of
magnetic quantifies (iuduclion, magnetic flux, inductance,
etc.) coincide with (hose of the cgsm system.
Let us now turn to the construction of units of (he elec
trical and magnetic quantities in the SI system. The main
part in the creation of this system was played by the cir
cumstance that widespread use was made in practical work
in electrical and radio engineering and in physics of the
so-called practical units such as the coulomb, ampere, volt
and joule. When these units were established, however,
they were not combined into a harmonious syslem that
would allow their direct use for electrostatic and electro
magnetic calculations. For this reason Ibe problem appeared
of introducing such faclors of proportionality into the
system that would make Us use possible in all fields of the
science of electricity and electromagnetism and, after
combination with mechanical, thermal and oilier units,
would creale a system covering all the fields of physics
and engineering. To make it possible lo relate the practical
units of electrical and magnetic quail (Kies to the mechanical
units having a ready unit of energy—the joule, and simulta
neously comply with the requirement (hat the units of
length and mass must he decimal system multiples or sub-
multiples of the cgs units the following condition has to be
observed:
1 J = 107 crg = 107 g'cm 2/s2 - 10a g (10f>)2 cm2/s2
Hence, it follows Ihat
a j (7.21)
166 UN ITS OF PHYSICAL QUANTITIES AND T H E IR DIMENSIONS
Systems have been proposed with various combinations of

the exponents a and 5, namely 107 g and 1 cm (Blondel’s
system), 10~u g and 109 cm (Maxwell’s system in which
the coefficient ju0 is equal to unity), etc. The greatest atten
tion was attracted to the system proposed by Giorgi in
which a — 3 and b ~ 2, i.c., 1 kg and 1 m. Both these units
are convenient for practical work and are directly repre
sented by international prototypes. Since the system was so
formed that a new unit had to be introduced into it (any
of the electrical or magnetic units, for example, the ampere,
volt or ohm), two new factors would inevitably appear in
the expressions for Coulomb’s law and electromagnetic
interaction instead of one that is present in each of the
cgse, cgsm and cgs systems.
With respect to the dimensions of the relevant units,
three possibilities existed here. One of the factors (in Cou
lomb’s law or the law of interaction of currents) could be
considered as a numerical factor deprived of a dimension
and the system of dimensions constructed in the same way
as in one of the two systems, cgse or cgsm, or one of the
electrical or magnetic units could be considered as the basic
one and the system of dimensions correspondingly constru
cted using not three, but four basic units*. It was the latter
way that was adopted in constructing the system of dimen
sions of the SI system. One of its advantages is the simpler
form acquired by the dimension formulas. As we have alrea
dy learned, the fourth quantity, the dimension of whose
unit is included among the basic ones, is the unit of current
intensity, the ampere. In formula (7.25) used to determine
the ampere, the constant fi0 is considered to have a dimen
sion, although its numerical value is fixed. If this quantity
were considered as a dimensionless one (the dimensions of I
and r cancel out), then the dimension of the unit of current
intensity would lie
i l
[I] = L*M2T-1 (7.22)
* Here, naturally, we do not take into consideration the dimen-

sions of temperature and light intensity, that are not included in the
dimensions of any of the electrical and magnetic units.
ELECTRICAL AN D MAGNETIC UNITS 11*7
whence the dimension of charge would he

1 1
[(?] — L 2M 2 (7.23)
This dimension differs from that of charge in the cgs system

[see formula (7.30)] by a factor whose dimension is the recipro
cal of that of velocity. Obviously, the units of current inten
sity and quantity of electricity will have the same dimen
sions (7.22) and (7.23) in the cgsm system.
Since the so-called rationalized form of writing the equa
tions of electromagnetism first proposed by 0. Heaviside
came into great favour in literature on electrical and radio
engineering, this form was adopted in constructing the SI
system. In the rationalized form the coefficient 4ji is written
in the denominators of the equations of the interaction of
electric charges (Coulomb’s law). As a result in a number of
equations that are relatively more frequently encountered
in practice, this coefficient disappears, and the equations
acquire a more symmetrical form. This relates first of all
to Maxwell’s equations describing the electromagnetic
field.
In accordance with the above, the following values of
the factors of proportionality in equations (7.11) to (7.13)
are used:
63—C4 -- 1 and C5—C6 -
The factor C8 will be designated p0 in the following.

The product of the factors C4C6î0 is so selected that when
the current intensities I ± and / 2 are measured in amperes,
lengths in metres and areas in square moires, the moment
will be measured in newton-metres (N-111). For convenience
of calculation let us take instead of equation (7.10) the
force of interaction of two parallel straight current condu
ctors. This is also expedient because of the fact that the
definition of the ampere adopted at the 9th Meeting of the
International Weights and Measures Congress in 1948 is
based on this interaction.
Let us determine the intensity of the magnetic field of
an infinitely long straight current conductor on the basis
108 UNITS o f p h y s i c a l q u a n titie s and th e ik DIMENSIONS
of the Biol, Savart and Laplace law (7.14):

A (7.24)
Zji r
Using this expression, we gel from Ampere’s formula,
wrillen wiHi account of the influence of Hie medium, the
force acting on a section of a conductor having a length /
and a current / 2, parallel to a conductor with the current I x:
j _ flQfXr I[f-J ^
If this force were written in the cgsm system (using the

non-ratioimlized form of writing the equations) we would
have the formula
/ — A. (7.25a)
When introducing the practical units, the ampere was
defined as 0.1 of the cgsm unit of current intensity. Assu
ming that I --r and / 2 - 1 A---0.1 cgsm, we get
/ ■- 2 X 10 " dyn 2 X 10'5 N
The SI system takes this relationship as the definition
of the ampere, without relating it to the cgsm unit. The
exact definition of the ampere was given in Sec. 1.5.
Formula (7.25) now makes it possible lo determine the
value of the coefficient p0 4-ji X 10-7. If, as is conditio
nally assumed in the SI system, the ampere is considered to
he a basic unit, then the dimension of the coefficient p0
will obviously be
Ip0] —LMT~2[ 2 (7.26)
where I is the symbol of the dimension of current intensity.
The coefficient p0 is known as the magnetic constant.
Although the name of p0 — N/A2—follows from formula
(7.25), the name it/m is generally used, where I t is the sym- .j
boi of the unit of induclance—the henry, which will he <
defined below. ;
It should be noted Hurt in accordance with the definition,
the number An X iCr7 is Ia ken as a precise one that must
not be changed when more accurate measurements are made. :
KLRCTRJCAL AND MAUNUTJO UNITS 169
The constant factor C\ iji equation (7.14) can be deter

mined in different ways. IJy .substituting iii the following
the designation l/e 0 for C7. Coulomb’s Jaw can be written as
1 QiQz (7.27)
/ •Inf'o y.rr-
As has been noted aI>ove, in the SI system the velocity
of light in a vacuum is equal to
__ 1_ _
(7.28)
TAoi'o
This relationship is equivalent to I be experimental deter
mination of the factor C., given above | formula (7.19)J.
Taking into account that c 3 X 10s m/s, we find that
1 1
t{) An X 10-7 X 9 X 1016 ~ An ;; 9 lu»
Us dimension is
[e0] I 'M 'TV!- (7.29)
Instead of the name of e0 following from formula (7.27),
viz., , the name F/m is used, where F is the symbol of
the unit of capacitance, the farad.
Thus, using the approximate value, we can write
c0 8.Sf> X l(FJi F m
This coefficient is called Ilie electric constant. It should
be noted that previously Ilie coefficients e0 and p0 regard
less of their numerical value (including p,{, in the cgse sys
tem and e0 in the cgsm system) were correspondingly called
the dielectric constant (permittivity) and magnetic permea
bility in a vacuum. These names are quite unsuccessful,
and at present their use has been discontinued. They may
sometimes be encountered in literature, however, especially
in comparatively old hooks.
7.3. Units of the CGS System

We shall begin a detailed analysis of the (mils of electri
cal and .magnetic quantities with the cgs (symmetrical or
(iaussian) system. This sequence is justified, firstly, by
historical considerations, since it was formed as a harmo-
170 UNITS OF PHY SICAL QUANTITIES ANJ) TJIEIR DIMENSIONS
nious system before the other ones, and secondly, by the fact
that ils construction is simpler and more consistent than
that of the SI system, which will be considered in the follo
wing section, together with the relationships between the
units of the two systems.
Elect?'ic charge (quantity of electricity). According to
Coulomb’s law, the unit of the quantity of electricity in
the cgs system (we shall omit the words “cgs system” in the
following definitions as self understood) is such a charge
that interacts with an equal charge at a distance of 1 cm
in a vacuum with a force of one dyne. From this definition
the dimension of the unit is as follows:
[(?] = Z/'!/2M 1/2r - 1 (7.30)
Surface density of charge. The surface density of a charge
is the quantity of electricity per unit of surface. According
to the formula
we shall have a surface density of charge equal to unity

with such a uniform distribution of the charge over the
surface of a conductor when there will be a unit of charge
per square centimetre.
The dimension of surface density is
7 3 /2M l / 2 r i _ i I ,
[g]= L M — =L (7.31a)
Jj
Intensity of electric field. The corresponding unit follows
from the definition of the field intensity
E ^-L (7.32)
where / is the force acting on the charge Q. The unit of the
intensity of an electric field (electric gradient*) is the inten
sity at such a point of the field in which aforce equal to
1 dyne actson a unitpositive charge.The sign ofthe charge
must be indicated because the field intensity is a vector, and
its direction has to be stated in its definition. The formula
* The name “electric gradient’1 is based on the similarity between
the field intensity and the potential (see below).
for the intensity also permits us to obtain its dimension,

namely,
_i 1
\E]r=L 2M 2T~i (7.32a)

Electric displacement. If interaction occurs not in a vacuum,
but in a certain medium, then the force of interaction will
decrease er times, where, as previously, er is the relative
permittivity of the medium. The product erE is called
the electric displacement or electric induction and is designa

ted D . Since er is dimensionless, the dimensions of D and E
coincide.
Electric flux. The electric flux dNj. through an element
of a surface dA is the product of the displacement, the area
of the element, and the cosine of the angle between the dire
ction of the displacement vector and a normal to the sur
face (Fig. 21):
dND= D dA cos (Z?, n) (7.33)
According to this definition, the unit of flux is the flux
through 1 cm2 of a surface perpendicular to the displace
ment vector at a displacement equal to unity.
Since according to the Gauss theorem the electric flux
through any closed surface surrounding a charge Q is
N D= 4nQ (7.34)
it is obvious that the unit of electric flux is equal to the
flux emitted from a charge equal to unity through a solid
angle equal to one steradian. It follows both from the Gauss
theorem and directly from the definition of the electric
172 UNITS OF PHYSICAL QUANT ITIES AND TH EIR DIMENSION S
flux that the dimension of the latter coincides with that ol*
charge:
|A'„| - [O] Lm M U2T~x (7.3:))
Potential. Potential is measured by the potential energy
possessed by a unit of charge placed in the given point of
a field. The unit of potential is the potential of such a point
of an electric held in which a unit positive charge possesses
potential energy equal to one erg
Ujb (7.30)
Q
The dimension formula of potential is
IU) LUZM\UT- l (7.37)
The unit of potential can also be used, naturally, for

measuring the difference of potentials, frequently called
voltage.
In this instance the unit of voltage can be defined as the
difference of potentials between two points, the transfer of
a unit charge between which is accompanied by the per
formance of work equal to one erg.
The electromotive force (e.in.f.) of a current source is
also measured in units of potential.
Dipole moment. /V dipole is a system consisting of two
egual charges opposite in sign, \ Q and —Q, placed a distance
I apart. The dipole moment is the product of the magnitude
of a charge and the distance between the charges. It is equal
to unity if this product is also unity. From the formula for
the dipole moment
p = Ql (7.38)
we get its dimension
\p\=--L^lzM m T~l (7.39)
The unit of the dipole moment can also be defined as the
moment of such a dipole that is acted upon by a mechanical
moment equal to unity in a homogeneous electric held with
an intensity equal to unity, when it is arranged perpendicular
to the held.
From the formula for the mechanical moment acting on

a dipole
/\
M —■pE sin (E , p) (7-40)
we get the same dimension as above.
Capacitance, The ratio of the charge of a conductor to
its potential determines the capacitance of the conductor
c = %- (7.41)
The dimension of capacitance is
IC} = =L (7.42)
L i / 2M 1/ 2T ~ 1
The unit of capacitance is the capacitance of a conductor
whose potential increases by a unit of potential when a unit
of charge is imparted to it.
Since the capacitance of a sphere in a vacuum is numeri
cally equal to its radius, then the capacitance of a sphere
with a radius of 1 cm can be taken as the unit of capacitance.
For this reason the cgs unit of capacitance is often called
“centimetre”.
Dielectric polarization. A dielectric in an electric field
is polarized, and each element of its volume is a dipole
having a definite dipole moment. By dielectric polarization
P is meant the dipole moment possessed by a unit of volume
—>
of a polarized dielectric. If the moment of the volume is p,
then
P -T (7-43)
Correspondingly
-> _i A .
[P\ = L (7.44)
Thus, the dimension of dielectricpolarization coincides
with that of fieldintensity and displacement.
The properties of a dielectric are characterized by two
related dimensionless quantities—the dielectric constant
(:relative permittivity) er and the dielectric susceptibility Xe.
The former was adequately discussed in the previous section.
174 I"NITS OF PHYSICAL QUANTITIES AND TH E IR DIMENSIONS
The latter is determined as the ratio between the dielectric

polarization and the field intensity
The relationship he tween er nnd Xe is given by the formula

8r= l + 4ji l e (7.46)
Current intensity. When charges move along a conductor,
we have to do with current intensity, similar to the flow
of a fluid or the heat flux and measured by the quantity of
electricity flowing through a cross section of the conductor
in a unit of time. In the general form the expression for
current intensity is
' = TT <7-«>
The unit of current intensity is that of a direct current aL
which a unit of charge flows through a cross section of a
conductor in a second. According to this definition, the
dimension of current intensity is
[7 ],.--LV2M il2T"2 (7.48)
Current density. The ratio of the current intensity to the
cross-sectional area of a conductor is called the current den
sity. The corresponding unit is a unit of current intensify
per square centimetre. Its dimension is
[/] = 4tt = (7.49)

Resistance. According to Ohm’s law, the current intensity
is proportional to the difference of potentials across the ends
of a conductor and inversely proportional to its resistance
1 = T< (7-50)
The unit of resistance is the resistance of a conductor
along which a current equal to a unit of current intensity
flows with a difference of potentials across the ends of this
conductor equal to a unit of potential. From the formula of
Ohm’s law we get its dimension

[It] = I r 1T (7.51)
Conductance. The reciprocal of resistance is called con
ductance:
(7.52)
This expression determines the unit of conductance and
iIs dimension
[CJ-Lr-1 (7.53)
Resistivity. The resistance of a homogeneous conductor
willi a constant cross section is expressed hy the formula
B (> - (7.54)
where I is the length and A the cross-sectional area of
the conductor. The factor p characterizing the properties
of the conductor is called resistivity. Its dimension is
[pi T (7.55)
The unit of resistivity is the resistivity of such a conducting
material, each centimetre of whose length with a cross sec
tion of 1 cm2 has a resistance equal to unity.
Conductivity. Similar to the definition of conductance,
the conductivity a is the reciprocal of resistivity. Accord
ing to the definition,
(7.50)
Magnetic induction. The main characteristic of a magnetic
field—the magnetic induction B -c a n be most clearly defined
according to the mechanical action resisted by an ele
ctric current in a magnetic field. Let us use for this purpose
formula (7.12) and assume in it that cp = 0° and A —- 1 cm2.
It should also be remembered that the factor of proportiona
lity C2 --- Me. In these conditions the unit of magnetic indu
ction can be defined as the induction of such a field in which
the maximum moment acting on a circuit with an area of
1 cm2 and through which there flows a current whose nume
rical value is equal to c (i.e., the velocity of light in a va
cuum, measured in cm/s) is 1 dyn-cm. This unit of induction
176 U N IT S o f p h y s ic a l q u a n t it ie s a n d t h e i r d i m e n s io n s
is called the gauss (Os). The gauss can also he defined as the
induction of such a field in which each centimetre of a
straight conductor arranged perpendicular to the field
and along which a current of c units flows is acted upon by
a force of one dyne. According to either of these definitions
the dimension of induction is
[5] = (7-57)
Magnetic, field intensity. The intensity of a magnetic
field canformally be determined fromformula (7.16) in
which the factor C8 is taken equal to unity, whence
*Hr (7<58)
With such a definition the unit of magnetic field intensity
is the intensity of a field in a vacuum with an induction
equal to one gauss. This unit is called an oersted (Oe). Since
the relative permeability is a dimensionless quantity, the
dimension of magnetic field intensity coincides with that of
induction.
Magnetic flux. If a magnetic field is depicted by force lines
whose density is proportional to the induction at the given
point of the field, then the total number of lines of force
penetrating the given surface can be characterized as the
magnetic flux. The latter is determined by the product of
the induction at the given point, an element of the area and
the cosine of the angle between the direction of the vector
of induction and a normal to the area
d® —BdA cos (B, n) (7.59)
The unit of magnetic flux—the maxwell (Mx)—is the flux
through an area of one square centimetre arranged perpendi
cular to a homogeneous magnetic field with an induction
of 1 Gs. The dimension of magnetic flux is
[<D]--.-L3/2M 1/27’~1 (7.60)
If a maguetic flux penetrates a circuit containing a certain
number of series-connected turns N, then sometimes use
has to be made of the concept flux linkage V, defined as the
product of the flux and the number of turns

¥ = N (7.61)
It is obvious that the dimension and unit of flux linkage are
the same as those of magnetic flux.
Magnetic moment. Formula (7.12) expressing the moment
acting on a circuit in a magnetic field includes the product
of the current intensity and the area of the circuit. This
product, that characterizes the given circuit and depends
neither on tiie external magnetic held nor on the orientation
of the circuit, together with a dimensional factor equal to
the velocity of light determines the so-called magnetic mo
ment of the circuit. According to this definition
Pm = 7 ^ 4 (7.62)
The magnetic moment is equal to the maximum mechanical
moment acting on the given circuit when if is placed in a
magnetic held with an induction of I Gs. The magnetic
moment is a vector quantity. The direction of this vector
is selected to coincide with a normal to the area of the
circuit if, when looking along this normal, we see the current
flowing clockwise through the circuit.
According to the definition of the magnetic moment, its
unit is the magnetic moment of a circuit acted upon by a
mechanical moment equal to 1 dyn*cm in a magnetic field
with an induction of 1 Gs. Introducing the angle between
the vector of induction and that of magnetic moment, we
can rewrite formula (7.12) as follows:
/\
M —Bpmsin (B, pm) (7.63)
The dimension of magnetic moment is
[pmJ = Z,5/2M ,/2r-! (7.64)
The concept of magnetic moment is applicable not only
to a circuit with a current, but also to a permanent magnet.
In the chapter devoted to the units of atomic physics we
shall also acquaint the reader with the magnetic moments
of elementary particles.
Magnetomotive force (circulation of magnetic field inten
sity). According to the law of the total current, the integral
12-1040
along a closed circuit of the scalar product II dl, where dl

is an element of the circuit, is proportional to the algebraic
sum of all the currents enclosed by the circuit:
(7.65)
The integral in the left-hand side is the circulation of the
magnetic field intensity, which is generally called the
magnetomotive force (m.m.f.) F. This name is connected with
the previously mentioned erroneous analogy between the
intensity of an electric field and that of a magnetic one. The
circulation along a closed circuit of the electric field inten
sity caused by the action of external forces of non-eleclric
origin is the electromotive force in the given circuit. It is
equal to the work done to move a unit of charge along the
circuit. The circulation of the intensity of a magnetic field
is not connected with any motion or with any work, so that
the name “magnetomotive force” is the same kind of anach
ronism as some other names that are still in use (horsepo
wer, etc.).
Formula (7.65) is true both for homogeneous and hetero
geneous media. With respect to the currents, such a direction
is selected as the positive one that forms an angle less than
90° with the positive direction of a normal to the selected
circuit.
The dimension of magnetomotive force follows from for
mula (7.65):
[F] = L m M in T-1 (7.66)
The unit of magnetomotive force—the gilbert (Gb)—is
defined as the magnetomotive force when passing once
around a conductor along which there flows a current of
c/4jt units. The concept of magnetomotive force is used in
calculating magnetic circuits. If we imagine a toroid (a short-
circuited solenoid) with a cross-sectional area A contai
ning N turns, then the magnetomotive force along the centre
line of the toroid will be —c
4n I N , where I is the current
flowing through the turns of the toroid. At the same time
the circulation of the magnetic field intensity is equal to
III, where I is the length of the centre line. Hence the field
intensity is H = ~ . Going over from the field inten

sity to the induction, we can determine the flux penetrat
ing the toroid
1 4ji IN F
C 1 I B.Jn
(7.67)
Hr A
where pr is the relative permeability of the medium fil
ling the toroid. The quantity in the denominator, namely,
_1_ _l_
Rm [xr A
(7.68)
is called the magnetic resistance (or reluctance), since formu
la (7.67) has the same appearance as Ohm’s law. The dimen
sion of reluctance is
[Rm] = L~' (7.69)
The unit of reluctance is that of a circuit in which the magne
tomotive force creates a flux of 1 maxwell. The reciprocal
of reluctance is called permeance.
Inductance and mutual inductance. Upon a change in
the magnetic flux linked with a given circuit, an e.m.f.
appears in the latter that is determined by Faraday’s law
(7.70)
If we have to do with a toroid or, what is the same, with

a solenoid whose length is quite great in comparison with
its diameter, then, using formula (7.67), we can write for
the flux linkage
_ 1 4ji/iV2
c 1 I
(7.71)
lb A
Assuming that the toroid is filled with a medium whose
relative permeability does not depend on the field intensity,
we can write the following instead of formula (7.70):
1 4ji/V2 dt
i — (7.72)
c2 J _ dt
a
12*
Formulas (7.71) and (7.72) relate to a particular case when

the flux whose change originates the e.m.f. of induction has
been created by the current in a toroid or long solenoid.
In the more general case of a circuit of any shape with any
number of arbitrarily arranged turns, it is possible, on the
basis of the law of Biot, Savart and Laplace, to express
the flux linkage with this circuit as
(7.73)
where the factor L depends on the configuration and dimen
sions of the conductors forming the circuit, and on the me
dium filling it. This factor is called the inductance of the
circuit (its previous name is the coefficient of self-induction).
Substitution for W in formula (7.70) of its value from for
mula (7.73) gives
(7.74)
In a more general form, if the inductance does not remain

constant, we should write
(7.75)
It follows from formula (7.71) that the inductance of a

toroid or a long solenoid is equal to
4 jx/V2
L = p,. I A (7.70)
Any of the formulas containing the inductance can be
used to determine its dimension and unit
[L\ = L (7.77)
The unit of inductance can be defined as the inductance
of a circuit linked with a flux of 1 maxwell when a current
equal to c units flows through it. According to another
definition, the unit of inductance is the inductance of a cir
cuit in which there appears an e.m.f. of induction equal to
unity when the current in the circuit changes by c2 units
per second. In accordance with its dimension, this unit of
inductance is sometimes called centimetre.
If wo have two circuits that are rather close to each other,

then with a current flowing through one of the circuits, part
of or the entire flux will he linked with the second circuit.
A change in the current in the first circuit will cause an e.m.f.
of induction to appear in the second one. The formula for
the flux linkage in one circuit depending on the current
in the other will have a form similar to formula (7.73):
^ 2 = = - M /1 (7.78)
where M, in contrast to L, is called the mutual inductance.
When a current flows through the second circuit, the flux
linkage in the first one will correspondingly be
¥ , = 4 a/ / 2 (7.78a)
the mutual inductance in both instances being the same.

It is clear from the above that the physical meaning of
inductance and mutual inductance is the same, and they
accordingly have the same dimensions and units.
Intensity of magnetization (magnetization). If a body is
placed in a magnetic field, each element of its volume
acquires a magnetic moment. If the body has ferromagnetic
properties, then the magnetization may remain after the
external source of the magnetic field has been removed.
The magnetic moment per unit of volume is called magne
tization:
J -pf - (7.79)
Its dimension is
_i \
[/] —L 2M 2T~l (7.80)
Themagnetic properties of a substance are characterized
by the relative permeability, which hasbeen defined pre
viously and according to its definition is a dimensionless
quantity.
The hysteresis properties of ferromagnetic materials are
described by the residual magnetic induction or remanence
and the coercive force 7/c, whose meaning will be clear
from Fig. 22. They are measured, naturally, in gausses

(Br) and oersteds (Hc).
Another characteristic of the magnetic properties of
a substance the magnetic susceptibility —is related to per
meability. It is determined as the ratio of the magnetiza
tion to the lield intensity
= 4 <7'81)
It is easy to see that %m is a dimensionless quantity. The

relative permeability and magnetic
susceptibility are related by the
expression
H.r = l + 4nxm (7.82)
7.4. Units of the SI System

It was previously (Secs. 1.5 and
7.2) shown that the construction
of a system of units which would
Flg. 22 include the practical units of cur
rent intensity, potential, charge,
work, power, etc., can be carried out in different ways,
by introducing an additional basic unit. When such a sys
tem, called the “absolute system of practical units”, was
developed, it was first intended to establish the unit of
permeability as the fourth basic unit. Thus a fourth ele
ment would have to appear in the dimension formulas—the
symbol of the independent dimension of the magnetic
constant p0. In this system (mks jx0), a development of
the mks one, the dimension formulas of all the units cor
respondingly included the dimension symbols of length
L, mass M , time T, and the constant (i0- The inclusion of
a fourth member in the dimension formulas was not novel,
since previously within the limits of the cgs system (cgse
and cgsm) a fourth member was also sometimes introduced —
e0 in the cgse system and f.i0 in the cgsm one. It is quite
obvious that in essence these systems, except lor the appea
rance of the dimension formulas, do not differ in any way
from the cgse and cgsm systems. It is clear that the dimen-
sion formulas in the mksjut0 system completely coincide

with those of the cgsm one.
Table 24, which gives the units of all the electrical and
magnetic quantities in common usage in different systems,
also gives the dimension formulas, the latter being pre
sented in three forms: cgse0> cgs (Gaussian) and cgsp().
The dimensions in the cgse and cgsm systems can be obtained
from the first and third ones if we omit the symbol e0 or
p0 respectively.
As we already know, the unit of current intensity, the
ampere, was selected as the fourth basic unit in establishing
the SI system. Correspondingly the fourth element in the
dimension formulas is the symbol of current intensity /.
For this reason the dimension formulas in the SI system
have a different appearance than those in the inks p0 one.
This is the only difference between the two systems, since
all the units in them are the same. With respect to the
conversion of the dimension formulas from one system to
the other, this can be done quite simply by substituting for
the unit of the given system its expression in the oilier one
in the corresponding dimension formulas. For purposes
of illustration formula (7.83) gives the dimension of the
unit of current intensity [which is a basic one (/) in the SI
system] in the inks system:
7 = [ /] = L1/2A/1/2r - V 01/2 (7.83)
If in all the dimension formulas of SI units given below

the expression contained in formula (7.83) is substituted
for /, then the dimension formulas in the mksp0 system
will be obtained.
The relationship between the ampere and the unit of
the cgs system can be established as follows. Two parallel
currents, each with an intensity of 1 A, act on each other
with a force of 2 X 10~7 N on a length equal to the distance
between the conductors. When written in the cgs system
this force of interaction will be
1 2 / , / 2Z
(7.84)
Assuming that /j = / 2 = / and l = r, and expressing the

force of interaction in dynes (10~5 N) with /-= 1 A, we gel
2 X 10~2 clyn - = — 2I2
Hence 1 A — 0.1c cgs units of current. Consequently we

approximately have t A = 3 X 10y cgs units (designated
cgs/)•
Electric charge (quantity of electricity). The unit of charge —
the coulomb (C)—is defined, according to formula (7.2),
as the quantity of electricity transported through the cross
section of a conductor by a direct current with an intensity
of one ampere. The relationship between the coulomb and
the cgs unit of charge is obviously the same as between the
relevant units of current intensity. The dimension of charge
in the Si system is
|(;]- TT (7.c33)
If should be noted that for measuring the capacity of storage
batteries the unit ampere-hour, equal to 3 (>0() °C, is used.
Potential (difference of potentials, voltage, electromotive
force). For determining this unit let us use the formula for
the power of a current
P-UI (7.8(5)
According to this formula the unit of potential difference —
the volt (V)~is defined as the difference of potentials across
the ends of a conductor in which a power of 1 watt is libe
rated when a current of 1 ampere flows through it. Its
dimension is
[f/l-M r^ 1 (7.87)
From formula (7.86). assuming that 1W 107 erg/s and
1 A —0.1c cgs/, we get
, v 107 1 , x
1 V ='o T 7 = W cgs u,llts (cgsU
It should be noted in passing that in electrical engineering
the name volt-ampere (VA) is frequently used instead of
watt for measuring the “apparent power”, i.e., the product
Uef I ef in the formula for the active power of an alternating
current P — Ucf I ef cos cp.
[ELECTRICAL AND MAGNETIC UNITS 185
Field intensity. The unit of held intensity can he deter

mined either from formula (7.32), or from the expression
for the held intensity of a point charge, or, finally, from
the relationship between the field intensity and the poten
tial
—grad U (7.88)
Any of these definitions gives the following dimension for
the unit of field intensity:
[E] —LMT~*I~l (7.89)
The unit of field intensity does not have a special name.
It can be called either newton per coulomb (N/C) or volt
per metre (V/m), the latter being common usage. The non-
system units volt per centimetre (V/crn), kilovolt per centi
metre (kV/cm), etc. are in great favour. Obviously
1 V/m = 407
10c ~ 3 x 104 c"-s unlts (cgs*)
Displacement (electric induction). The electric field vec
tor D is determined differently in the cgs and SI systems.
It was shown above that in the cgs system the relationship
between D and E is
E=—
and, consequently, the dimensions of both vectors coincide.
Matters are different in the SI system. Here E and D are
related by the expression
E=*— (7.90)
and, accordingly,
ID] = \E\ [e0| —L~2T I (7.91)
Different dimensions of two quantities within the limits
of a single system are ail indication that these quantities
have a different physical meaning. We would like to remind
the reader that in general, whereas different quantities
may sometimes have the same dimensions in one system
or in different systems, quantities of the same physical
nature can have different dimensions only in different
ISO UNITS OF PHYSICAL QUANTITIES AND T H E IR DIMENSIONS
systems. For this reason the physical definitions of the

vector D in the cgs and SI systems may differ, since the
dimensions of E and D coincide in the cgs, and differ in
the SI system.
Let us consider these definitions with the aid of the follow
ing example (Fig. 23). Suppose we have two identical plane
capacitors connected in parallel that have been charged
and disconnected from the source of voltage. The held
Fig. 23
intensities in both capacitors will naturally be the same,

as will also be the displacements. Let the space between the
plates of one of the capacitors be filled with a dielectric
having a relative permittivity er (Fig. 23b). It will be
convenient to have a liquid dielectric for our further dis
cussion. The difference of potentials between the plates
of the capacitors will decrease, but remain the same for
both | capacitors since they are interconnected. For this
reason the field intensity in both capacitors will also be
the same. The charges on the capacitor plates, however,
will now be different, and, accordingly, the values of the
displacement vectors D will also be different. Assume that
before connection of the capacitors the field intensities
in them were E 0, and the displacement D 0. After introduc-
lion of the dielectric the field intensity will become E .

The displacements in the capacitors will become Dx =
= srE and D 2 = E in the cgs system, and Dx = e0srE
and D 2 = s0E in the SI system.
If we now disconnect the capacitors from each other and
remove the dielectric from capacitor Cu then the field
intensity in it will grow er times (Fig. 23c), while the dis
placement will not change. If the new intensity in this
capacitor is E \ then we can write:
in the cgs system
E' = Di (7.92)
and in the SI system
E' —- e0Z>, (7.92a)
Before removal of the dielectric, the field intensity E
in capacitor Cx may be considered to consist of the inten
sities of two fields—that of the charge on the plates (obvi
ously equal to E') and that of the bound charges of the dielec
tric. After removal of the dielectric there remains only the
field of the free charges on the capacitor plates.
Let us consider both capacitors before their disconnection
as a single electrostatic system. We can now, within the
limits of the cgs system, define the displacement vector
as the field intensity of free charges (i.e., without taking
into consideration the bound charges of the dielectric)
with such an arrangement of these charges on conductors
that is due to the presence of a dielectric. Indeed, according
to expression (7.92), the displacement is the field of dis
placed charges whose redistribution between the capacitors
was caused by the introduction of a dielectric into the
capacitor.
To define the vector D in the SI system, let us introduce
so-called “Mie plates” inside the dielectric, i.e., two small
and very thin flat conductors first placed together. A charge
will be induced on these plates whose density will depend
on the value of D at the given point and on the orientation
of the plates. The density of the charge will obviously be
maximum when the plane of the Mie plates is perpendicular
to the direction of the lines of force of vector D . In our
example this direction is parallel to the capacitor plates,
and the density of the induced charge will be equal to that

of the charge on the capacitor plates, since when the ratio
nalized form of writing equations is used, the displacement
in a plane capacitor is equal to the density of the charge
on its plates
D ô (7.93)
The charge induced on the Mie plates can be measured
if we first move them slightly apart and then remove them
from the dielectric. In the general case of a non-uniform
field the density of this charge, of course, will not equal
the density of the charge on the conductors, but it will
equal D, regardless of the distribution of the field. Thus,
in the SI system, displacement can be defined as the maxi
mum density of a charge induced on Mie plates in a given
point of a field. The fact that the density of the induced
charge depends on the orientation of the Mie plates (this
is why the maximum density must be indicated) reflects
the vector nature of displacement D.
The different nature of the physical definitions of the
vector D leads to a number of inconveniences in setting
out the course of physics and related subjects. Many scien
tists, including Academician M. A. Leontovich and pro
fessors I. G. Klyatskin and L. B. Slepyan objected to this
division of concepts. Without going any deeper info their
serious arguments, we shall only indicate that the homoge
neous nature of the vectors E and D could be ensured in
the SI system if we retained the relationship between E
and D in the form D — erE, and introduced the coefficient
e0 in the expressions for calculating D according to a given
distribution of charges. For example, for a point charge we
should write D ——----instead
Asis0 r 2
of D — —--------
4 ji
%
r2
-.
The unit of displacement in the SI system and its rela
tionship with the cgs unit can be obtained if we use any
expression for D , for example, equation (7.93). According
to the latter, the unit of displacement is the displacement
in a plane capacitor with a charge density on its plates
of 1 coulomb per square metre (C/m2). In the cgs system
D = 12jt x 105 cgs units (cgs^)
In conversion we have already used here the SI unit of

surface charge density C/m2, which is equal to 3 X 105 cgs
units. The electric flux, defined as the product of the mag
nitude of the displacement vector, the area, and the cosine
of the angle between the direction of the vector D and a nor
mal to the surface (see Fig. 21), has a dimension coinciding
with that of charge:
[NjA ^ T I (7.94)
The dielectric constant (relative permittivity) is defined
in the same way as in the cgs syslern. Here, however, the
following remark should be made. In literature on electrical
engineering, besides the relative permittivity, use is made
of the absolute permittivity 8, determined by the expression
8--^ 808r (7.95)
The dimension of absolute permittivity coincides with
that of the electric constant e0 and its unit is also
designated F/m
[s0l = ZraM-1714/ 2 (7.95a)
—
V
Dipolemoment. The formula p — Ql determines its
dimension:
[p]^LTI (7.96)
and its unit, coulomb-metre (C-in)
lC - m - - 3 x lO n cgs units (cgs->)
v
Dielectric polarization is the dipole moment of a unit
of volume of a polarized dielectric
Its dimension is
\ ~P] =,- L~2T I (7.97)
and its unit is coulomb per square metre (C/m2)
1 C/m2= 3 X 105 cgs units (cgs-»)
Dielectric susceptibility is determined by the ratio

p
We shall obtain the relationship between %e and
er from the following expressions:
D = e0E + P = e0E + leE = e0erE (7.98)
and, consequently,
e0 (er —l) = xe (7.99)
or
er —1 + — (7.100)
e0
Since the ratio %e!ô has no dimension, then the dimension
of %e coincides with that of the electric constant, and its
unit is also designated F/m (farad per metre).
Upon comparing expression (7.100) with the similar
expression in the cgs system
—1 + dJt Xe
we shall find that the unit of dielectric susceptibility
1 F/m = 9 X 109 cgs units (cgsXc)
Capacitance. The unit of capacitance—the farad (F) —
is the capacitance of a conductor whose potential increases
by one volt when a charge of 1 coulomb is imparted to it.
Since C = —, its dimension is
[C] = L r (7.101)
The relationship between the units is
o 4r2
1 F = ■jQ8 ■= 9 x 1011 cgs units (cgsc)
In practical work fractional units are generally employed,
namely, the microfarad (pF) and the picofarad (pF).
Resistance. The unit of resistance—the ohm (f2)—is the
resistance of a conductor in which a current of one ampere
flows with a difference of potentials of one volt across its
ends. OhmV law determines its dimension:
[R\=L*MT-*I-* (7.102)
ELECTRICAL AND MAGNETIC UNITS l9j
It should be noted that the product RC has the dimension

of time in both systems. In a circuit including a capacitor
and a resistor, the product RC characterizes the time con
stant of charge attenuation. It is easy to see that
1
I Q ^ 9 x ion CSS units (cSsn)
Conductance. The unit of conductance is obviously the
conductance of a conductor whose resistance is one ohm.
This unit is called the siemens (S). In literature the name
mho (reciprocal ohm) and the designation Q"1 are some
times encountered, although they are not recommended by
the relevant standards. The dimension of conductance is
the reciprocal of that of resistance
{G]--=L-*M-lrT*P (7.103)
Resistivity p is measured by the unit Q-m
9x-^ -9 cgs units (cgsp)
Its dimension is
[ p j - M r 3/ “2 (7.104)
In practical work the resistivity is frequently measured
in the units Q -mm2/m and Q -cm, which are obviously equal
to 1 X 10~6 and 1 X 10~2 Q-m, respectively.
The reciprocal of resistivity—conductivity—is measured
by a unit that can be called siemens per metre (S/m).
Magnetic induction. The unit of magnetic induction—
the tesla (T)—is the induction of such a field in which each
metre of conductor with a current of one ampere and arrang
ed perpendicular to the direction of the vector of induction
is acted upon by a force of one newton. From this definition
we get the dimension of induction
[Z?] = MT~2I~X (7.105)
Substitution of the above units in the expression for induc
tion connected with this definition, but written in the cgs
system, gives
I) _ fc 10* d y n X 3 x lO™ c m / s .
II ~ 100 c m X 3 x 109 c g s j U
Thus
IT —104Gs
Magnetic flux. The unit of magnetic flux —(.he weber
(Wb)—is deli nod as the flux with an induction of 1 T through
an area of 1 m2 arranged perpendicular to the vector of
induction. Hence its dimension is
[<D] = L2MT~-I~l (7.106)
and the relationship between the units is
lWb —IT x lm 2-- 104Gs x 104 cm2 —108 Mx
Magnetic field intensity. To determine the unit of mag
netic field intensity it will be convenient to make use of
any of the corollaries of the Biot, Savart and Laplace law
that give an expression for the intensity of the magnetic
field of a current for specific circuits. Let us take for this
purpose the formula for the intensity of the magnetic field
at the centre of a circular current
« = w <7' “ >7>
According to this formula the field intensity will be equal
to unity if a current of 2 A flows around a ring with a radius
of one metre, or, which is the same, a current of one ampere
flows around a ring with a radius of 0.5 m. This unit has
no special name. In accordance with its dimension in the
SI system it is called ampere per metre (A/m). It was pro-
jiosed to call this unit the lenz (in honour of E. Lenz).
The dimension of magnetic field intensity is
[H ]^L~U (7.108)
To establish the relationship between the units ampere
per metre and oersted let us rewrite equation (7.107) in the
cgs system:
(7-109>
and insert the corresponding values, converting them to
cgs units
1 A/m = I f x Io T x ^ T ' 4jt x 10~30e = 1-26 x 10"20e

ELECTRICAL AND MAGNETIC. UNITS 193
In measuring the magnetic field of the earth, celestial

bodies and outer space, a unit of magnetic field intensity
called the gamma (y) is used. It is equal to 10-5 Oe. Hence
1 A/m = 1.26 x 103 y
It is worthwhile noting here that whereas in the cgs
system the dimensions of the vectors B and II coincide,
in the SI system they differ. A similar event occurred in
electrostatics when we considered the vectors E and D.
The objections that were raised against the lack of homo
geneity of the vectors E and D observed in the SI system
relate to an equal degree to the vectors B and H . This
discrepancy could be eliminated quite easily if the magnetic
constant p 0 were introduced into the equation for the mag
netic field intensity. If this were done the Biot, Savart
and Laplace law could be written as
I dl s i n cp
7*2 '
while the relationship between B and II would be the

same as in the cgs system, i.e.,
B - \KrH
Magnetic moment. The unit of magnetic moment can be
determined in two different ways, using either the expres
sion for the mechanical moment acting on a circuit with
current in a magnetic field, or the direct expression for
the magnetic moment of a circuit. According to the first
definition the unit of magnetic moment is the moment of
a circuit that is subjected to a maximum torque of one
newton-metre in a field with an induction of one tesla, and
according to the second—the moment of a plane circuit
with an area of one square metre through which a current
of one ampere flows. Both definitions give the same dimen
sion formula
[pm\= L H (7.110)
The unit of magnetic moment has no special name and is
designated A»m2 (ampere-square metre). If we substitute
for it an equivalent designation N >m/T, it is easy to obtain
13-1040
194 UNITS OF PHYSICAL QUANTITIES ANI) TH E IR DIMENSIONS
the relationship between the SI and cgs units

N-m 105dyn x 102 c r a _ /,A3dyn«cm
lA -m 2= 1 T WG~s “ 1U Gs
Magnetomotive force. The circulation of the magnetic
field intensity in the SI system is written as
F= § H d l cos ( i O l ) = 2 / (7.111)
The unit of magnetomotive force is the circulation of the

magnetic field intensity when a current of one ampere flows
once around a circuit. The dimension of magnetomotive
force coincides with that of current intensity and its unit
is also called the ampere. Since when calculating magnetic
circuits the total magnetomotive force is equal to the cur
rent intensity in each conductor multiplied by the number
of turns, the magnetomotive force is frequently expressed
in ampere-turns (At)
1A = 1At = 3— 5 x 3 X 10» = 1.26 Gb
Reluctance (magnetic resistance). The unit of reluctance

is defined from the law of a magnetic circuit [formula (7.67)1
as the reluctance of a magnetic circuit in which a magne
tomotive force of 1 A creates a flux of 1 Wb. Formula (7.68)
determines its dimension
[R ^^L ^M ^P (7.112)
The relationship between the units is
1A/Wb = l ? s r = 1-26 x 10-8 Gb/Mx
Inductance and mutual inductance. To determine the unit
and dimension use can be made either of the expression
for the relationship between the current in a circuit and
the flux linking with it
(7.113)
or of the expression for the e.m.f. of inductance
(7.114)
Equation (7.114) is written on the assumption that the

inductance is constant. According to equation (7.113) the
unit of inductance—the henry (H)—is defined as the induc
tance of a circuit that will be linked with a flux of 1 Wb
when a current of 1 A flows through it. According to expres
sion (7.114) the henry is the inductance of a circuit in which
an inductance e.m.f. of one volt is induced when the cur
rent flowing through it uniformly changes by one ampere
a second. Both definitions give the dimension
[L] = L2MT“2/" 2 (7.115)
Comparison with formulas (7.73) and (7.74) written in the
cgs system gives the relationship
1H = 109 cgs units (cgSL —centimetres of inductance)
The same units are used to measure mutual inductance.
Intensity of magnetization (magnetization). According to
formula (7.79), the unit will be such a magnetization when
each cubic metre has a magnetic moment of one A *m2.
The name of this unit is accordingly ampere per metre (A/m)
and coincides with that of the unit of field intensity. In the
same way its dimension is
[/] = Zr1/ (7.116)
Upon rewriting formula (7.79) in the form
J= T i r (7 J 1 7 >
we can easily compare the ampere per metre with the cgs
unit
. A. 3 X 109 x 104 .n_o .. . .
1 A / m = 3 X 1010 X 10* = 1 0 CgS UDltS (CgS' )
It will be helpful to draw attention here to the following
circumstance. Notwithstanding the fact that the units
of magnetic field intensity and magnetization coincide in
dimension and even in name, the relationship between
these units and the corresponding cgs units is different.
This is explained by the fact that in one case (field inten
sity) the equations are different in the rationalized and
unrationalized forms, and in the other they are the same.
13*
The given example once more illustrates the circumstance

previously noted that the complex name of a derived unit
can say nothing of its actual dimension if no indication is
given of the specific defining relationship used to establish
the given unit.
The magnetic properties of a substance—relative per
meability, remanence (residual magnetic induction), and
coercive force—require no special explanation. It should
only be noted that in publications on electrical engineering,
besides the relative permeability pT and the magnetic
constant p0i there is also used their product
HoHr = H (7.118)
called the absolute permeability.
Magnetic susceptibility. Equation (7.81) also defines mag
netic susceptibility in the SI system. Since J and II have
the same dimension, then %m, as in the cgs system, is a
dimensionless quantity. The rationalized form of the equ
ations, however, leads to the following relationship between
pr and xm
r = H Xm (7.119)
As a result the SI unit of magnetic susceptibility is
1/4n of the cgs unit.
7.5. On the So-called Wave Resistance of a Vacuum

In the previous section, using the example of the units
of magnetic field intensity and of magnetization, whose
dimensions and designations (A/m) coincide, an illustration
was given of what was previously said about the absence
of a single-valued relationship between the dimension for
mula of a unit and its concrete magnitude. This can be
especially clearly illustrated by considerating the units
and numerical values of a combined constant called the
wave or characteristic resistance of a vacuum.
Upon the propagation of an electromagnetic wave in
a medium with a relative permittivity and permeability
of er and pr, the amplitude and instantaneous values of the
electric and magnetic field intensities obey the relationship
Vti&T E = H (7*120)
This expression can be written as
The ratio EIH is generally called the wave resistance of a

medium, since there is a formal analogy between equation
(7.121) and Ohm’s law. For a vacuum
<7-,22>
It is this quantity that is generally known as the wave
or characteristic resistance of a vacuum. Let us consider
the value of R x in different systems of units. In the cgs
system, where e0 = jn0 = 1 and have no dimensions,
Rx = 1 and is also a dimensionless quantity. Ic should be
remembered that in this system the dimension of resistance
is L~i T. In the cgse system e0 = 1 and p0 = 1/c2. In this
system Rx = He and its dimension coincides with that of
resistance. In the cgsm system j l i 0 = 1 and e0 = 1/c2. Cor
respondingly Rx = c. The dimension of Rx coincides with
that of resistance in this sysLem too.
Let us consider, finally, the value of Rx in the SI system.
Substitution of the values
\in = x 10~7H/m
and
6° = 4 n x 10-7 x 9 x 1(H«
in equation (7.120) gives
/fx = 120jt(H/F) 2 (7.123)

Replacing H/m by V*s/A-m and F/m by A-s/V-m, we can
write
Rx = 120jx V/A (7.123a)
Since the ratio volt/ampere defines the unit of resistance,
the ohm, then it is assumed that the “wave resistance of
a vacuum” is 120jc = 377 Q.
If, however, we use the inks (Li0 system in its unrationa-

lizecl form, in which the basic units are the same as in the
SI system, and the unit of resistance, the ohm, is defined
in the same way as the volt-ampere (since regardless of
the form of writing the equations Ohm’s law has the same
appearance), then, taking into account that in this instance
I
u0 = 10'7 H/m and e0 -- ^—T
r
7-QF/m we find that
9 x 109
Rx — 30 V/ A (7.123b)
i.e., 30 Q.
The contradiction between the values of R x determined
in different ways is explained by the fact that the name of
a complex unit is not at all a definition of this unit. In par
ticular, in the example considered above the name V/A
obtained as a result of the corresponding transformation of
units or as the ratio of the units of electric and magnetic
field intensities (V/m and A/m) cannot be interpreted as
the unit of resistance. For this reason the concept “wave
resistance of a vacuum” itself appears to be deprived of
physical meaning, although it may sometimes be conve
nient to use one symbol for the expression Y mV£o in cal
culations. It should be noted here that if we relate the
vectors E and B* instead of the vectors E and H, then
instead of formula (7.120) we shall have
B
Y CqCr E (7.124)
V PoPr
or
E
- 7L = - 7L ^ b (7*125)
V eoPo Verpr
This expression, as can be easily seen, does not change
when the units are changed.
7.6. International Units

As we have previously mentioned, the practical units that
served as the basis of the SI system did not first form a single
system, but made up an isolated group of units connected
* It will be useful to note that the relationship between the vec
tors E and B or, correspondingly, D and H is more logical from the
viewpoint of Maxwell’s equations.
ELECTRICAL AND MAGNETIC UNITS iy9
to one another by several relationships. The introduction

of these units played an important part in the development
of the techniques of electrical and magnetic measurements,
as a result of which soon after its appearance the practical
system acquired international recognition. Much work was
done to establish standards of the practical units of resi
stance, current intensity and potential difference; origi
nally these standards or prototypes were intended to serve
for reproduction of the ohm, ampere and volt, defined as
109, 0.1 and 108 of the respective units of the cgsm system.
It was later found, as could have been foreseen, by the
way, that there are insignificant, but nevertheless noticeable
discrepancies between the established standards and their
prototypes based on the absolute system. It was then deci
ded, as was done previously with the metre and kilogram,
to adopt the prototypes as legal international units of
electrical quantities. These international units were defined
as follows:
the international ohm—the resistance of a column of mer
cury 106.300 cm long with a mass of 14.4521 g and with
an identical cross section along its entire length, measured
at the melting point of ice and with an unvarying current;
the international ampere—the intensity of an unvarying
current that deposits 0.00111800 grams of silver by electro
lysis from a silver nitrate solution in one second;
the international volt—the electrical potential or elec
tromotive force that induces a current of one international
ampere in a conductor with a resistance of one international
ohm;
the international watt—the power of an unvarying current
of one international ampere at a difference of potentials
of one volt.
The remaining international units, the same as the inter
national volt and watt, are determined from the correspond
ing basic international units.
The considerably improved accuracy of electrical and
magnetic measurements made it possible to perform the
reverse transition and, having established in a definite way
the exact formulation of one of the units (as we already
know, the ampere), construct the mksp,0 system of units
that formed the basis of the SI system now adopted in many
20Q UNITS OE PHYSICAL QUANTITIES AND THE IK DIM ENSIONS
countries. To distinguish them from the international units

given above, the units of the mks|r0 system were called
“absolute1 to underline the fact that they were constructed
according to the same principle as those of the cgs system.
To allow the results of measurements made in international
units, during the time these units were still in use, to be
converted to mksp,0 units, the following relationships were
established between them:
1 mean international ampere = 0.99985 abs. ampere
1 mean international ohm = 1.00049 abs. ohm
1 mean international coulomb = 0.99985 abs. coulomb
1 mean international volt = 1.00034 abs. volt
1 mean international henry = 1.00049 abs. henry
1 mean international farad —0.99951abs. farad
1 mean international weber = 1.00034 abs. weber
1 mean international watt = 1.00019 abs. watt
The adjective “mean11 is due to the fact that when estab
lishing the relationship between international and absolute
units, it was found that there is a slight discrepancy be
tween the prototypes of the international units kept in
different countries, so that the mean value of these pro
totypes has been taken for purposes of comparison. For
example, the following relationship existed between the
international units adopted in the USSR and the mean
international units:
1 USSR int. ohm ~ 1.000010 mean int. ohm
1 USSR int. volt = 1.0000075 mean int. volt
At present the international units have been completely
discarded and replaced by “absolute11 units, i.e., units of
the SI system.
The definition of a basic unit of this system—the am pere-
through mechanical units with the establishment of an
exact value of the coefficient r 0 in the defining relationship
has made it possible to include the practical electrical
and magnetic units into the general system of units of phy
sical quantities.
CHAPTER EIGHT
UNITS OF RADIATION
8.1. Scale of Electromagnetic Waves

The field of investigated electromagnetic waves extends
almost without interruptions from waves with a length
of thousands of kilometres radiated by low-frequency elec
trical machines to the short-wave y-radiation of radioactive
elements and cosmic rays. Different ranges of this spectrum
have different properties, propagate differently and mani
fest themselves in different ways. The narrow band of wave
lengths from 0.38 to 0.76 micron is perceptible by our eyes;
within certain ranges radiation is capable of calling forth
chemical reactions, the photoeffect, and ionization of gases.
Radiation with the greatest wavelengths can be detected
with the aid of electromagnetic oscillating circuits. For
this reason together with general characteristics of radiation,
first of all from the viewpoint of its energy, there are spe
cific characteristics for separate ranges of the spectrum of
electromagnetic weaves.
Wavelengths and the frequencies corresponding to them
are measured in the usual units of length and frequency.
It is quite natural that in the range of long waves the units
of length are the metre and centimetre, while light and
shorter waves are measured in microns, angstroms and
X-units. Frequencies are generally measured in herz, while
kiloherz and megaherz are used for radio waves. In addi
tion to wavelengths and frequencies, in spectrometry use
is often made of the wave number a, which shows the number
of waves per unit of length. Obviously
1
( 8 . 1)
202 UN ITS OF PHYSICAL QUANTITIES AND TH E IR DIMENSION S
where X is the wavelength. The wave number is frequently

determined as
( 8 . 2)
The units of wave number are the reciprocal metre, re

ciprocal centimetre, reciprocal micron, etc.
8.2. Characteristics of Radiant Energy
Quantities characterizing the energy aspect of the radi

ation of electromagnetic waves are measured by the general
units of energy used to measure energy, volume density
of energy, energy flux, etc. The names of some of these
quantities reflect the fact that they are the result of extended
usage of the concepts employed in illumination engineering,
although they may relate to such ranges of the spectrum
that are not perceived by our eye.
Radiant flux (radiant energy flux) <Dr is the amount of
radiant energy passing in a given direction in a unit of
time. Both with respect to its physical meaning and to its
units and dimension, radiant flux is absolutely identical
to energy flux considered in the chapter on acoustic units.
We shall remind the reader that the units and dimension
of energy flux coincide with those of power. It should be
noted, however, that together with the units W and erg/s,
the heat units cal/s, kcal/s, etc. are also used to measure
radiant flux.
Radiant flux density (irradiance) d$>!dA is the flux per
unit of surface area. Here several concepts have to be dis
tinguished, although their units and dimension coincide.
Energy flow 8 is the radiant flux in a given direction per
unit of surface area perpendicular to the direction of radi
ation. The energy flow of electromagnetic waves is a vector
(Poynting vector):
S = ( E x H ) (SI) (8.3)
or
S = £ ( E x H ) (cgs) (8.3a)
UNITS OF RADIATION 203
The source of radiation is characterized by the radiant

emittance R r, i.e., the total radiant llux emitted from a unit
of surface area of the source.
Radiant illumination Er measures the density of the
radiant flux incident on a given surface. It is easy to see
that with the same intensity of radiation the radiant illu
mination may be different depending on the orientation of
the surface onto which the radiation falls. With a given
intensity S the radiant illumination will be proportional
to the sine of the angle between the direction of the flux
and that of a normal to the surface onto which the flux
is impinging.
The dimensions of all three quantities S, R r and Er
coincide, namely,
lS]--=lRr] = [Er] = MT-* (8.4)
The units in the SI and cgs systems are correspondingly
W/m2 and erg/(s-cm2). In addition to the system units, the
heat units cal/(s-cin2), kcal/(h-m2), etc., are used, as in
measuring radiant flux.
The total quantity of radiant energy impinging during
a certain time onto a unit of surface area is measured by the
radiant quantity of illumination / / r, determined by the
expression
t
HT= ^ E Tdt (8.5)
0
Its dimension is
[Hr] = M T-2 (8.6)
Besides radiant emittance, a source of radiation is cha
racterized by the radiant intensity and the radiance. The
radiant intensity I r is defined as the radiant flux of a source
per unit of solid angle in the given direction.For the same
source, the radiant intensity may differ in different direc
tions. The dimension of radiant intensity coincides with
that of radiant flux, i.e., with that of power, since in the
SI and cgs systems solid angle is a dimensionless quantity.
The name of the units of radiant intensity contains the unit
of solid angle steradian. The corresponding units are W/sr
and erg/sr.
ZU4 UNITS OF PHYSICAL QUANTITIES AND THE1K DIMENSIONS
Radiance Lr is the radiant intensity per unit of area

of the projection of the surface of the source onto a direc
tion perpendicular to the direction of propagation of the
radiation. According to this definition
It (8.7)
Here
dAr = dA cos a
where dA is the area of an element of the surface, and a
is the angle between the direction of radiation and that
of a normal to the surface.
If the radiation of a source of light complies with the
Lambert law, according to which
/ r = / r0 cos a (8.8)
where / r0 is the radiant intensity in a direction perpendicu
lar to the surface of the source, then the radiance of the
source is the same in all directions. Such sources are called
Lambert ones.
The dimension of radiance is the same as that of radiant
flux density:
[Ir] = M T-3 (8.9)
Its units are W/(m2-sr) and erg/(s-cm2-sr).
Radiant energy density u. The radiant energy per unit
of volume is called the radiant energy density. The radiant
energy density (see Sec. 4.4) is measured in J/m3, erg/cm3,
etc.
The radiant energy density is of special interest if the
radiation is concentrated in a closed space. Here the radi
ation obeys the laws of radiation of a black body, in par
ticular the Stcfan-Boltzmann law, according to which the
volume density of radiation is proportional to the fourth
power of the absolute temperature. If a small (in comparison
with the total surface) hole is made in the shell containing
the radiation, then this hole will be a black emitter whose
radiant emittance is related to the radiant energy density
by the expression
r> =•=t1 UC
Rr ( 8 . 10)
4
UNITS UT RADIATION 205
where c is the velocity of light in a vacuum. According

to the Stefan-Boltzmann law
Rr = oT4 ( 8 . 11)
where a is a constant of this law,

a - 5.669 x 10'8 W/(ms-°K) - 5.669 x 10"5erg/(s*cm2.°K)
Together with the radiant energy characteristics listed
above that have an integral nature, that is, that do not
relate to a definite range of the radiation spectrum, of
great significance are spectral characteristics that are in
essence functions of distribution of the given quantity by
wavelength or by frequency.
Since the radiation of a source is not perfectly monochro
matic, but is distributed in some manner along the spec
trum, the action of the radiation may be quite diverse.
In some instances we use the features of distribution of the
given source, in others we convert the radiation of one
spectral composition into that of another (for example, the
conversion of ultraviolet radiation into visible light in
luminescent lamps), and, finally, we sometimes have to
ensure protection from a definite part of radiation, etc.
Since the concept of the function of distribution was con
sidered in sufficient detail in Sec. 4.5, here we shall only
give mathematical expressions for the corresponding spec
tral characteristics, the dimension formulas and units.
Spectral radiant flux density along the wavelength is deter
mined as
<8-*2>
Its dimension is
l®rX] = LMT-* (8.13)
Its units in the SI and cgs systems are respectively W/m
and erg/(s-cm). In spectroscopy the flux is generally related
to the interval of wavelengths measured in the units em
ployed for the given region of the spectrum. For example, the
units W/A or erg/s -A are used for the visible and adjacent
regions of the spectrum.
Spectral radiant flux density along the frequency is ex

pressed as
G>rv = 5 (8.14)
Its dimension is
[<Drv] = L*MT~2 (8.15)
In the following we shall not specially indicate whether
the distribution is given by wavelength or frequency for
spectral distributions, since this is obvious from the mathe
matical definition.
The spectral density of quantities determined by the
radiant flux density (spectral density of energy flow, radiant
emittance, radiant illumination) is
equal to
dEr
dX ; Erkz dX
(8.16)
o __ d S m d R r ^ jp dEr
n , , v "5v“ »
(8.17)
[Sk] = [Rrk] = [Erk] = (8.18)
[Sv] = {Rrv] = [Erv] = MT~* (8.19)
- The units of S , R r), and ET%are W/m3
Flg' 4 or erg/(s-cm3), and those of Sv, Rrv
and Erv are J/m2 or erg/cm2.
The spectral density of the radiant emittance of a black
body, M rx, is shown in Fig. 24. The dimension of the spec
tral distribution of the radiant intensity coincides with
that of the spectral distribution of the radiant flux. With
respect to the relevant units, then in contrast to the units
of a>rx and Orv, their names show that they are related
to a unit of solid angle, which is also reflected in the deno
minator of the symbols of these units.
In the same way the dimension of the spectral distribu
tion of radiance coincides with that of the radiant flux
density (i.e., energy flow, radiant emittance and radiant
illumination), while the units are obtained from the cor
responding ones by relating them to a unit of solid angle.
The dimension of the spectral distributions of the radiant

energy density is
[ud = Ir*MT-* (8 . 20)
[Mv| = t w r l (8.21)
It is general knowledge that for a black body ux and
Uy are determined by Planck’s formula:
S n he 1
( 8 . 22)
ew r _ i
Snhv3 1
Uy
c3 hv (8.23)
ehT- l
where k = Planck’s constant
k — Boltzmann’s constant
T = absolute temperature
e = base of natural logarithms.
The spectral distribution of the radiant emittance of
a black body can be obtained from formulas (8.22) and
(8.23) by multiplying them by c/4.
8.3. Illumination Engineering Units

Measurements of light have the feature that direct per
ception plays a very great part in them. Thus such mea
surements, strictly speaking, are not quite objective. Since
in measurements of light we are interested only in the part
of the total radiant flux that directly acts on our eye, the
customary energy characteristics are not sufficient. Indeed,
among the tremendous region of investigated electromag
netic oscillations only a narrow band of the visible spectrum
with wavelengths ranging approximately from 0.38 to
0.76 micron are “optically valuable” for us or, as is said,
have adequate visibility.
Various technical means are available that allow us to
detect and measure radiation of electromagnetic waves of
any range from long ones used in radio engineering to the
shortest ones registered by counters of penetrating radi
ations, but, whatever the power of a radiation, we are
“blind” with respect to it if its wavelengths are beyond the
limits of the interval indicated above. Moreover, even

within this interval the sensitivity of our eye is different
and, consequently, different regions of the visible spectrum
have different visibility.
Practical illumination engineering poses many questions:
what spectral composition of light should be considered as
the most “natural” one, how can sources with a different
spectral composition be compared, and many others. It is
obviously essential to come to an agreement on some single
methods to be used for comparing and measuring quantities
that should characterize sources of light and the conditions
of illumination.
It would seem most expedient to turn to natural sun
light, taking it as the prototype for purposes of comparison.
It is easy to see that such a concept as natural sunlight is,
however, very ambiguous. The time of the year, time of the
day, geographical latitude, weather, altitude above sea
level, purity of the atmosphere—all these factors change the
quantitative and qualitative composition of sunlight in
very broad limits. For this reason we have to come to an
agreement on the selection of an artificial source of light
that could be taken as an international prototype. Many
such prototypes were proposed at various times (the Hefner
candle, carcel, Viole standard, etc.), which at present have
only a historical significance. The main drawback of these
prototypes was the difficulty of reproducing them. It was
obviously desirable to select such a source whose light
radiation would be determined by physical laws as general
as possible.
Since a black body is a universal radiator, its radiation
was taken as the prototype. The temperature of the radiator
must be fixed as accurately as possible, because radiation
sharply grows with temperature. The temperature of soli
dification of platinum (2 042°K) was taken as such a tem
perature. The basic illumination engineering unit included
among the basic units of the SI system is the unit of lumi
nous intensity, the candela (cd), which is l/60th of the
radiation emitted by one square centimetre of a surface
of a perfect radiator (black body) at the temperature of
solidification of platinum. The international candle pre
viously in use is equal to 1,005 cd.
The basic unit candela serves to determine all the other

illumination engineering units. Since luminous intensity
is included among the basic quantities, its symbol J
appears in the dimension formulas.
Luminous flux is determined by the product of the lumi
nousintensity and the solid angle formed by the flux.
In the general case ofnon-uniform radiation
<D =j/(K 2 (8.24)
£2
With uniform radiation within the limits of the angle
<D= IQ (8.24a)
With uniform radiation in all directions
0 = 4jt7 (8.24b)
Before the introduction of the SI system the basic quan
tity in illumination engineering was the luminous flux,
defined as the power of luminous radiation evaluated in
terms of its visual effect. This definition underlines the
subjective, physiological nature of illumination engineering
quantities.
The unit of luminous flux is the lumen (lrn)—-the flux
within a solid angle of one steradian with a luminous inten
sity of one candela. This unit is a basic one in the syslem
of units of light based on the centimetre, gram and second,
owing to which this system is generally designated cgsl.
The dimension of luminous flux coincides with that of lu
minous intensity:
[®] = J (8.25)
Quantity of light is the product of the luminous flux
and the duration of its action
Q = j 0> dt (8.26)
t
Its dimension is
IQ] = TJ (8.27)
Its unit is the lumen-second (lm-s).
14-1040
210 UNITS OF PHY SICAL QUANTITIES AND T H E IR DIM ENSIONS
Similarly to the radiant energy quantities measured by the

radiant flux density, the corresponding illumination engi
neering quantities and their units can be defined. Since these
definitions are absolutely similar to those of their radiant
energy counterparts, we shall limit ourselves to the for
mulas of the defining relationships and to the definitions
of the units.
Luminous emittance
do
dA
(8.28)
The unit of luminous emittance is the luminous emittance
of a source, each square metre of which produces a luminous
flux of one lumen. This unit is called lumen per square
metre (lm/m2). It was previously called the radlux. The
unit in the cgsl system, the lumen per square centimetre
(lm/cm2), is obviously 104 times greater than a lm/m2.
This unit was previously called the radphot.
Luminous flux intensity, the same as luminous emittance,
is measured in lm/m2 and lm/cm2.
Illumination
dO
dA
(8.29)
The unit of illumination, the lux (lx), is the illumination
of a surface each square metre of which receives a luminous
flux of one lumen. In the cgsl system the relevant unit is
the phot—the illumination of a surface, a square centimetre
of which receives a flux of one lumen. Hence 1 lx = 10“4 ph.
Using the expression for the illumination of a surface by
a source of light with an intensity of I candelas at a dis
tance of r metres from the illuminated surface
£ = — cosa (8.30)
where a is the angle between the direction of the luminous
flux and a normal to the illuminated surface, we can define
the lux as the illumination of a surface at a distance of
1 metre from a source with a luminous intensity of 1 candela
and arranged perpendicular to the incident light.
Luminance. The unit of luminance, the nit (nt), is the
luminance of a source each square metre of whose radiating
UNITS OP RADIATION 211
surface has in the given direction a luminous intensity

equal to one candela. The cgsl unit, the stilb (sb), equal
to one candela per square centimetre, is 104 times greater
than the nit.
Sometimes special units are employed to measure non
emitting surfaces. If a surface diffuses light perfectly in
all directions, without absorbing it at all, such a surface
has the properties of a Lambert source whose luminance,
the same in all directions, is equal to
L=~ E <8‘31)
The luminance of a perfectly white surface whose illumina
tion is equal to one phot is called a lambert (L). It follows
from formula (8.31) that 1 L = ~ sb=0.318 sb. The
luminance of the same surface with an illumination of one
lux is sometimes called an apostilb (ash). Accordingly,
1 asb = 10-4 L = 3.18 X 10-5 sb = 0.318 nt.
The units of all the quantities listed above (emittance,
intensity, illumination, luminance) coincide in each of the
systems, the only feature being that the dimension for
mulas in the SI system include the symbol of the dimension
of luminous intensity (J), and in the cgsl system—that of
luminous flux (O).These dimensions are, respectively,
[R] = [E]= [L[ = L -zj = Zr2O (8.32)
In addition to the quantities listed above, use is also
made of candela-second — the product of luminous intensity
and the duration of illumination
C= lt (8.33)
and the quantity of illumination — the product of illumina

tion and its duration
H Ê t (8.34)
Candela-second, as its name implies, is measured in cd-s,
and quantity of illumination in lx-s and ph*s.
14*
212 UNITS OE PHYSICAL QUANTITIES AND THE1H DIMENSIONS
8.4, Relationship between Subjective

and Objective Characteristics of Light
The quantities whose units were considered in the pre
vious two sections sharply differ from each other in the
way of registering them. If the radiant energy quantities
can be measured objectively with the aid of the relevant
instruments, then in the final run the human eye is the
main “instrument” by means of which illumination engi
neering quantities can be measured.
The question appears of how to bring into agreement sub
jective quantities appraised by arbitrary perception, and
direct energy measurements. For this purpose it is obviously
essential to take account only of the “valuable” part, and
not the total energy radiated by a source of light, since any
source, especially a heat one, radiates the predominate
part of its energy beyond the limits of the visible part of
the spectrum. Having selected a definite narrow part of
the spectrum, we should measure the energy radiated in
this part, and the luminous flux obtained with this energy.
The task is complicated by the fact that the measurements
have to be combined with subjective observations, and
since the sensitivity to different colours varies appreciably
in different people, the measurements have to be performed
with the employment of a great number of observers so as
to obtain sufficiently substantiated mean statistical values.
Investigations have shown that the “average eye” reacts
differently to different regions of the spectrum. The sen
sitivity of the eye grows beginning from the shortest waves
(about 0.4 p), reaches its maximum at a wavelength of about
0.554 p and then decreases again. This relationship is cha
racterized by a special quantity named luminous efficiency.
By absolute luminous efficiency is meant the ratio of the
luminous flux (i.e., the power appraised by our eye) to the
corresponding true, total power of radiant energy
»] = l £ : (8.35)
where rj is the luminous efficiency, and Or the power of
radiant energy.
Customarily Or is measured in watts, and since O is
measured in lumens, then the unit of luminous efficiency
UN ITS'OF RADIATION 213
is lm/W. The ratio of the total luminous flux of white light

to the corresponding radiant flux is generally called the
total luminous efficiency, while the corresponding ratio for
light having a definite wavelength is called the monochro
matic efficiency.
Luminous efficiency is a special quantity that makes pos
sible the conversion of radiant energy quantities into light
n* v.
ones. For this reason luminous efficiency is often selected

as a basic quantity having its own dimension (rj).
In this instance the dimension of luminous flux will be
[O] = [(Dr] [r)] - I W T - 3r) (8.36)
Relative luminous efficiency. As we have already noted,
the luminous efficiency differs in different regions of the
spectrum. The ratio of the luminous efficiency of a given
wavelength to the maximum efficiency is called the relative
luminous efficiency
% (8.37)
Vx rj m a x
Figure 25 shows a curve of the spectral sensitivity of the
eye, the wavelengths in microns being laid off along the
axis of abscissas, and the relevant absolute and relative
luminous efficiencies along the axis of ordinates. This curve
has been plotted using the data of Table 43. A glance at

this table shows that the maximum luminous efficiency,
at a wavelength of X = 0.554 p, lying in the green region
of the spectrum, is 683 Im/YV.
A source that would give up all its energy in the form
of radiation only with a wavelength of 0.554 p would be
the most economical one. Such a source, however, would
not suit us at all, because all the objects surrounding us
would be coloured only green and would differ from one
another only in that some would be brighter and others
darker. It would be the best to have such a source that would
radiate energy only in the visible region, and with such
a distribution by wavelengths that would correspond to
the conditional “mean sunlight”. If we take as a unit the
efficiency of a perfect source, i.e., such a source that radiates
only light with a wavelength of 0.554 p, then the efficiency
of a perfect “daylight” source would be 0.35. A source of
heat radiation that is the closest to sunlight in the com
position of its radiation is a black body at a temperature
of about 6 000°K. Its efficiency is about 0.14. The efficiency
of incandescent lamps is about 0.02, and that of luminescent
lamps about 0.06.
Mechanical equivalent of light. As mentioned above, for
the maximum luminous efficiency we have r)ma* =
= 683 lm/W. The reciprocal quantity of r\max is called
the mechanical equivalent of light
Mi = — = 1.466 x 10~3 W/lm (8.38)
“Hmax
This is in essence the minimum mechanical equivalent of

light, i.e., the minimum power in watts that is capable of
creating a flux of one lumen in the region of the spectrum
that is best perceived by the eye.
8.5. Units of Parameters of Optical Instruments

In this section we shall deal with the units of quantities
characterizing the optical properties of instruments. In
essence they should be related to the group of geometrical
quantities, but since they are encountered in optics, we
have found it more expedient to include them in the chapter

relating to the theory of radiation.
Lens power. If a plane wave (parallel rays) impinges on
the lens of an instrument and the lens imparts to it a cur
vature with a radius /, then we say that the lens has a power
of
1 (8.39)
/
The unit of lens power is the power of such a lens that imparts
to a plane wave a curvature with a radius of one metre.
This unit is called the dioptre, and depending on the direc

tion of the radius of curvature, is considered to be either
positive (converging rays) or negative (diverging rays).
The dimension of the dioptre is
[D] = L~l (8.40)
Principal focal length. The quantity /, which is the reci
procal of lens power, is called the principal focal length
of a lens and is usually measured in metres or centimetres.
With a thin lens the principal focal length is the distance
from the lens to the principal focus, i.e., to the point in
which the rays gather that impinge on the lens parallel
to its principal optical axis (Fig. 26).
The principal focus of a diverging lens is the point at
which the continuations of the diverging rays obtained
when a beam of parallel rays impinges on the lens will
intersect (Fig. 27).
For a complicated centred optical system the principal
focal length is measured from the principal focus, i.e.,
the point of real or virtual intersection of the rays leaving
216 UNITS OP PHYSICAL QUANTITIES AND T H E IR DIMENSIONS
the system when they enter it parallel to the principal

optical axis, to the principal plane—the plane in which
Fig. 27
the directions of the incident and emergent rays intersect

(Fig. 28).
Relative aperture (/ number) is the ratio of the principal
focal length of a lens to the diameter of its entrance pupil
i
F ig . 28
(aperture). It is generally written as a fraction with / in

the numerator and the ratio in the denominator. Thus,
it is said that a camera has a relative aperture (or / number)
of //2.8. This quantity is an abstract dimensionless one.
8.6. Units of Optical Properties of a Substance

Refractive index n. Upon the refraction of light at the
interface between two isotropic media, the ratio of the
sine of the angle of incidence to the sine of the angle of
refraction (the angles are measured from a perpendicular
to the interface) remains constant. This ratio is called the
refractive index of the second medium with respect to the
first one, or the relative refractive index.
When light enters a given medium from a vacuum, we
call this ratio the absolute refractive index, thus assuming
the refractive index of a vacuum to be equal to unity. It
follows from the definition that the refractive index is an
abstract (dimensionless) quantity.
Absorption factor is the ratio of the energy absorbed by
a body to the total energy impinging on it. This factor is
also a dimensionless quantity.
Light factors. When a luminous flux impinges on the
surface of a body, part of the flux is directly reflected
(according to the law of reflection), part is more or less
uniformly diffused in all directions, part is absorbed, and
part is transmitted through the body. The ratios of these
luminous fluxes to the total incident flux are respectively
called the reflection factor p, the diffusion factor a, the absorp
tion factor a, and the transmission factor x (transparency).
It is good practice to relate the latter two factors to a unit
of layer thickness. In particular, the absorption factor
related to a unit of length can be determined from the
formula
/ = (8.41)
where J 0 is the intensity of the incident light, and I the
intensity of the light passing through the thickness x .
The dimension of the factor a is
[a] = L"1 (8.42)
since the exponent must be dimensionless.
CHAPTER NINE
SELECTED UNITS
OF ATOMIC PHYSICS
9.1. Introduction
The progress of atomic physics gave birth to a great
number of specific methods for measuring the properties
of atomic particles, the quantities characterizing the pro
cesses which they participate in, etc. It was often found
convenient to introduce special units, partly based on units
of the cgs system, partly of a mixed nature, and sometimes
not directly related to any definite system.
Lately a tendency has appeared in literature on the sub
ject to use units of one of the general systems (most fre
quently the SI) for all such measurements, but the predo
minant majority of scientific articles retain the units that
were always used in atomic physics. Without undertaking
the task of dealing with all these units, we shall consider
the most important and widely used of them.
9.2. Basic Properties of Atomic

and Elementary Particles
Mass. The mass of particles can be measured either abso
lutely or relatively. By absolute measurement we mean
the use of one of the generally adopted units of mass (kg, g)
and by relative measurement—comparison of a given mass
with that of a particle conventionally accepted as a unit.
Such a unit is the atomic mass unit (ainu), which has under
gone certain changes during a number of years. Previously
in chemistry the atomic mass unit was taken as one-sixteenth
of the atomic weight of oxygen in its natural state, and
SELECTED UNITS OF ATOMIC PHYSICS 219
in physics as one-sixteenth of the mass of the lightest isotope

of oxygen whose mass number is sixteen. It should be
remembered that the mass number is an integer equal to
the total number of nucleons (i.e., protons and neutrons)
in the nucleus.
Since natural oxygen contains three stable isotopes with
mass numbers of 16, 17 and 18, and whose content is 99.76%,
0.04% and 0.20% respectively, then the atomic mass
unit used by chemists was 1.000272 times greater than
that used by physicists.
The use of the physical atomic mass unit defined above
had a number of incoveniences due to the fact that the
precise determination of atomic masses was experimentally
related not to atoms of oxygen, but to atoms of carbon.
For this reason in 1961 the International Union of Pure
and Applied Physics and the International Union of Pure
and Applied Chemistry decided to establish the atomic
mass unit (both in physics and chemistry) as one-twelfth
of the mass of the carbon isotope with a mass number of 12
(C*2). This unit is equal to 1.0003179 old “oxygen” physical
units. It is very close to the old chemical mass unit, differing
from the latter only by several units in the fifth decimal
place.
The atomic mass unit is equal to 1.6604 X 10~27 kg.
The atomic weights of elements, molecular weights (relative
molecular masses) of chemical substances and the masses
of nuclei are determined relative to the atomic mass unit.
The masses of elementary particles are generally related
to the mass of an electron me, equal to 9.109 X 10“31 kg
or 5.486 x 10"4 amu.
Charge. Atomic and elementary particles are either de
prived of a charge, or have a positive or negative charge
which is a multiple of that of an electron. The latter is
equal to 1.6021 X 10“19 C = 4.803 x 1 0 '10 cgSQ.
Moment of momentum (angular momentum) of micropar
ticles obeys the laws of quantum mechanics, according to
which it can have only definite discrete values, determined
by the expression
L - & V H I + 1) ( 9. 1)
220 UNITS OF PHYSICAL QUANTITIES AND T H E IR DIM ENSIONS
where h is the Planck constant, and / the quantum number

of the moment of momentum. The Planck constant h =
= 6.626 X 10"34 J -s = 6.626 x 10-27 erg-s.
The ratio - = 1.0545 x 10-34 J -s = 1.0545 X 10~27 erg-s
that is generally used in quantum mechanics instead
of h is designated h. The quantum number / may be an
integer, a half-integer ân odd multiple of , or zero.
For an electron the quantum number of the moment of
momentum is designated s = — and is called the spin
number. Hence the own moment of momentum of an elec
tron is
L = ~ ) X - = 0.913 x 10'34 J-s = 0 .9 l3 x 10-27 erg-s
(9.2)
The quantity
serves in atomic physics as the unit of the moment of mo

mentum (angular momentum).
Magnetic moment. In the classical Bohr theory an elec
tron revolving in a circular orbit about a nucleus forms
a closed current that, consequently, has its own magnetic
moment. Quantum mechanics, while renouncing the illu
strative model representations (the “orbit1’ of an electron
in an atom, the “revolving electron”), retains such quan
tities as the moment of momentum considered above and,
correspondingly, the magnetic moment.
In the classical model, the magnetic moment of an atom
of hydrogen in the normal (unexcited) state is easily cal
culated as follows. The ratio of the charge of an electron
to the period of its revolution in an atom is the “current
intensity”
/= (9.4)
According to Bohr’s postulate
m(da«= -J; (9.5)
where a0 is the radius of the orbit (the so-called Bohr

radius). Consequently
and the magnetic moment (designated here (iB) is
(9.6)
The magnetic moment determined by formula (9.6) is

called the Bohr magneton and serves as a unit of magnetic
moment. Its value is
jxB = 9.273 x 10‘24 A-m2—9.273 x 10"21 dyn-cm/Gs (9.7)
The magnetic moments of nuclear particles are measured
with the aid of the so-called nuclear magneton (p-^), which
is determined by the same formula (9.6), but with the
mass of a proton substituted for that of an electron, the
former being 1 836 times greater than the latter. Hence
the nuclear magneton is
|i* = 5.051 X 10“27 A-m2—5.051 X 10'24 dyn-cm/Gs (9.8)
It should be noted here that the magnetic moments of
nuclei are not integer multiples of the nuclear magneton,
but are calculated by a quite complicated formula. In
particular, the magnetic moment of a proton is
= 2.7928pjv —1.4105 x 10 26 A-m2—
—1.4105 x 10~23 dyn-cm/Gs
Dipole moment. Polarization. The electric charges in
molecules may be distributed unsymmetrically, as a result
of which the molecule as a whole acquires an electric dipole
moment. The dipole moment is measured either in cgs or SI
units (see Secs. 7.3 and 7.4), or in a special unit, the debye
(D), equal to 10“18 cgs units or 3.336 xlO -30 C-m.
Atoms and molecules, even when not having their own
dipole moment, can acquire it under the action of an exter
nal field as a result of electronic polarization.
The ratio of the acquired dipole moment to the field

intensity is called polarization a. According to the definition
The dimension of a in the cgs system is

[a ] = Z® (9.10)
and its name is cubic centimetre.
In the SI system the relevant dimension is
[a] = (9.11)
From the definition of a according to formula (9.9),
upon inserting p — 1 C *m and E = 1 V/m and making
the corresponding substitutions, it is easy to find that the
SI unit of polarization is 9 X 1015 times greater than the
cgs unit. The same relationship can be found from the
dimension formulas. Polarization is related to relative
permittivity (if the latter is determined only by electronic
polarization) by the expression
t-1 (9.12)
N
where N is the concentration of molecules of the given

substance.
Lifetimes. Many elementary and atomic particles are
not stable and after a certain time either disintegrate, or
pass over into another state. To characterize the stability
of atomic radioactive nuclei, the concept half-life is used.
This is the time during which half the initial number of
atoms disintegrate. Since the change in the number of
radioactive atoms follows the exponential law
N ^ N 0e~kt (9.13)
where N 0 is the initial number of atoms, N the number of
undecayed atoms after the time t has elapsed, and K is the
so-called decay or disintegration constant, then the half-life
will be determined by the expression
~§~— ( 9 . 14)
SELECTED UNITS OP ATOMIC PHYSICS 223
whence
log? __ 0.693
(9.15)
X ““ X
To characterize the stability of atoms in an excited
state, the concept of mean lifetime r is used, found from
the exponential law
N = , N 0 e - ^ (9.16)
The mean lifetime is equal to the time during which the
number of atoms in an excited state will be reduced to
He of the original number. The half-life and the mean
lifetime are obviously related by the following expression
T —0.693t (9.17)
Linear dimensions. In quantum mechanics such concepts
as “the radius of an electron orbit”, the radius of an ele
mentary particle (for example, of an electron), etc., have
no meaning. It is often convenient, however, to introduce
definite linear scales whose capacity is filled by quantities
obtained on the basis of classical calculations. The most
widespread of these are the “classical electron radius”
determined by the expression
ro= i = 2-818x10' 13 cm <9-18)

and “the radius of the first Bohr orbit”
a 0 = — = 0.5292 x 10'8 cm (9.19)
In addition, in nuclear physics use is made of a unit of

length called the fermi, equal to 10~13 cm.
9.3. Effective Interaction Cross Sections

The classical kinetic theory of gases introduced the con
cept of the free path, relating it to the concept of the cross
section of colliding particles. Atomic physics has extended
the concept of cross section and has simultaneously divided
it, with the establishment of the concept of effective cross
section relative to a concrete process of interaction of atoms,
ions, molecules, nuclear particles, etc. The concept of

effective cross section with respect to a process will be best
explained using the following semiclassical diagram, which
we shall consider with respect to a concrete example of the
excitation of an atom by an
electron impact (Fig. 29). Let
an electron having a given
velocity move perpendicular
to the plane of the drawing
toward the atom with an
impact parameter r. By im
pact parameter is meant the
length of a perpendicular
erected from the centre of the
atom to the vector of the
initial velocity of the elect
ron. Let, further, the proba
bility of excitation of the
atom with the given impact
parameter be p(r). Let us depict a ring confined
within the radii r and r 4 dr and separate on it a part
equal to
do -- i (r) 2nrdr (9.20)
The quantity that we get by integration of do with res
pect to all the values of r from 0 to oo, namely,
a —2ji j p(r)r dr (9.21)

o
is called the effective cross section of excitation of an atom
by an electron having a given velocity. That a has the
dimension of area can be seen from the defining expression.
With respect to the physical meaning of a, it can be easily
seen from the definition that the effective cross section is
the section an atom must have for excitation to take place
with a probability of 100% upon each impact of an electron.
The concept of effective cross section is exceedingly
widely used in atomic and nuclear physics and in the fields
of physics investigating macroscopic processes connected
with the interaction of atomic particles. It is employed
to describe quantitatively all kinds of elastic and inelastic

processes of interaction.
Various units are used to measure effective cross sections.
In nuclear physics the corresponding unit is the barn (b),
1 b = 10~28 m2 = 10"24 cm2. In the physics of atomic col
lisions use is made of the cm2, less frequently the m2, and
the units al and nal (where a0 is the radius of the first Bohr
orbit):
al = 0.2800 X 10~1G cm2
na02-0.8797 x l 0 - 16 cm2
In scientific literature the term reduced effective cross
section is sometimes used. This is the sum of the corres
ponding effective cross sections of all the atoms or molecules
contained in 1 cm3 at 0°C and 1 mm Hg. Since in these con
ditions the number of molecules in 1 cm3 is 3.54 X 1016,
then we shall obtain the reduced effective cross section by
multiplying by this number the effective cross section
measured in cm2/(cm3-mm Ilg). The unit of reduced effective
cross section is usually designated by cm2/(cm3-mm Hg).
9.4. Units of Energy in Atomic Physics

Electron-volt. If an electron travels through a potential
difference without losing any energy on its way, it acquires
the kinetic energy
^ - = eU [(9.22)
(assuming that v0 = 0).
When the potential difference is 1 V, the energy
will be
1 eV = 1.60207 x lO'19 B x 1 V = 1.G0207 x 10~19 J
= 1.60207 x 10'12 erg (9.23)
It is very often convenient to use this energy as a unit
for measuring not only the energy of electrons, but also
that of other particles or energy levels in atoms and mole
cules. This unit of energy is called the electron-volt (eV).
Units that are 103, 106 and 109 times greater are frequently
used—keV, MeV, and GeV.
15—1040
The velocity acquired by an electron after travelling

through a potential difference of U volts can also be deter
mined by formula (9.22)
v - } / — = 5.932 x 105 V u m/s (9.24)
Thus, the velocity of an electron uniquely determines the
potential difference it acquires. This is why it is often said
that “an electron has a velocity of U volts”, meaning that
it has such a velocity which it would acquire if it travelled
through a potential difference of U volts. The velocity of
an electron in volts is converted to velocity in m/s or
crn/s by means of formula (9.24).
Relationship between electron-volt and degree Kelvin. If
a gas is at a temperature 71, then the mean kinetic energy
of translational motion of its molecules is equal to
kT (9.25)
If an elementary charge were connected with each mole
cule, then the molecules could acquire their energy by
travelling through a potential difference U determined by
the relationship
—kT —eU (9.26)
If U — 1 V, then the corresponding temperature is
T 2 x 1 eV _ 2 x 1 . 6 0 2 x 10-19n 700oTZ /n
1 “ V> - ~ 3 x / T = 3 X 1.380 X 10“23 - ' K ^
Many equations of statistical physics, thermodynamics,
spectroscopy, etc., include the exponent eEl]lT, where E is
the energy of transition from one state to another. When
measuring this energy in electron-volts, it will be conve
nient to write the denominator in electron-volts too. The
latter will be equal to one electron-volt if T ^ 11 600°K.
Relationship between electron-volt and calorie per mole.
If each of the molecules contained in one mole acquires
an energy of one electron-volt, then the total energy of all
the molecules will increase by
1 eV X N a - 1.602 x lO"19 x 6.023 x 10*> J = 9.749 x 104 J
—23 053 cal (9.28)
Relationship between energy measured in electron-volts

and length of a light wave. Each spectrum line is characte
rized by a definite wavelength or frequency and, consequent
ly, a definite quantum of energy
hv - ^ (9.29)
For this reason it is possible lo establish the relationship
between the wavelength of a given spectrum line measured
in angstroms and the energy corresponding to it, measured
in electron-volts. The importance of this grows because
of the fact that an atom is frequently excited to a higher
energy level by an electron impact, upon the transition
from which to the normal state it radiates a quantum of
energy.
From the relationship
e U = --h v ^ ~ (9.30)
it is easy to obtain
W = (12 395 ± 2) AV (9.30a)
Relationship between velocity of electron measured in elec
tron-volts and the length of its De Broglie wave. The following
wavelength is connected with an electron moving with
a velocity v:
m (,v
(9.31)
By expressing the velocity of the electron in electron-
volts (see Sec. 9.24) and substituting the corresponding
values for h and mCf we get
* 1 2 .2 6 /n 0 4 x
l = w (0.31a)
Here U is measured in volts, and X in angstroms. It is

convenient to use tha app.’oximito expression
(9.31b)
a glance at which shows that an electron with an energy of
150 eV hasâ De Broglie wavelength of 1 A.
15 *
Relationship between mass and energy. The theory of rela

tivity gives the following relationship between mass and
energy:
E ~m c2 (9.32)
It is known that the measured difference between the
mass of an atomic nucleus and the sum of the masses of the
protons and neutrons forming it can be used to compute the
nucleon binding energy in a nucleus. Below are given the
most favoured approximate relationships between the units
of mass and energy.
For macroscopic bodies
1 icg= 9 x 10lG J
1 g—9 x 1020 erg
The energy corresponding to one atomic mass unit is
1 anm — 1.493 x 1(T10 J - 931.7 MeV (9.33)
For elementary particles
1 ji?*=8.18 x 10"14 3 -0 .5 1 1 MeV
1 mp~ 1.50 x 10'10 J - 938 MeV
Unit of energy rydberg. The spectrum lines of atoms
similar to hydrogen are arranged in a series satisfying
the formula
°= R i k — k) <9-34>
where a = wave number of the given line
nx and n2 — quantum number of the energy levels between
which the transition is accompanied by the
radiation of the corresponding quantum
R = so-called Rydberg constant.
For an infinitely massive atomic nucleus
Roo - 109 737.3 cm"1
For light atoms the values of Rydberg’s constant are
somewhat smaller. For example, for a hydrogen atom
Rn - 109 677 ern-l (9,34a)
Multiplying both parts of equation (9.34) by he, we get

the value of the energy of an emitted quantum. The product
in the right-hand part of the equation
\RV - Rch (9.35)
is called the rydberg (Ry) and is used as a measure of the
energy of electron levels. Assuming that w3 = 1 and n2 = oo
we can define the rydberg as the energy that would have to
be spent to ionize an atom of hydrogen if its mass were
equal to infinity. Substitution of values for /?, c and h yields
1 Ry = 13.60 eV (9.36)
The actual energy of ionization of a hydrogen atom computed
using the value of R H is equal to 13.57 eV.
9.5. Ionizing Radiation Units

Atomic particles (electrons, atoms, ions, nuclear par
ticles) and photons with a sufficiently great energy are
capable of causing ionization of a gas when absorbed in it.
This ability determines the experimental methods of regi
stering such radiation and its quantitative characteristics.
This is why in addition to radiant energy units determining
the power of radiation, the radiant flux, etc., use is also
made of certain specific units, in particular such that are
employed to measure the ability of the given radiation to
produce a definite ionization of a gas. Some of these units
have been constructed on the basis of SI units, and others
on the basis of cgs ones.
Particle or quantum flux is the number of particles or
quanta passing through a given surface in a unit of time.
The flux is generally related to a second and is correspon
dingly determined by the number of particles (a, p, etc.)
or quanta per second. The particle flux passing through
a unit of area perpendicular to its direction is called the
flux density. Depending on the system of units used for
the measurement, the^flux is related to one square metre
or centimetre. The flux and flux density may be measured
not by the number of particles, but by the energy transferred
(watts or ergs per second or, correspondingly, watts per
square metre or ergs per second per square centimetre). In
this case they are respectively called the radiant flux and
radiant flux density. Both these quantities and their units
completely coincide with the radiant energy characteristics
considered in Sec. 8.2.
We have already considered the radiant energy density
and its units. In investigation of ionizing radiation this
quantity acquires a special importance, since it characte
rizes the energy absorbed in a unit of volume as a result
of the transition of radiant energy into other forms. The
total quantity of absorbed energy is measured in the common
energy units (J, erg), while the density of absorbed radiant
energy is measured in the same units as radiant energy
density (J/m3, erg/cm3). In measuring the ionization caused
by radiation, however, a more important quantity is the
relation of the absorbed energy not to the volume, but to
the mass of the absorbing substance. This is easy to under
stand if we consider absorption in a gas. Since ionization
occurs upon the interaction of the emitted particles or
quanta with atoms or molecules of the gas, then it is obvious
that when the pressure is halved, the volume of the gas
must be doubled to obtain the same ionization. The energy
absorbed by a unit of mass of a given substance is called
the absorbed radiation dose D . Its dimension is
[D] = L*T~* (9.37)
and J/kg and erg/g are its units, 1 J/kg = 104 erg/g. The
rad, a unit equal to 10“2 J/kg or 102 erg/g is also used. The
absorbed radiation dose related to the duration of absorp
tion is called the power of absorbed radiation dose p. Its
dimension is
[p]=--L2T~* (9.38)
and its units are W/kg and erg/(s-kg).
To describe radiation from the viewpoint of the ionization
it produces, a quantity called radiation exposure is used.
In the SI system the corresponding unit is the coulomb per
kilogram (C/kg)—an exposure producing in one kilogram
of dry air a number of ions whose total charge is one cou
lomb of each sign. The cgs unit, the roentgen (r), is defined
as the exposure to X rays or gamma rays upon which, as
a result of complete ionizing absorption, ions with a tota
SELEGTE DUNITS OF ATOMIC PHYSICS 231
charge of 1 cgs unit of each sign are formed in one cubic

centimetre of air in standard conditions. This corresponds
to 2.082 X 10° pairs of ions in 1 cm3 or 1.6 X 1012 pairs
of ions in one gram of air.
Since the mass of a cubic centimetre of air at 0°C and
760 mm Ilg is equal to 1.293 X 10~6 kg, then, with a view
to the relationship between the coulomb and the cgs unit
of charge, we get 1 r = 2.58 X 10 4 C/kg and 1 C/kg =
= 3.88 X 103 r. In practical work use is generally made
of the sub multiple units microcoulomb per kilogram
(juiC/kg), milliroentgen (mr), and smaller ones.
The units of radiation exposure serve to form the units
of dose rate—coulomb per kilogram per second and roentgen
per second.
The measurement of a radiation dose according to its
ionization ability makes it possible to establish a physical
equivalent of the unit of dose. Taking into account the
mean energy spent for the ionization of a molecule of air
(about 33 eV), it can be computed that 1 r is equivalent
to 85 erg/g. This quantity is called the physical roentgen
equivalent (designated rep). In appraising the biological
action of radiation the man roentgen equivalent, designated
rem, is employed.
The rad, equal to 100 erg/g, is used together with the
roentgen to measure the total energy absorbed by a unit
of mass of a substance.
A special unit, the einstein (E), sometimes used in the
investigation of photochemical processes, can be included
to a certain extent among the units of ionizing radiation.
This unit is defined as N A hv, where NA is Avogadro’s num
ber and hv is the energy of a quantum. It follows from this
definition that the einstein unit is not a general unit of
energy, but has different values depending on the wave
length of light.
9.6. Units of Radioactivity

The main process that is to be measured and registered
in radioactive conversions is disintegration (decay) accom
panied by the emission of alpha or beta particles, neutrons
and gamma quanta. For this reason the unit characterizing
232 U NITS OF PH Y SIC A L Q U A N T I T I E S _*AND T H E IR D IM E N S IO N S
the activity of a radioactive source is the disintegration per

second (d/s). The rutherjord (Rd) is a unit of radioactivity
equal to 106 d/s.
In addition to the units disintegration per second and
rutherford, a unit of radioactivity the curie (c), now defined
as the quantity of any radioactive isotope in which the
number of disintegrations per second is 3.7 X 1010, is also
in use, although fractions of this unit are in greater favour.
Previously the curie was defined as the radioactivity asso
ciated with the quantity of radon in equilibrium with one
gram of radium. This can be explained as follows.
If radium is placed in a closed vessel, then initially the
quantity of radon (radium emanation), which is a product
of radium disintegration, will grow, but since radon itself
also decays (with a half-life of 3.82 days), then equilibrium
will finally set in between the newly appearing radon and
the decaying one. The number of disintegrations per second
will remain practically constant, if no consideration is
taken of the change in the mass of the radium itself, which
occurs very slowly, with a half-life of about 1 600 years.
For this reason the radioactivity of radium can be compared
with that of radon in equilibrium with a certain amount of
radium. This was the origin of the previous definition of the
curie. The quantity of radon corresponding to a radioacti
vity of 1 curie has a mass of 6.51 X 10~6 g and contains
1.78 X 1016 atoms. The alpha particles emitted by radon
(not counting the products of its further decay) are capable
of creating an ionization saturation current of 0.92 mA
in air.
To measure the concentration of a radioactive prepara
tion, use is sometimes made of the unit ernan, a concentra
tion of 10~10 curies per litre of fluid.
Formerly a unit of concentration, the mache, was used,
which is equal to 3.46 emans, and is defined as the quantity
of radioactive emanation (alpha-particles) of radon that
sets up an ionization saturation current of 10~3 cgs units
of current in air. This unit is now obsolete.
The particles emitted by a radioactive preparation form
a flux measured by the number of particles per second. The
number of particles per unit of surface area (square metre
or square centimetre) is the particle flux density.
9.7. Ionization, Recombination

and Mobility Coefficients
The properties of electrons, ions, atoms and other particles

are described by various quantities inherent in the given
particles and characterizing separate acts of interaction
of these particles with one another, with quanta of radiation,
etc. These quantities include, in particular, the effective
cross sections considered above. To describe phenomena
in which a great number of particles are involved, however,
it is often convenient to use mean macroscopic quantities.
This is encountered, for example, in the kinetic theory
of gases when describing transfer phenomena (diffusion,
viscosity, heat conductivity), i.e., phenomena characterized
by macroscopic coefficients whose values can be found with
the aid of the molecular theory.
In the present section we shall give several such quantities
and their units as applied to the motion of charged
particles in a gas.
Volume electronic ionization coefficient. When moving
in an electric field, an electron acquires the ability of ioniz
ing a gas. The mean number of ionizations N t performed
by an electron on a unit of its path in the direction of the
field is called the volume electronic ionization coefficient
or the first Townsend coefficient. The latter name is due to
the fact that this coefficient was introduced by F. Townsend
in his theory of semi-self-maintained discharge in a gas.
The number N t is measured in units that are reciprocals of
the unit of length (m '1, cm"1).
Similar coefficients with the same units can be introduced
for describing ionization by other particles (for example,
by ions).
Recombination coefficient. If a gas contains charged par
ticles of both signs with concentrations of n+ and then
the process of recombination of these particles into neutral
atoms or molecules may occur. The number of such^recom-
binations occurring in a unit of time in a unit of volume is
determined by the equation
Nr = An+n_ (9.39)
where A is the recombination coefficient. Its dimension

[A] = L*T~l (9.40)
determines its units, namely, m3/s and cm3/s.
Mobility coefficient (mobility). The velocity of a charged
particle moving in a certain medium in an electric field
will be established at a definite mean level owing to nume
rous collisions. There arc distinguished the chaotic undi
rected velocity and the directed or drift velocity along the
direction of the field. The latter determines the passage of
an electric current. In the general case the directed (drift)
velocity u may depend in a complicated way on the field
intensity. In definite conditions the directed velocity will be
proportional to the field intensity E :
u = bE (9.41)
where b is the mobility coefficient or, as it is called more
frequently, the mobility of a given charged particle.
Einstein showed that the mobility is related to the dif-
fusivity D by the equation
D _ kT
b e (9.42)
where k = Boltzmann’s constant
e ~ charge of an electron
T = absolute temperature.
Equation (9.31), in particular, is complied with upon
the motion of electrons in a metal, thus ensuring the appli
cability of Ohm’s law to metals. Mobility can be defined
as the mean directed velocity acquired by a particle when
moving in a field whose intensity is unity. The dimension
of mobility in the SI system is
[b] = L M ^ T 2I (9.43)
and in the cgs system
[b] = L v zM ~ m (9.43a)
In practical work mobility is measured in the units m2/V *s
and cm2/V -s.
With other conditions equal, mobility, the same as diffu-
sivity, is inversely proportional to the density of a gas or
the reduced pressure. This is why the concept of reduced

mobility is frequently used, defined by the expression
=y (9-44)
The reduced mobility is usually related either to 1 mm Ilg
or to a standard atmosphere.
9.8. Natural Systems of Units

It has been indicated more than once in this book that
a single-valued relationship exists between the number
of basic units and Ihc number of universal constants, i.e.,
the greater the number of basic units, the more constants
there are in the formulas of physical laws and definitions.
By equating the gravitational constant to unity while
simultaneously retaining the inertial constant equal to
unity, we reduced the number of basic units in systems of
geometrical and mechanical units from three to two. By
equating Boltzmann’s constant to unity, we make the unit
of temperature a derived one. In systems of electrical and
magnetic units we can further reduce the number of basic
units by equatiug to unity the electric and magnetic constants
in a system constructed according to the principle of the SI,
or the velocity of light in a system constructed according
to the principle of the cgs system. Thus we remain with
two units, one of which, light intensity, reflects the spe
cific physiological nature of perception of light, while the
other may be either the unit of length or that of time, as we
wish.
To what extent have we used all the possibilities of
reducing the number of universal constants? Although the
total number of such constants is comparatively great, it
can be proved, however, by analysing the origin of the
relevant equations, that as a result of the reduction of the
number of units made above, almost all the constants will
become equal either to unity or to a dimensionless constant
number obtained as the result of a mathematical operation.
Nevertheless, even after the reduction of the number of
basic units of all the quantities to one (leaving aside the
unit of luminous intensity as not directly related to the
common physical quantities), we shall still have “unused”

constants. These are the Planck constant h and the charge
of an electron e. It is easy to see that in a system with one
basic unit—length—the dimensions of these constants
will be
ly = L 2 (9.45)
[e\ = L (9.46)
It is possible to deal with another constant, for example, A,
and so select the unit of length that the Planck constant
will be equal to unity. Here we shall obtain a system that
is in general dimensionless, i.e., such a system in which
we are completely deprived of freedom in selecting the
dimension of any unit whatsoever.
Instead of equating the Planck constant to unity, we can
do this with the charge of an electron; thus the Planck
constant will be uniquely determined. We can, finally,
equate to unity both the Planck constant and the charge
of an electron, but in this instance a different constant
will appear, the velocity of light, that will now differ
from unity.
Systems in which the greatest possible number of uni
versal constants have been equated to unity are called
natural systems. The system described above in which
h — 1 was proposed by Max Planck. In this system the unit
of length is found to be 4.02 X 10~33 cm, of mass 5.43 X
X 10~5 g, and of time 1.34 X 10“43 s. In addition to
Planck’s system, other natural systems were proposed, in
which different universal constants are equated to unity.
For example, in the system proposed by D. Hartree, the
charge and mass of an electron, the radius of the first Bohr
orbit, and the Planck constant divided by 2n (i.e., the
constant Ti) are all equated to unity. In this system the
unit of time is 2.419 X 1 0 '17 s, the unit of energy is
4.359 X 10~u erg, etc.
The impossibility of equating to unity all the universal
constants is due to the fact that there are definite relation
ships between some of them. For instance, the charge of an
electron, the Planck constant and the velocity of light
form a dimensionless combination, the so-called fine
SELECTED units op atomic physics 23?
structure constant
e2 1
he 137
(9.47)
It follows from this value of a that in the Planck system
the charge of an electron will he —. _ . In the same
1/211.137
way the values of the constants in the laws of Stefan-Boltz-
mann, Wien and others (see Appendix 4) will be given.
Although in natural systems all the units are quite far
from the dimensions of quantities generally encountered
in practical work, these systems are employed with great
success in theoretical physics, since they result in an exceed
ingly great simplification of the fundamental equations,
freeing them of superfluous factors.
It should be noted in conclusion that natural systems
are sometimes considered as systems in which constants
equated to unity are used as the basic units, and this allows
us to construct a system of dimensions whose formulas
will contain the conditional symbols of the dimensions
of these constants.
APPENDIX 1
Logarithmic Units
In Chapter Six we considered logarithmic units describing sound

intensity— the hel, its tenth fraction—the decibel, and the neper.
The frequency characteristic of musical intervals was also constructed
to a logarithmic scale. The use of a logarithmic scale, however, is not
at all limited to acoustics. A physical quantity sometimes changes
within such broad lim its that it is practically impossible to show it
using a linear scale. For example, in modern vacuum engineering
during the process of evacuation of an apparatus the pressure of a gas
may change from atmospheric to 1 0 “y- 1 0 -11 atm, and in some labo
ratory investigations to 1 0 ~13- 1 (H 5 atm. It is useless to attempt to
show how this process proceeds with time in a linear scale of pressures.
Similar instances are encountered in astronomy and in many other
fields of physics and related sciences. The use of a logarithmic scale
permits processes and laws to be shown with a practically unlimited
range of the values of the quantity we are interested in, and both
small and great values w ill be shown with suflicient clarity.
The reasons for using a logarithmic scale, however, are much more
numerous. Quite often the essence of a phenomenon itself points to
the expediency of describing it with the aid of logarithmic units. We
have already mentioned the logarithmic nature of the psychophysical
perception of the loudness and pitch of sound. This also relates, to the
same extent, to the perception of other external stim uli that comply
satisfactorily with the Weber-Fechner law, according to which per
ception is proportional to the logarithm of the stimulus.
For each stimulus there exists a minimum ratio of two values of
the quantity characterizing it that can be detected by the corres
ponding sensory organ. For example, two brightnesses can be distin
guished by the human eye if their ratio is about 2.5. This ratio deter
mined the logarithmic scale for measuring the “brightness” of stars,
the “stellar magnitude”. We have put the word “brightness” in quota
tion marks because, owing to the great remoteness of stars, what is
actually meant here is the illumination created by a given star at the
boundary of the atmosphere. For this reason the human eye perceives
stars as luminescent points of different brightness. Photoelectric
registration makes it possible to introduce fractional values of stellar
magnitudes.- The brightest stars and, naturally, the Moon and the
Sun, are described by negative values of the stellar magnitude
(—12.54 and —26.59 stellar magnitude).
Another field of application of a logarithmic scale are processes
in which a change in a quantity is proportional to the quantity itself.
Such processes include the absorption of light by a homogeneous
medium, the aperiodic discharge of a capacitor onto a resistor, the
LOGARITHMIC UNITS 239
attenuation of a signal along a transmission line, and a chemical or

nuclear chain reaction. In the first examples the relevant quantity
decreases with distance or time, and in the last one it grows. In the
general form the law of change of the corresponding quantity can
be written as
A--=A0a^ (A.l)
where j40 = initial value of a given quantity
A —- its value at a value of the argument (distance, time,
etc.) equal to z
k = factor describing the “rate” of the given process (absorp
tion, attenuation, amplification, etc.)
a •— base of logarithms used to describe the given process.
The factor k may be either negative (absorption, attenuation) or
positive (amplification, development). The value of a is quite fre
quently taken equal to the base of natural logarithms c, but any other
number may be used for this purpose, for example 1 0 or 2 , the fac
tor k being selected accordingly.
To characterize the pitch of sound, as indicated in Chapter Six,
a logarithmic scale with a base 2 is employed. The same scale is used
to describe radioactive decay when the half-life is used as a unit of
time. The level of sound intensity is measured either in bels (with
1 0 as the base of logarithms), or in decibels (the base of logarithms
is 1(\ f 1 0 = 1 .259+ ), or in nepers (the base of logarithms is the number
e = 2 .7 1 8 + ) .
The spreading of the use of logarithmic units was not smooth, but
was accompanied by certain confusion. Whereas bels, decibels and
nepers served to measure the difference in sound intensity levels in
acoustics, in electrical and radio engineering when describing atte
nuation along a transmission line decibels were used to measure the
power level, and nepers the change in the field intensity level. Since
the power is proportional to the square of the field intensity amplitude,
then for the ratio of two powers we have
or, using logarithms,
logio — ■= 2 log I0 4 L = - 2 x 0.4343 log, -§L (A.3)

*2 ^2
For this reason, if an acoustics 1 B was equal to 2.303 n [see formula
(6.16)] or 1 dB = 0.2303 n, then in electrical engineering 1 dB =
= 0.1151 n or 1 n = 8 . 6 8 6 dB.
The dual nature of logarithmic units partly spreads to the decibels
themselves, which were begun to be used to measure both a change
in power, and a change in field intensity, voltage, etc. This confusion
led to the idea of introducing a common logarithmic unit whose use
should be accompanied each time by an indication of the quantity it
relates to. A number of proposals were made on the nature and name
of this unit. The most favoured was a unit named the decilog, nume-
240 APPENDICES
rically coinciding with the decibel, but applied to any quantity with
the corresponding indication made. The use of decilogs made it pos
sible to substitute addition and subtraction for m ultiplication and
division, and even the final result could be expressed directly in deci
logs. With respect to the decibel, it was decided to retain it only for
measuring power levels. With such a definition of the decilog it can
be said, for example, that one decilog of current intensity is equal to
one decilog of voltage minus one decilog of resistance. According to
what has been said above, the decilog can be defined either as 1 0 com
mon logarithms of the given quantity, or as the logarithm of this
quantity with the base When writing down quantities measured
in decilogs, a subscript should be used indicating the unit in question.
For instance, power measured in kilowatts and written down in deci
logs should be designated dlgkW.
Let us illustrate the above with an example, and determine the
power of a current at a voltage of 2 kV and a current of 10 A:
®= d l g k w 3.01 dlgky - \ - 10 dlgA ~ 13.01 dlgkw
A special binary logarithmic unit, the bit (the name is derived from
the words binary digiiL employed in the theory of information, stands
somewhat apart. If the given information is determined from a pos
sible number n of equally probable events, then the measure of this
information is given by the expression
N —log 2 n (A.4)
Let us consider the following problem as an example. Suppose we
have a deck of 32 cards containing only “number” cards from 1 (ace)
to eight. How many questions w ill have to be asked with the answer
being only “yes” or “no” to guess a card that someone is thinking of?
Each answer, obviously, w ill reduce the indeterminacy to one-half
of the original number. Assume that the seven of spades is the card
in question. The following scheme of questionsTmd answers could be
used, for example, to guess it:
Question Answer
1. A black suit? Yes
2. A club? No
3. An even number? No
4. An ace or three? No
5. The seven of spades? Yes
The only alternative to the last question is “The five of spades”,
and the answer “No” would show that the card in question is the
seven of spades.
It “can readily be seen th at in any case five questions are sufficient
to get The correct answer. This can be expressed by the statement that
the-knowledge of a definite card from among a total number of 32 con
tains information amounting to 5 bits. The knowing of the square of
a chess-board on which a given figure is contains, obviously, N =
= log?, 64 = 6 bits, etc.
APPENDIX 2
Measuring the Density of a Liquid with an Areometer
The density of a liquid can be measured with the aid of an areo

meter. A schematic view of such an instrument is shown in Fig. 30.
Its lower part is provided with weight 1 that
keeps it in a vertical position when immersed
in the liquid being measured. Narrow tube 2 is
calibrated according to the density being mea
sured, since the depth of submergence depends on
the density of the liquid. At present these cali
brations directly show the density (usually in
grams per cubic centimetre). Previously, how
ever, conditional scales were used and the
density was determined in degrees of the
relevant scale. The calibrations were marked
on the tube at equal distances from one, an
other. If an areometer is immersed up to? the
mark conditionally taken as zero into a liquid
with a density p0, the volume of its submerged
part being V, and then immersed into a liquid Fig. 30
with a density p, the level changing by n d ivi
sions and the volume of the narrow tube between two divisions being
p, then
9 = 9olH k m (A‘5>
where the plus sign corresponds to a lighter and the minus sign to
a heavier liquid. A definite relationship N = Vlv was established
for different areometers. Here the following formula w ill serve to
convert density into degrees (divisions of the scale) or into relative
density, with the density p0 taken as unity
p=
r N db—n (A. 6 )
x '
In the greatest favour was the Baume areometer (hydrometer), in
which N = 144.3, and the density of water at 15°C was taken as the
unit of density. The density was obtained in Baume degrees (°Be).
10-1040
APPENDIX 3
pH Index
The activity of electrolyte solutions depends on the concentration

of ions in them. This relationship, however, is not quite single-valued
owing to the interaction between ions. For this reason concentration
can serve to describe the activity of a solution only when it is greatly
diluted. At high concentrations the concept of equivalent concentra
tion is introduced, which is the product of the actual concentration
and an activity factor less than unity. Since both the actual and the
equivalent concentrations of ions can change within very broad lim its,
a logarithmic scale is used. The index measured in this scale (desig
nated pH) is equal to the common logarithm, with sign reversed, of
the activity or equivalent concentration of hydrogen ions measured
in gram-equivalents per litre. Since the concentration of hydrogen
ions in water (and chemically neutral solutions) is 1 0 '7, then for water
pH = 7. In acid solutions the hydrogen ion concentration is higher,
and accordingly pH <c 7, and in alkaline solutions, on the contrary,
pH > 7.
APPENDIX 4
Constants
The present appendix gives the values of the most important uni
versal constants. Those of them whose meaning is sufficiently obvious
are given without explanations. For others either a reference to the
corresponding formulas in the present book is given, or the origin
and physical meaning of the constant are explained. In addition, since
some of the constants are interrelated, the formulas are given showing
the relationships between them.
The numerical values of the constants are given with such a number
of digits that in the event of their more accurate determination, the
change w ill be not more than by unity in the next to last significant
digit. All values are given in the SI and cgs systems, and in indivi
dual cases in some non-system units.
Velocity of light
c = 2.997925 X 108 m/s = 2.997925 X 101° cm /s
Since a number of expressions, in particular some equations of
electromagnetism in the cgs system, contain the velocity of light
to the powers 2 , — 1 and — 2 , we give the corresponding values with
an accuracy of unity in the last digit:
c* - 8.98726 X 1016 m 2 /s* =.=8.98726 X 10*° cm 2 /s 2
— = 3.33572 x 10- 9 s/m = 3.33572 x 10-n s/cm

c
1.11270 x 10-17 S2/m2= 1.11270 x IO- 2 1 s 2 /cm 2
Avogadro's number—the number of molecules in a kilomole or

mole
Na = 6.0225 X 10*6 km ole-i = 6.0225 X 10*3 m ole-i
Gravitational constant
G= 6.670x 10-ii N -m 2 /kg 2 — 6.670x 10~ 8 dyn.cm 2/g 2
Charge of an electron
1.60210 X 10-1° C -4 .8 0 3 0 X lO'io cgsQ
Mass of an electron
= 9.1091 X 10-31 kg = 9.1091 X 10* :28 g
Mass of a proton
mp = 1.67252 X 10-27 kg = 1.67252 x 10-2i g
16*
244 APPENDICES
Mass of a neutron
mn = 1.67482 X 10' 27 kg = 1.67482 X 10~ 24 g
Faraday's constant (faraday)—the quantity of electricity which,
when flowing through an electrolyte, causes one gram-equivalent of
substance to be liberated at each electrode
F = eNA = 9.6487 X 104 C = 2.8926 X 1014 CgsQ
Planck's constant
/* = 6.62517x lO-34 J-s = 6.62517x 10-27 erg>s
h --~ - Jl = 1.0544x 1 0 -3 4 J .s = 1.0544x 10-” erg-s
2k
The constant H is sometimes called the Dirac constant.
Fine-structure constant. The investigation of the spectrum lines
of hydrogen with the aid of instruments having a high resolution has
shown that these lines possess a fine structure, i.e., consist of several
lines very close to one another.
The fine structure of the lines is explained when account is taken
of the theory of relativity and the own magnetic moment of an elec
tron. The additional energy causing the splitting of the lines is deter
mined by an expression including the dimensionless factor a called
the fine-structure constant and numerically equal to
a—
he = 7.2970 x 10~ 3
Its reciprocal is
— = 137.039
a
Ratio of the charge of an electron to its mass
— = 1.75880 x 1011 C/kg = 5.27274 X 10” cgsQ/g
Compton wavelength. When X-rays are scattered on free electrons

the wavelength changes, owing to the exchange of energy and impulse
between a photon and an electron (the Compton effect). This change
is determined by the formula
AA, = (1 — cos 0) (A.7)
where 0 is the angle of deviation of a photon from its in itial
direction, and A0 is the Compton wavelength:
X0= — = 2.42621 X 10-12 m = 2.42621 X 10-1® cm

mec
Sometimes a quantity obtained by dividing A0 by 2 n is used in
equations:
X= = 3.86144 X 10-13 m = 3.86144 X 10-” cm
Rydberg constant. Formula (9.34) determines the wave numbers
of spectrum lines of atoms similar to hydrogen. The Rydberg con-
CONSTANTS 245
stant R in this formula changes somewhat for different atoms owing

to the difference in the masses of their nuclei. For an infinitely mas
sive nucleus
R„ = - = 1.0973731 X 10? m -i = 1.0973731 X 105 cm- i
4Jt«3c
Bohr radius is the radius of the orbit of an electron in a hydrogen
atom in the normal state according to Bohr’s “classical” theory
ft*
flo = — —5 —5.29187 x 10-n m = 5,29187 X 10~ 9 cm = 0.529187 A
tnee2
Bohr magneton. A definition of the Bohr magneton was given in
Sec. 9.2 [formula (9.6)]. Its more accurate value in the customarily
used units is
jiB = 9.2732 x 10-24 J .T -i = 9.2732 X IO- 2 1 erg-Gs-i
Standard volume of a gas is the volume of a kilomole or mole of
the gas in standard conditions (0°C and 1 atm):
V0= 22.414 m3/kmole (1/mole) = 2.2414 x 103 cm3/mole
Universal gas constant. According to the Clapeyron-Mendeleev
equation (5.2), the universal gas constant can be determined by
the expression
R= ~ (A.8 )
Upon inserting the values of p and T corresponding to standard
conditions, we get the value of R:
R —8.3143 x 103 J/(kmole* deg) = 9.3143 x 107 erg/(mole-deg) =
= 1.9858 cal/(mole» deg) = 8.2053 x 10“ 2 l*atm/(mole*deg)
Boltzmann constant can be determined as the ratio of the univer
sal gas constant to Avogadro’s number
A-= A ^ 1.3805 X 10-23 j/deg = 1.3805 x 10-™ erg/deg
™A
Constant in the Stefan-Boltzmann law [formulas (5.4) and (8.11)]
0 = 5.669 x 10“ 8 W/(m 2 «deg4) = 5.669 x 10-5 erg/(cm2• deg4)
Constant in the Wien displacement law [formula (5.5)]

6 = 2.8978x 10“ 3 m* deg = 0.28978 cm*deg
The Planck radiation law makes it possible to express the con

stants o and b through the constants h, k and c:
a 12nAA (A.9)
1.0823
c2h3
ch
b= 4.9651k (A.10)
(the factors 1.0823 and 4.9651 are obtained as a result of the corres
ponding mathematical operations).
APPENDIX 5
Tables
Table 1 is a summary of the units of geometrical and mechanical

quantities. Table 24 is a similar summary of electrical and magnetic
units. The units and their dimensions are given in four systems. Tab
le 25 gives the equations of electromagnetism, also in four systems.
Tables 2-23 and 26-45 arc conversion tables for units of various quan
tities in different systems, and partly for non-system units. The con
version factors are given, as a rale, with an accuracy to the third
significant digit, an accuracy to the fourth digit being used only for
the units of time. Tables 46-53 are auxiliary ones and contain some
British and U.S. units, units not approved by the relevant standards,
scales of hardness and wind velocity, symbols of units and decimal
prefixes, etc. Tables 54-59 are purely illustrative and have been
included to show the order of the values obtained when measuring
some properties of materials in units of different systems or in the
most widespread non-system units.
LIST OF TABLES
1. Summary Table of Geomet 21. Ileat Transfer Coefficient

rical and Mechanical Units 22. Frequency Interval
Conversion Tables: 23. Musical Intervals
2. Length 24. Summary Table of Elec
3. Area trical and Magnetic Units
4. Volume 25. Equations of Electro
5. Solid Angle magnetism in Different
6 . Angle Systems of Units
7. Time Conversion Tables:
8 . V elocity 26. Charge
9. Acceleration 27. Field Intensity
10. Angular Velocity 28. Surface Density of
11. Mass Charge
12. Force 29. Volume Density of
13. Pressure Charge
14. Work and Energy 30. Displacement
15. Power 31. Displacement Flux
16. Moment of Inertia 32. Potential (Potential DL
17. Moduli of Elasticity and ffercnce)
Shear 33. Capacitance
18. Reduced Pressure and 34. Current Intensity
Concentration 35. Resistance
19. Specific Heat 36. R esistivity
20. Thermal Conductivity 37. Magnetic Induction
TABLES 247
38. Magnetic Flux 50. Temperature Points

39. Magnetic Field Intensity 51. Symbols of Units
40. Magnetomotive Force 52. Prefixes for Multiples and
41. Inductance and Mutual Submultiples of Units
Inductance 53. Symbols of Physical Quan
42. Luminance tities
43. Values of R elative and Abso 54. Modulus of E lasticity
lute Luminous Efficiency at (Young’s Modulus) for Se
Different Wavelengths lected Materials
44. Relationship between Elec 55. Viscosity of Selected Liquids
tron-Volt and Other Units at 20°C
45. Conversion Table—Effective 56. Specific Heats of Selected
Gross Section Substances
46. Beaufort Scale 57. Thermal Conductivities of
47. Hardness Scales Selected Materials
48. Some Most Frequently En 58. Acoustic R esistivity of Se
countered British and' U.S. lected Media
Units 59. R esistivity of Selected Con
49. Some Units and Names of ductors
Units Having a Limited Use
or Not Introduced Officially
TABLE 1. Summary Table of Geometrical and Mechanical Units 248
APPENDICES
Table 1 (continued) TABLED
249
g on on
** In SI and cgs systems. 2. s g a o °2
* In mk(forcc)s system. a c« 8g>
CD* C
*30 *se-
<s d- O
0 CD CD Sg
c
p-o
O
aO §!
CO
g-©
oXT pt-i pco
Q t*3
II
-h
> 0-1 O-
5L
t- t-ii
Si!
a
3CD
P
5 tr-1 I 3
CO p?
1 >1 1
M
3 C2 3 3
'c o '33 co
z 3
CO
3“
1o
o *v a 0-1
S ^,0
3 ffq 3CO
*^3
<J
*& . *cT
33 0<! 3
Table 1 (concluded)
OQ OQ *r
O
Q
OQ
*-
PS2 saiarx
09Z
Table 1 (continued)
SSDiaNLaddV
252 A P P E N D IC E S
TABLE 2. Conversion Table—Length
km | m cm mm u
1 km — 1 103 105 106 109

1 m = 10-3 1 102 103 10«
1 cm = 10-5 1 0 '2 1 10 10«
1 mm = 1 0 -6 10-3 io -i 1 10*
i n 10-9 1 0 -6 10-4 10-3 1
1 nm = 1 0 -1 2 10-9 10-7 10-9 1 0 -6
i A 10-13 1 0 -1 0 1 0 -8 10-7 10-4

1 xu ~ 10-16 10-13 1 0 -1 1 1 0 -io 10-7
1 inch = 2.54x10-5 2.54x10-2 2.54 25.4 2.54x101
1 foot — 3.05X10-4 0.305 30.5 3.05X102 3.05x105
1 mile (na- = 1.85 1.85X103 1.85x105 1.85X106 1.85x109
utical)
TABLE 3. Conversion Table—Area
km2 ha a [m2
1 km 2 = 1 100 104 10«

1 ha = 1 0 -2 1 lO2 104
la = 10-4 1 0 -2 1 102
1 m2 = 1 0 -6 10-4 lO' 2 1
1 cm 2 = 1 0 -1 0 1 0 -8 1 0 -6 10-4
1 mm 2 = 1 0 -1 2 1 0 -1 0 1 0 -8 l ()-6
1 sq. inch = 6.45X10-10 6 .4 5 x 1 0 -8 6.45X 10-6 6 .4 5 x 1 0 -4
1 sq. foot — 9 .2 9 x 1 0 -8 9 .2 9 x 1 0 -6 9.29X 10-4 9 .2 9 x 1 0 -2
1 aero = 4 .0 5 x 1 0 -3 0.405 40.5 4 .0 5 x 1 0 *
1 sq. m ile (nau~ — 3.43 3.43X102 3.43X104 3.43x106
tical)
TABLES 253
mile
nra A XU inch foot (nautical)
1012 1013 1016 3.94x104 3.28x103 0.540

10» 1 0 io 10W 39.4 3.28 5.40X 10-4
lO? 108 io n 0.394 3 .2 8 x 1 0 -2 5 .4 0 x 1 0 -6
10« 107 1010 3 .9 4 x 1 0 -2 3 .2 8 x 1 0 -3 5 .4 0 x 1 0 -7
103 104 107 3 .9 4 x 1 0 -5 3 .2 8 x 1 0 -8 5 .4 0 x 1 0 -1 0
1 10 10* 3.94X 10-8 3.28X 10-9 5.40x10-13
0 .1 1 1033.94X 10-9 3 .28x10-10 5.40x10-14
10-4 10-3 1 3.94x10-12 3.28X10-13 5.40x10-17
2.54x10? 2 .5 4x1 0 8 2.54X1011 1 8.33X 10-2 1.37X 10-5
3.05x108 3.05X109 3.05X1012 12 1 1.65X 10-4
1.85X1012 1.85x1013 1.85X1016 7.29 x 1 0 4 6.08X103 1
cm2 mm2 sq. inch sq. foot acre sq. mile

(nautical)
1 0 io ] 012 1.55x10* 1 .08x107 2.47X102 0.292

108 10io 1.55X107 1 .08x106 2.47 2 .9 2 x 1 0 -3
106 108 1.55X106 1 .08x103 2 .4 7 x 1 0 -2 2 .9 2 x 1 0 -5
104 10« 1.55X103 1 0 .8 2 .4 7 x 1 0 -4 2 .9 2 x 1 0 -7
1 102 0.155 1.08X 10-3 2.47X 10-8 2.92X 10-H
1 0 -2 1 1.5 5 x 1 0 -3 1 .0 8 x 1 0 -5 2.47X10-10 2.9 2 x 1 0 -1 3
6.45 6 .4 5 x 1 0 -2 1 6 .9 4 x 1 0 -3 1.5 9 x 1 0 -7 1 . 8 8 x 1 0 -1 0
9.29X102 9.29x104 1.44X102 1 2 .3 0 x 1 0 -5 2 .7 1 x 1 0 -8
4.05X107 4 .0 5 x 1 0 9 6.27X106 4.36X 104 1 1.18X 10-3
3.43X1010 3.43x1012 5 .3 2 x 1 0 * 3.69x107 8.47x102 1
*54 APPENDICES
«M O *I*Ir-I«
O OC5
gallon ^ ^^^S
1 UK
x
0CM ? J
*O0CMx x^x^N.^,
0cM CO
CM CMCM00
(M iO 05
1 1 1 1
cu. foot
CO^ ^
o o o o
^
xOx x00xooxOi^ S '
C CO
lOLOlOl>- o
CO00COLO
«!#» (M
© IIiO 051 -M
cu. inch
o o o o
x s x x ^ xcoxn
-O*r-l O-O «r-t t'~-t'~-
CO COCO ^ C M*
©r- to
© o
w
S o S ^
rH-rH^-h CO X
LO
O X X
S •00L
T"H
OO
CM^
■«* o
eo
o
so <o ÔCOfOLO
O
Ô
eo ^ ^ X X
^ ^— 100LO
CM
(N 1
«
TABLE 4. Conversion Table—Volume
s3
T
S 2 2 C^
O^r
th
•f15<me1o
1O
C
O n a OO■
to •rH
a ■*H ©
—t C
VO fC
00LO
OL
o o X X X
O
th’cSIN?
11 II 11 II 11 II 11
G
O
cr- *g
o o ri in
a «.2 . s s
a - § a g gp
T-l-r-lT-l
TABLES 255
IO T# Cl oo co*^ n n
OOO <o
1 1 1 1 I
tH*“« 0
otHotHotHotHt
oH
vA A NXA V 2?o ZT;L COO
O H< CM th
0
8
year
th lO X th th th X X X X X th
COt h 00 00 ^
05 th tH00 t
LO
COTHtH CDO^CO H
thC3tH|> 05
CM COTHtHCMTH
COCM
O OOH O1 Ot^H
H thi M
0 X tXh “?X 05 X 0 0
tHTH
^ 0
tH COto 00
COth 0 ©o o
COTH CO
week
CM x x x ^ S
1 T|' CM COth CM>H cM
^ 0 0
t^TlTlTl rH
0
V
,
?
„
COC
lO M05
05 iOO• lQ
bn cotJ X X X 0 ^ 0 0 THOilO
co LO05 *^H th th th
00 0
|
th CO
CMTf 50 to T* (M
1
0 0 0
1 ! _ 1 1 1
CO^ "H^ 7 7 ? oth t
o-ht-h
o cM
J ^XXX ^
day
^ 0 0 0 x x x ^ t,^ -
O tHLO05 tH tH th i>O co
tH00 O lOvf
th tHCO 05 C
rH CO
W1 1U31h eIo I O I I> th CO
05 ti thh ^0
H LM
COtC5 CZ5
h th
C5
tH
rev
^ X X X th d X X X
0 00 00 C M LO iQ lO T* <M
t- co r- CM CM CM o o o
(Nsfl>
10 W 50 u) (o
O OH O O x x ^ ^ g x
THT TH THOtH ^^ A
X X o-H X X X vf SJ £ 5 -g
% c—CO t>-
COCO^ O cvi d CMt-I 00
<CM
= > c6 «
TH
^
COCO
CO 0
CM
eo 1 hi eo eo e1o *o
O O O O sF 1 cm
TH th tH th 0 iO 0 ot1H O Oo
- Xra X X Xsji d TJ tHtHTH
TABLE 6. Conversion Table-*Angle
NtjH
fH C— CHO 0 L O A
min
V CO T VJH V jH X «oX X X
CO TH CM uo LO t>. CO 00 ^
C
COO sP OC
•O ^M
C1M 1 C M C1O to1
0 0 0 Q 0TH 0 TH th lO
CO T HtH tH ^ th
0
tC ^ X X X X X
10 t'- 00 O
COt'- CO Q
05O 05 05
tHCM CO
COTf lO
<5
THotHToHoTH
CM CO CM HI «e
O O O 00 th o X X X X
tH^-h^-h t>- 0th 0
th 0 th w co CO vfO00
■*<C
toO
rad
^ XXX ^ " X X X •C
LQtHLO O tH t- t"- ” 06^ co co
05 00 lo m lo
th CM th th th
(met
1 revolution
ric sec.) =
=
=
=
(circle) =
angle) =
IS (gon) =
=
=
l c (metric
=
=
=
=
=
=
(right
1 week
min.)
1 year
1 min
1 day
1 rad
1 L.
1 S
1 h
lcc
1"
1°
V
256 APPENDICES
TABLE 8 . Conversion Table—Velocity

m/s m/min cm/s km/h knot
1 m /s = 1 60 102 3.6 1.94

1 m/min = 1.67x10-2 1 1.67 6x 1 0 - 2 3 .2 4 x 1 0 -2
1 cro/s - 1 0 -2 0 .6 1 3 .6 x 1 0 -2 1.94x10-2
1 km/h = 0.278 16.7 27.8 1 0.540
1 knot = 0.514 30.9 51.4 1.85 1
TABLE 9. Conversion Table —Acceleration

m/s2 cm/s2 8
1 m /s 2 = 1 102 0 .1 0 2
1 cm /s 2 = 1 0 -2 1 1 .0 2 x 1 0 -3
1 8 = 9.81 9.81X102 1
TABLE 10. Conversion Table—Angular Velocity

rad/s rev/s | rev/min °/sec
1 rad/s = 1 0.159 9.55 57.3

1 rev/s = 6.28 1 60 3 .6 x102
(rps) .
1 rev/min = 0.105 1.67x10-2. 1 6
(rpm)
1 °/s = 1.75X10-2 2.78x10-3 0.167 1
TABLE 11. Conversion Table—Mass
kg g i (turn) ton
1 kg = 1 103 0 .1 0 2 10-3
1 g = 10-3 1 1.02X10-4 1 0 -6
1 i (turn) = 9.81 9.81x103 1 9 .8 x 1 0 -3
1 ton = 103 10« 102 1
TABLE 12. Conversion Table—Force
N dyn kgf sn
IN = i 105 0 .1 0 2 10-3
1 dyn = 10-5 1 1 .0 2 x 1 0 - 8 1 0 -8
1 kgf = 9.81 9.81X105 1 9.81x10-3
1sn (stbene) = 10* 108 102 1
TABLE 13. Conversion Table—Pressure
1/2
17—1040
TABLES
257
TABLE 15. Conversion Table—Power 258
APPENDICES
TABLES 259
TABLE 16. Conversion Table—Moment of Inertia

kg-m 2 g e m 2 i •m 2
1 kg-m2 1 107 0.102

1 g*cm2 10-7 1 1 .02X10-8
1 i-m 2 (tum-in2) 9.81 9.81x107 1
TABLE 17. Conversion Table-Moduli of Elasticity and Shear

| N /m 2 d y n /cm 2 k g f/rn 2 |I k g f/cm 2 k g f/m m 2
1 N/m2 I 10 0.102 1.02x10-6 1.02x10-7
1 dyn/cm2 - 0.1 1 1.02x10-2 1.02x10-6 1.02x10-8
1 kgf/rn2 9.81 98.1 1 10-4 10'6
1 kgf/cm2 9.81 X 104 9.81x105 10* 1 10-2
1 kgf/mm2 -- 9.81x106 9.81x107 106 102 1
TABLE 18. Conversion Table—Reduced Pressure and

Concentration
cin-;;
molc/l
nr* i-i (k moJe/m*)
1 N/m 2 2.60 < 10-6 2.00 x 10” 2.00 XJ OH 4 .4 2x10-7

1 dyn/cm2 2.00X10*9 2.00x1016 2.00x1043 4 .4 2 \1 0 -8
1 atm 2.09x1025 2 .0 9 x 1 0 -- 2.09x1049 4 .4 0 X 1 0 -2
1 imn Hg 8.54 < 10— 3 .5 4 x l0 i» 3.54/J 0 4 6 5.87 x 10-5
TABLE 19. Conversion Table—Specific Heat
J/kg-dcg erg/g-deg keal/kg •deg

(cal/g- deg)
1 J/kg-deg 1 104 2 .3 9 x 1 0 -4
1 erg/g'deg = 10-4 1 2 .3 9 x 1 0 -8
1 keal/kg-deg (cal/g-deg) = 4.19X103 4.19x107 1
17*
260 APPENDICES
TABLE 20. Conversion Table—Thermal Conductivity

W/m ■deg erg/cm •deg kcal/m- h- deg cal/cm-s-deg
1 W/m-deg = 1 106 0.860 2 .3 9 X 1 0 -3

1 erg/cm *s* deg — 10-5 1 8 . 6 0 x l 0 “6 2 .3 9 x 1 0 -8
1 kcal/m *h* deg = 1 .1 6 1.16X 10® 1 2 .7 8 X 1 0 -3
1 cal/cm*s*deg = 4 .1 9 X 1 0 2 4 .1 9 X 1 0 7 3 .6 x 1 0 2 1
TABLE 21. Conversion Table—Heat Transfer Coefficient
bo
bo o> bo
T033 'O 0J3
T
bo m V)
TC3D C) a
cj
oe
1 eo
s bo o
£ <v o
1 W/m2-deg = 1 lO3 0.860 2 .3 9 x 1 0 -5

1 erg/cm2*s* deg = 10-3 1 8.6 0 x 1 0 -4 2 .3 9 x 1 0 -8
1 kcal/m2*h* deg = 1.16 1.16x10* 1 2 .7 8 x 1 0 -5
1 cal/cm2*s*deg = 4.19 4.19X 107 3.60x 1 0 4 1
TABLE 22. Conversion Table—Frequency Interval
savart octave millioctave cent
1 savart = 1 3.32x10-3 3.32 3.98

1 octavo — 301 1 1000 1200
1 millioctave = 0.301 lO"3 1 1.2
1 cent = 0.251 8.33x10-4 0.833 1
TABLES 261
TABLE 23. Musical Intervals

Natural scale Tempered scale
Name Name of interval Fre
of tone relative to “C” quency Inter Inter Inter I ter-
relative val in val in val in val in
to that savarts cents savarts cents
of “C”
C Unison 1 0 0 0 0
D Major tone 9/8 51.1 204 50.2 200
E Major third 5/4 96.9 386 100.4 400
F Fourth 4/3 125.0 498 125.4 500
G Fifth 3/2 176.1 702 175.6 700
A Major sixth 5/3 221.9 884 225.8 900
H( B) Seventh 15/8 273.0 1088 276.0 1100
C Octave 2 301.0 1200 301.0 1200
TABLE 24. Summary Table of Electrical and Magnetic Units
Defining relationships in systems Dimension

Quantity Symbol
SI and cgsm cgs SI
Quantity of elec Q Q=*t Q=r Y~ftr TI

tricity (charge)
Intensity of elec E f lu r-3/-i

trical held ' Q
Electric displace D D=s0«rE D = trE L--T1

ment
Electric flux N j y . -1)A TI
Potential V U =
EP L2A/T-3J-1
Q
—> —►
Dipole moment P P= = Ql LTI
Surface density of a Q L -2TI

charge A'
Volume density Q
P P= V
L-3T1
of charge
Q
Capacitance C c -
u L-2AI-1T4J2
—> —
> V
Dielectric polari V V— V L-2TI
zation
Absolute permit c e —8oe r L-3A/-1T4I2

tivity
er ~ 1
Dielectric sus Xc=e0 (er- l ) Xe in
L-ZM -1T4I*
ceptibility
f 2jir/
Current intensity I
I= y V-o^r1 - 4 - I
1
TABLES 263
Name and sym

formulas in systems bol of unit in
systems
cgsô cgs CgS £o SI cgs
L H2M U 2^~\/2 L 3 /2 M H 2 T-i L3/2A;l/2 r -lpl/2 coulomb -

(C)
L l / 2 M l / 2 T - 2 tL1 / 2 7- l / 2 iUl/2 r- le-- 1/2 volt pri -
me tie
L- 3 / 2 M l J 2 ^ - i / 2 L- U 2 h iU 2 T- \ L 1/ “A/1/ 2y1—1eJ / 2 coulomb —
per sq.
metre
L U 2 A [ l / 2 ll ~ i / 2 L3/2A/l/2 r—l L3/2tV;l/2 r - l e l/2 coulomb —
(Q
L 3 / 2 M U 2 T- 2 l l i / 2 X.1/ 2jv/ 1/ 2r—1 Ll/2A/l/2 T- l e~ l/2 volt (V) -
i ,3/2m 1/2j1-1 /2 L5/2A/1/2T-1 L5/2A/l/2 r~ l £^l/2 coulomb- -

metre
L-3 /2 Afl/2 tl- l / 2 L- l / 2 Ml/2 r- l L- l / 2 Ml/2 r- l e l/2 coulomb ■
per sq.
metre
(C/mii)
L- 5/2^1/2^-172 L~ 3/2Ml/2 r—1 L- 3 /2 3fl/2 T- ic- l / 2 coulomb
per cu.
metre '
(C/im<)
L Leo farad (F) centi
metre
(cm)
£—a/2^1/2^—1/2 L~ 1/2m1/2T--1 coulomb
per sq.
metre
(C/rri2)
L-22’2 ^ 1 1 eo farad per —
metre
(F/m)
L”2T’2jx^~1 1 co farad per -
metre
(F/m)
L ^ 2 M U 2 T ~ 1 ]1- 1 J 2 L3/2;Ul/2 r^2 p3/2Ail/2 T- 2 gl/2 ampere (A) -
Sym Defining relationships in systems Dimension

Quantity bol
SI and cgs in [ cgs SI
1
Current density J L~2J
A
U
Resistance R R= L2MT-3J-2
1
Conductance G G=
1 L-2M-1T3J2
R
Magnetic induc B M T-zi-l

tion B = 1 iT b = c TT
Magnetic flux <D 0>= BA L2MT-21-1
Intensity of mag H 11= —— tf = — L~U

netic field POP/ Pr
->
Magnetic moment Pm v M 1 L21
B c
Magnetomotive —
c 4irX I
F 1
fe. *1
W
II
force
Inductance and L(M) c^r c2% i L2MT-21-2
Jl
1!
1
mutual induc 1 dl/dt L ~- J ~ d I Jd t

tance
Magnetization J Pm
J = L-n
V
Absolute permea P P = PoPr Pr LMT-2j-a

bility
Magnetic suscep Xm 7 _ Pr - 1 1
tibility 1
Notes. 1. All defining relationships are given for the simplest cases —homogene-
2. The dimensions in the cgs pq and cgs eo systems differ from the dimen-
the dimensions po and eq•
TABLES 2b5
Table 24 ( continued)
Name and sym
formulas in systems bol of unit in
systems
cgs ho cgs 1 cgs e0 SI cgs
l -3 /2 U \ l 2 T- \ y - H 2 L- l / 2 Ml/2T- 2 L- l / 2 Ml/2 T- 2 e(l/2 ampere -

per sq.
metre
(A/m2)
L-lT L -m -1 ohm (Q) -

LT-ljlO
LT-l LT-leo siemens -

13.o
(S)
L- l / 2 Ml/2 r- V l / 2 L—l/2Ml/2 r ~ 1 L-3 /2 Ml/2e- 1/2 tesla (T) gauss

(Os)
L3/2Jvll/2 r- l lxl/2 L 3/2M 1/2T- 1 Ll/2A/l/2e- l / 2 max

wober
(Wb)well
(Mx)
L- l / 2 3/l/2 T- l jl- l / 2 L- l / 2 .v l/2 r - l Ll/2M1/2r- 2 eJ/2 ampere per oersted
metre (Oe)
(A/m)
L5/2^1/2r-iM- l / 2 L5/2iUl/2 r- l L7/2Ml/2 r- 2 e U2 ampere-
sq. metre
(A •m2)
Ll/2^ 1/2r- l L3/2Ml/2 r- 2 € L/2 ampere (A) gilbert
Ll/2 Afl/2 T- lj i- l/2 ampere- (Ob,
turn (At)
L jlio L L-1T26"1 henry (H) centi
metre
(cm)
L-l/2 Ml/2 T-lM- l / 2 Ll/2Jwl/2 T- 2 e l/2 ampere —

1,— U 2 j \ i U 2 > T ~ 1 per metre
(A/m)
MO 1 henry per
metre
(H/m)
1 l 1
ous fields, non-varying currents (except for the e.m.f. of induction), etc.
sion formulas of the same units in the cgsc and cgs in systems hy the addition of
266 A PPE N D IC E S
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s £o 2^ SO +j■* S.d
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U fa & o S ft 5 ft
TABLES 267
e
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<m k j
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£ £U s> 1! S £
c3 -Oft | | ^
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cr Cc:ft 'S
ft V c .5
O o o J +c-»
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£ £ UC -S £« o
£ «-■ c T3
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co Kft* opj+> ->
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fit A II «M
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g o
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q
« o o L- -
sS cO cS 13 S A
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ft • PC ft ft ft
18*
Table 25 (continued)
268
APPENDICES
TABLES 269
Table 25 (concluded) 270
APPENDICES
TABLES :\i\
TABLE 26. Conversion Table—Charge
c cgs (cgsc) runmi
1 c = 1 3 x l0 » 0.1
1 cgs (cgse) = 3 .3 4 x lO -io 1 3.34X 10 11
1 cgsm = 10 3X1010 1
TABLE 27. Conversion T**ble—Field Intensity
V/m Y/cm cgs (cgsc) cgsm
1 V/m = 1 10-2 3.34x1 0 -* 10«

1 V/cm — 102 1 3.34x10-3 108
legs (cgse) ~ 3X104 3x102 1 3 x l0 io
1 cgsm = 10-o 10-8 3 .3 4 X 10-11 1
TABLE 28. Conversion Table—Surface Density of Charge
C/m2 cgs (cgse) cgsm
1 C/m2 1 3X105 10-5

1 cgs (cgse) = 3 .3 4 x 1 0 -6 1 3.34x10*11
1 cgsm — 105 3 X l0io 1
TABLE 29. Conversion Table—Volume Density of Charge
G/m3 cgs (cgsc) cgsm
1 C/m3 - 1 3X103 10-7

cgs (cgse) = 3.34 x 1 0 -4 1 3.34x10-11
cgsm = 10’ 3x1010 1
272 APPENDICES
TABLE 30. Conversion Table—Displacement
C/m2 cgs (cgse) cgsm
1 C/m2 = 1 3 .7 7 X 1 0 6 1 .2 6 X 1 0 -4
1 cgs (cgse) = 2 .6 5 x 1 0 -7 1 3 .3 4x10-11
1 cgsm = 7 .9 6 x 1 0 3 3X101® 1
TABLE 31. Conversion Table—Displacement Flux
c cgs (cgsc) cgsm
1 C = 1 3.77x1010 1.26
1 cgs (cgse) = 2.65x10-11 1 3.34x10-11
1 cgsm = 0.796 3x1010 1
TABLE 32. Conversion Table—Potential (Potential Difference)
V cgs (cgse) cgsm
1 V = 1 3 .3 4 x 1 0 -3 108
1 c g s (c g se ) = 300 1 3X101°
1 cgsm — 1 0 -8 3 . 3 4 X 1 0 - 11 1
TABLE 33. Conversion Table—Capaci tance
F cm (cgs, cgsc) cgsm
1F = 1 8.99x1011 10-9
1 cm (cgs, cgse) -- i.lix lo -ii 1 1.11X10-21
1 cgsm UP 8.99x1020 1
TABLES 273
TABLE 34. Conversion Table—Current Intensity
A cgs (cgse) cgsm
1 A 1 3x100 0.1
1 cgs (cgse) - 3.34X10-10 1 3.34X 10 - 11
1 cgsm 10 3X1010 1
TABLE 35. Conversion Table—Resistance
Q cgs (cgse) cgsm
i a = 1 l.llxio-^ 109
1 cgs (cgse) 8 .9 9 X 1 0 1 1 1 8.99x1020
1 cgsm = 10-9 1.11x10-21 1
TABLE 36. Conversion Table—Resistivity
£>• mm2
S2- m Q-cm cgs (cgse) cgsm
m
lQ * m 1 102 106 1. 1 1 X 10-10 io n

1Q • cm — 10-2 1 104 1.1 1 x 1 0 -1 2 109
, a -m m 2
1------------ — 10-6 10-4 1 I .llx l0 - 1 « 105
m
1 cgs (cgse) -r_- 8 .9 9 x 1 0 9 8.99x1011 8.99x 1 0 1 0 1 8.99x 1 0 * 0
1cgsm 10- u 10-9 10-5 1.1 1 x 1 0 -2 1 1
TABLE 37. Conversion Table—Magnetic Induction
T Gs cgse
1 T = 1 104 3.34x10"?
1 Gs = 10-4 1 3.34x10-11
1 cgse — 3x100 3x10io 1
274 APPENDICES
TABLE 38. Conversion Table—Magnetic Flux
Wb Mx cgse
1 Wb = 1 108 3 .34X 10" 8

1 Mx = 108 1 3 .3 4X 10-H
1 cgse = 3X102 3X1010 1
TABLE 39. Conversion Table—Magnetic Field Intensity
A/m Oe cgsc At/cm
1 A/m = 1 1.26x10-2 3.77X108 10-2

1 Oe = 79.6 1 3x1010 0.796
1 cgse = 2.65x10-9 3.34X 10-H 1 2.65x10-11
1 At/cm = 102 1.26 3.77x1010 1
(ampere-turn
per centi
metre)
TABLE 40. Conversion Table—Magnetomotive Force
A Gb cgse
1 A = 1 1 .2 6 3.7 7 X 1 0 1 0
1 Gb = 0 .7 9 6 1 3X1010
1 cgse — 2 .65x10-11 3 .3 4 x 1 0 -1 1 1
TABLE 41. Conversion Table—Inductance and Mutual Inductance
H cm (cgs, cgsm) cgsc
1 H = 1 10» ■l.llxio-12
1 cm (cgs,
cgsm) = 10-9 1 1.11x10-21
1 cgse = 8.99x1011 8.99x102° l
Cables 275
TABLE 42. Conversion Table—Luminance*

nt Sb asb L
1 nit = 1 10-4 3.14 3.14X10-4

1 stilb = 104 1 3.14X104 3.14
1 apostilb = 0.318 3.18x10-5 1 10-4
1 lambert = 3.18x103 0.318 104 1
* The values of the stilb, apostilb and lambert are frequently given in
tables not on the basis of the SI candela, but on the basis of the interna
tional candle, which is 1.005 times greater (sec Sec. 8.3). Here the fac
tors converting the nit into Sb, asb and L should be divided by 1.005
(9.95x10-5 Sb, 3.13 asb and 3.13x10-4 L), and the factors converting
the stilb, apostilb and lambert into nt multiplied by 1.005 ( 1.005x 104 nt,
0.320 nt and 3.20x 10*1nt).
TABLE 43. Values of Relative and Absolute Luminous
EfYicicncy at Different Wavelengths
Y (A) T)<v(lm/W) | Y (A) ri ?(lm/W)
VY
3 800 0.00004 0.03 5 800 0.870 594

4 000 0.0004 0.27 6 000 0.631 431
4 200 0.004 0.73 6 200 0.381 260
4 400 0.023 15.7 6 400 0.175 120
4 600 0.060 41.0 6 600 0.061 41.7
4 800 0.139 90.2 6 800 0.017 11.6
5 000 0.323 221 7 000 0.0041 2.8
5 200 0.710 485 7 200 0.00105 0.72
5 400 0.954 652 7 400 0.00025 0.17
5 600 0.995 680 7 600 0.00006 0.04
TABLE 44. Relationship between Electron-Volt and Other Units
Defining Reciprocal expres

relationship Unit 1 eV = sing given unit
in eV
f J 1 .6 0 X 1 0 -19 6.24x 10 18
1!
I erg 1 . 6 0 x l 0 “12 6.24X1011

°K 7 .7 3 x 1 0 s 1.29 X 1 0 -4
i kT=eV
kT = eU °K 1 .1 6 x 1 0 4 8 .6 2 X 1 0 -5
Q --eU N A
cal/mole 2 .31X 104 4.3 6 x 1 0 -5
(kcal/kmole)
* - = e U cm'1 8 .0 7 x 1 0 3 1 .2 4 x 1 0 -4
VY
h\ eU S'1 2 .4 2X 1014 4.13X 10-16
1 amu c ' Z ^ e U e 1.07 X10~9 9.32X 108
m ec2 = e U me 1.95X 10-6 5.11X 105
R ch = eU Ry
7.35X 10-2 13.6
TABLE 45. Conversion Table—Effective Cross Section 276
APPENDICES
TABLES 277
TABLE 46. Beaufort Scale
Beaufort Beaufort Velocity, m/s

number Velocity, m/s number
0 0 -0.5 7 12.5-15.2
1 0 .6 -1 .7 8 15.3-18.2
2 1.8 -3 ,3 9 18.3-21.5
3 3 .4 -5 .2 10 21.6-25.1
4 5 .3 -7 .4 11 25.2-29.0
5 7 .5 -9 .8 12 > 2 9 .0
6 9.9-12.4
TABLE 47. Hardness Scales
Hardness number Hardness number

Mineral Mohs1 Brcit- Mineral Mohs’ Brcit-
scale haupt’s scale haupt’s
scale scale
Talc 1 1 Hornblende 7
Gypsum 2 2 Felspar 6 8
Mica — 3 Quartz 7 9
Lime felspar 3 4 Topaz 8 10
Fluorite 4 5 Corundum 9 11
Apatite 5 6 Diamond 10 12
TABLE 48. Some Most Frequently Encountered British

and U.S. Units
Acre— a unit of area, equal to 4040.86 m2.

Barrel— a unit of volume (capacity). There arc distinguished
a dry barrel, equal to 115.028 1, and a petroleum oil
barrel, equal to 158.988 1.
B.t.u. (British thermal unit) —a unit of work or energy, equal
to 1.055X 103 J.
Bushel— a unit of volume (capacity). A U.K. bushel equals
36.36871, a U.S. bushel equals 35.2393 1.
Gallon— a unit of volume (capacity). A U.K. gallon equals
4.546091, a U.S. gallon equals 3.78543 1.
278 APPENDICES
Table 48 (concluded)
Grain — a unit of weight, equal to 0.0648 g.

Mil — a thousandth of an inch, equal to 2.54 p.
Mile ( U . K . ) - 1609.344 m.
Ounce (Avoirdupois)— a unit of mass (1/16 U.K. pound), equal
to 28.3495 g.
Pound —a unit of mass, equal to 453.5924 g.
Poundal—a unit of force, equal to 1,38255 x 104 dyn.
Quart —a unit of volume (capacity). A U.K. quart equals
1.13650 1, a U.S. dry quart equals 1.1012 1, and a liquid
quart equals 0,94633 1.
Ton— a unit of mass. A U.K. (long) ton equals 2240 pounds or
1010 kg; a U.S. (short) ton equals 2000 pounds or 907.2 kg.
Yard— a unit of length, equal to 3 feet or 0.9144 m.
TABLE 49. Some Units and Names of Units Having a Limited

Use or Not Introduced Officially
Acohm — acoustic ohm— cgs unit of acoustic resistance.

B iot— cgsm unit of current intensity, equal to 10 A.
E otvos— unit of the gradient of the force of gravity, equal to a
change in the acceleration of gravity by 1 cin/s2 per
centimetre.
Franklin— cgs unit of charge, equal to 3 . 3 4 x l 0 10C.
Gal — unit of acceleration, equal to 1 cm /s2.
Inerta— unit of mass in the mk(Force)s system, equal to 9,81 kg.
Kayser —unit of wave number, equal to cm”1.
Kilopond— name of kilogram-force used in German literature.
Lenz — unit of magnetic field intensity, equal to 1 A/m.
Magn— unit of “absolute1’ permeability, equal to 107/4 jx F/m.
Mes— unit of velocity, equal to 1 m/s.
Mho— unit of conductivity. The same as siemens.
Pascal — unit of pressure equal to 1 N/m 2.
Point (of the compass) — a unit of angle used in navigation,
equal to 1/32 of a circle, i. c., 11.25°.
Rhe— cgs unit of fluidity. The fluidity of a liquid having a vis
cosity of 1 poise.
Uranium u n it— a unit of a -a c tiv ity — the activity of the oxide
U30 8 with a density of surface coating of 20 m g/cm 2.
A uranium unit creates an ionization current in air having
a density of 5.78 X 10~13 A/cm2.
TABLES 279
TABLE 50. Temperature Points

A number of temperature points supplementing the reference
points given in Sec. 5.3 have been selected for practical reproduc
tion of separate sections of the thermodynamic temperature scale.
These supplementary temperature points, the same as the reference
ones, are the points of equilibrium of two or three phases of a given
substance. A number of such points are given below, it being indi
cated between what phases equilibrium serves for establishing the
given point. In the equilibrium of two phases the pressure is equal
to a standard atmosphere. The temperatures arc given in °C.
Carbon dioxide — solid Mercury— liquid and

and vapour . . . . - 7 8 .5 vapour .................... 356.58
Mercury— solid and Aluminium—solid and
liquid .................... - 3 8 .8 7 liquid .................... 650.1
Ice and water . . . . 0.000 Copper— solid and li
Diphenyl oxide— trip- quid ........................ 1083
le p o i n t .................... 26.88 Nickel —solid and li
Benzoic acid— triple quid ........................ 1453
point ........................ 122.36 Cobalt — solid and li
Indium— solid and quid ........................ 1492
liquid .................... 156.61 Palladium — solid and
Naphthalene — liquid liquid .................... 1 552
and vapour . . . . 218.0 Platinum — solid and
T in— solid and liquid 231.91 liquid .................... 1768
Benzophenone—liquid Rhodium— solid and
and vapour . . . . 305.9 liquid .................... 1 960
Cadmium— solid and Iridium — solid and
liquid .................... 321.03 liquid .................... 2 443
Lead—solid and liquid 327.3
TABLE 51. Symbols of Units
Unit Symbol Unit Symbol Unit Symbol
Ampere A Bar bar Coulomb C

Angstrom A Barn b Curie c
Apostilb ash Bel B Day d
Are a Bit bit Debye D
Atmosphere Bohr mag Decibel dB
(standard) atm neton Pb Degree °K (deg)
Atmosphere Calorie cal Dioptre D
(technical) at Candela cd Dyne dyn
Atomic mass Cent c Einstein E
unit amu Centimetre cm Electron-volt eV
Unit Symbol Unit Symbol Unit Symbol
Erg erg Mile (nau- Rad rad

Farad F tical) n. mile Radian rad
Fermi f Millimetre Roentgen r
Gal G of mercury mm II g Rutherford Rd
Gamma y Millimetre Rydberg Hy
Gauss Gs of water mm II20 Savart Sav
Gilbert. Gb Minute min Second s
Gram g Minute (an- Second, an
Henry H / gular n
gular)
Hertz Hz Mole mole Siemens S
Horsepower hp Neper n Ste radian sr
Hour h Newton N Sthenc sn
Joule J Nit nt Stilb Sb
Kilogram kg Octavo ocl. Stokes St
Knot kn Oersted Oe Tesla T
Lambert L Ohm Q Ton ton
Litre 1 Parsec pc Torr torr
Lumen 1m Phon P Volt V
Lux lx Phot ph Watt W
Maxwell Mx Picze pz Weber wb
Metre m Poise P X unit XU
Micron (mi Quintal q
crometre) p (pm)
TABLE 52. Prefixes for Multiples and Submultiplcs of Units
Name Multiple of Symbol Example

basic unit
Tera 1012 T tcrajoule TJ

Giga 1()» G giganewton GN
Mega 106 M megaohm (megohm) M
Kilo 103 k kilogauss kGs
Hecto 102 h hectowatt hW
Dcca 10 da decalitre dal
Deci 0.1 d decimetre dm
Centi 10-*2 c ccntipoise cP
M illi 10-3 m milliampero mA
Micro 10-6 P microvolt pV
Nano 10-9 n nanosecond ns
Pico 10-12 P picofarad pF
Fern to 10— 15 f fern tog ram fg
Atto 10-13 a attocoulomb aC
TABLES 281
TABLE 53. Symbols of Physical Q uantities

The present table includes quantities encountered in the general
course of physics and related subjects. The symbols used are gene
rally those recommended by international organizations for various
branches of science and engineering, or have been taken from the
most well known textbooks on physics.
According to existing rules, capital and lower case letters may
be interchanged where this is expedient and does not lead to
confusion.
Quantity Symbol Quanti (y Symbol
Acceleration a Dielectric constant

Acceleration, angular a (relative perm itti
Action S, H vity) d
A ctivity of radio- Difference of poten
active source A tials //
Amplitude A Displacement h
Angle a, q>, if), 0 Efficiency h
Area A E led ric coiisl nn 1 (per
Avogadro’s number m illiv ily nf vaeu
Boltzmann’s constant /r M i l l) eo
Capacitance (' Energy Et W
Charge, electric O Energy, free F
Charge of election e Energy, internal U
Coefficient, friction Cfn f Energy, kinetic Ek
Coefficient, heat Energy, potential
transfer a Entropy
Coefficient, recombi Faraday’s constant F
nation A Eield intensity, elec
Concentration u tric.ii 1 F.
Conductance G Field intensity, mag
Curvature P netic //
Curvature, Gaussian I< Fluidity T
Density P Flux, electric ND
Density, charge, l i Flux, luminous O
near T Flux, magnetic (p
Density, charge, sur Flux, radiant CD,
face a Flux linkage xy
Density, charge, vo Force F,P< Q, R
lume P Force, electrom otive %
Density, electric cur Force, magnetomotive F
rent J Frequency v, /
Density, energy e Frequency, angular (a
Density, energy, vo Gas constant li
lume u Gravitational con
Diameter D, d stant G
19—1040
Table 53 (continued)
Quantity Symbol Quantity Symbol
Heat flow 0 Moment of inertia

Heat of phase conver (polar) h
sion (fusion, evapo
ration) H, r Moment, magnetic Pm
Illum ination E Moment of m om en-,
Impedance Z turn (angular mo
Impulse p mentum) X
Inductance L Moment, statical s
Inductance, mutual M Momentum p (mv)
Induction, magnetic B Musical interval 1Im
Intensity, current I Number n, N
Intensity, luminous I Period T, T
Intensity, sound I Permeability, relative Hr
Length I Perm ittivity, relative 8r
Length of free path I pH index pH
Length of light wave % Planck constant h
Level, sound loudness Ln Planck-Dirac constant h
T
Level, sound pressure
Luminance / Polarization, dielec
Luminous efficiency, tric ~p
absolute n Polarization, of mo
Luminous efficiency, lecule a
relative V Potential V
Luminous omittance R Power P
Magnetic constant Power, lens D
(permeability of Poynting vector S
vacuum) Ho
Pressure P
Magnetization J Quantity of heat Q
Mass m Quantity of illu m i
Mass, electron me nation H
Mass, relative mole Quantity of light Q
cular (molecular Radiation dose D
weight) M Radius r
Mass flow v0™
m Reflection factor P
M obility b Refraction index n
Modulus, elasticity Reluctance Pm
(Young’s modulus) E Resistance, acoustic Pa
Modulus, shear G Resistance, electrical R
Resistance, mechani Pm
Moment, dipole P cal, of acoustic
Moment of force M system
Moment of inertia R esistivity P
(axial) Jz Solid angle Q
Moment of inertia
(dynamic) I
TABLES 283
Quantity Symbol Quantity Symbol
Specific heat c V elocity (speed) V

Spin number s Velocity, angular (0
Surface tension o Velocity, of light c
Susceptibility, dielec Viscosity, dynamic 'll
tric X(e Viscosity, kinematic V
Susceptibility, magne Volume V
tic Xm Volumetric flow rate Qv
Temperature /°, 0 Wave number V, 0
Temperature, absolute T Weight G
Thermal diffusitivity a Weight, specific y
Time t, X Work w
TABLE 54. Modulus of E lasticity (Young’s Modulus)

for Selected Materials (Mean Rounded off Values)
Material E Material E
Aluminium 7 Rubber 0 .5
Copper 12 Quartz 5
Steel 20 Lead 1.6
To obtain the values of the inodulus in N/m 2 thie numbers in

the E column sh()uld be m ultif died by 1010, in dyn/cm 2— by
1011, in kgf/m 2— by 10», in kgf/ cm2— by 105 and i;n kgf /mm2—
by 103.
TABLE 55. Viscosity of Selected Liquids at 20°C
Viscosity in Viscosity in
Liquid Liquid
Ether 0.23 Ethyl alcohol 1.19

Methyl alcohol 0.59 Mercury 1.59
Benzene 0.65 Glycerine 850
Water 1.01
19*
284 APPENDICES
TABLE 56. Specific Heats of Selected Substances

Specific heat Specific heat
Substance J/(kg-dcg)x cal/(g- deg) Substance J/(kg* dcg)x cal/(g* deg)
X 10-2 X 10-2
Aluminium 8.8 0.21 Water 41.9 1.00

Iron 4.6 0.11 Quartz 8.4 0.20
Copper 3.8 0.091 Glass 6.3 0.15
Germanium 3.1 0.074 Mercury 1.3 0.033
Tungsten 1.5 0.036
TABLE 57. Thermal Conductivities of Selected Materials
Thermal conduc Thermal conduc

tivity tivity
Material bJD be Material obH bo

T3
o>
otuo T3 703)
0tuD> ooj g oG3 oCb o
(S Td O 6 £
■o sO B
e co A s CO is
Copper 390 0.92 330 Cement 2.9 7x10-3 2.5

Aluminium 210 0.51 190 Brick 1.7 4.1x10-3 1.4
Graphite 130 0.30 110 Glass 0.84 2x10-3 0.7
Brass 110 0.26 94 Water 0.63 1.5x10-3 0.54
Tungsten 76 0.18 65 Cotton 0.25 6x10-4 0.22
Steel 46 0.11 40 Wood 0.21 5x10-4 0.18
Mercury 6.7 1 .6 x l0 - 2 5.8 Felt 0.038 9x10-5 0.032
TABLE 58. Acoustic R esistivity of Selected Media
Acoustic Acoustic
Medium resistivity Medium resistivity
in g/s- cm2 in g/s -cm2
Air (in standard Steel 4 .1 x 1 0 6

conditions) 43 Copper 3 .2 x 1 0 6
Mercury 2.0x106 Rubber 2 .9 x 1 0 3
Water 1.4x106
TABLES 285
TABLE 59. R esistivity of Selected Conductors

Resistivity R esistivity
Conductor Conductor
QX arC£> Cgs X 1018 OX ^ cgs X 1018
X length “ X length
Bismuth 120 130 Molybdenum 4.8 5.3

Nichrome 110 120 Aluminium 3.2 3.6
Manganine 43 48 Lead 2.1 2.3
Steel 20 22 Copper 1.8 2.0
Tantalum 15 17 Silver 1.6 1.8
Brass 8 8.9
To obtain the values of the resistivity in Q-m, Q-cm and £2— , the
numbers in the column Q x should be m ultiplied respectively by
10-8, 10~6 and lu-2.
BIBLIOGRAPHY
1. Aristov, E. M. Fizicheskie velichinij i edinitsy ikh izmereniya (Phy

sical Quantities and Their Units), Sudpromgiz, Leningrad, 1963.
2. Beklemishev, A. V. Mery i edinilsy fizicheskikh velichin (Measures
and Units of Physical Quantities), Gostekhizdat, Moscow, 1954.
3. Boguslavsky, M. G., P. P. Kremlevsky, B. N. Oleinik, E. N. Che-
churina, K. P. Shirokov. Tablitsy perevoda edinits izmereniy (Con
version Tables of Units of Measurement), Standartgiz, Moscow,
1963.
4. Bridgman, P. \V. Dimensional Analysis, New Haven, Yale Uni
versity Press, 1932.
5. Burdun, G. D. Edinitsy fizicheskikh velichin (Units of Physical
Quantities), Izd. Komiteta standartov, Moscow, 1967.
6. Burdun, G. D ., N. V. Kalashnikov, L. R. Stotsky. Mezhdunarodna-
ya sistema edinits (International System of Units), “Vysshaya
shkola” , Moscow, 1964.
7. Chertov, A. G. Edinilsy izmereniya fizicheskikh velichin (Units
of Physical Quantities), “Vysshaya shkola” , Moscow, 1960.
8. Chertov, A. G. Mezhdunarodnaya sistema edinits izmereniy (Inter
national System of Units), Rosvuzizdat, Moscow, 1963.
9. Davydov, V. V. Primenenie novoy Mezhdunarodnoy sistemy edi
nits v tekhnike (Employment of New International System of
Units in Engineering), “Transport11, Moscow, 1964.
10. Danilov, N. I. Edinitsy izmereniy (Units of Measurement), Uch-
pedgiz, Moscow, 1961.
11. Ginkin, G. G. Logarifmy, detsibely, detsilogi (Logarithms, Deci
bels, Decilogs), Gosenergoizdat, Moscow-Leningrad, 1962.
12. Gordov, A. N. Temperaturnye shkaly (Temperature Scales), Izd.
Komiteta standartov, Moscow, 1966.
13. Jerrard, H. G. and D. B. McNeill. A Dictionary of Scientific Units,
London, Chapman & Hall, 1964.
14. Kalantarov, P. L. Edinitsy izmereniy elektricheskikh i magnitnykh
velichin (Units of Electrical and Magnetic Quantities), Gosener
goizdat, Leningrad-Moscow, 1948.
15. Kalashnikov, N. V ., L. R. Stotsky et al. Edinitsy izmereniy i
oboznacheniya fiziko-tekhnicheskikh velichin (Units and Symbols
of Physical and Engineering Quantities), “Nedra” , Moscow, 1966.
16. Khvolson, 0 . D. Ob absolyutnykh edinitsakh, v osobennosti magnit
nykh i elektricheskikh (On Absolute Units, Especially Magnetic
and Electrical Ones), Saint Petersburg, 1887,
17. Kogan, B. Yu. Razmernost1 fzicheskoy velichiny (Dimension of
a Physical Quantity), “Nauka”, Moscow, 1968.
18. Lisenkov A. A. Mezhdunarodnaya sistema edinits (International
System of Units), ’’Nauka” , Moscow, 1966.
BIBLIOGRAPHY 287
19. Malikov, M. F. Osnovy metrologii (Fundamentals of Metrology),

Izd, Komiteta po delam mer i priborov, Moscow, 1949.
20. Malikov, S. F. Prakticheskie elektricheskie edinitsy (mezhdunarod-
nye i absolyutnye) [Practical Electrical Units (International and
Absolute)], Gosenergoizdat, Moscow-Leningrad, 1948.
21. Malitsky, A. N. Edinitsy izmereniya elektricheskikh i magnitnylh
velichin (Units of Electrical and Magnetic Quantities), MGU,
1961.
22. Savenko, V. G. Mezhdunarodnaya sistema edinits (SI) [Interna
tional System of Units (SI)], Izd. Leningradskogo Elektrotekh-
nicheskogo Institute Svyazi, 1965.
23. Sedov, L. T. Melody podobiya i razmernosti v mekhanike (Methods
of Similarity and Dimensions in Mechanics), “Nauka” , Moscow,
1967.
24. Sena, L. A. Edinitsy izmereniya fizicheskikh velichin (Units of
Physical Quantities), Gostekhizdat, 1951.
25. Shirokov, K. P. (editor). O vnedrenii mezhdunarodnoy sistemy
edinits (On the Introduction of the International System of Units),
collected articles, Izd. Komiteta standartov, Moscow, 1965.
26. Stille, U. Messen und Rechnen in der Physik, Fridr. Vieweg &.
Sohn, Braunschweig, 1961.
27. Some of the most important USSR State Standards (GOST):
GOST 9867-61—The International System of Units.
GOST 7664-61—Mechanical Units.
GOST 8550-61—Thermal Units.
GOST 7932-56—Illumination Engineering Units.
GOST 8849-58—Acoustic Units.
INDEX
Cable, 81
ABRAHAM, M., 156 Cable-length, 81
Acceleration, 22f, 46, 91 f Caloric, 41, 101, 132
angular, 92f per hour, 103, 133
conversion table, 256 mean, 132
norma1, 52 per minute, 133
Acohm, 278 per second, 133
Acre, 82, 277 Candela, 40, 208f
Action, 107 Candela-second, 211
Ammeter, 16 Candle,
Ampere, 37f, 154, 167, 168, 194 Hefner, 208
international, 199 international, 208
per metre, 192, 195 Capacitance, 173, 190
Ampere-hour, 184 conversion table, 272
Ampere-turn, 194 Capacity, heat, 135
Angle, 61, 82ff specific, 135
conversion table, 255 Carat, 96
right, 83 Carcel, 208
solid, 84 ff CELSIUS, A., 128
conversion table, 254 Cent, 147, 260
Angstrom, 80 Centimetre, 24, 36, 80, 173, 180
Aperture, relative, 216 cubic, 82, 222
Apostilb, 211, 275 inverse, 88
Are, 82 reciprocal, 202
Area, 20, 25f, 46, 49, 81 f per second, 91
conversion table, 252 per second per second, 92
Areometer, 241 square, 81
Atmosphere, Ccntipoisc, 115
standard, 41, 99 Centner, 96
technical, 99 Charge,
Atomic unit of mass, 111 electric, 170, 181, 219
Audibility threshold, 149f conversion table, 271
electron, 243
magnetic, 156
Bar, 99, 143 Circulation, magnetic field inten
Barn, 225 sity, 177f
Barrel, 277 CLAPEYRON, B .P .E ., 123
Bel, 146, 239 Coefficient (s),
Biot, 278 bulk compression, 112
BIOT, J. B., 157 diffusion, 118
Bit, 240 extension, 112
Black body, 206f first Townsend, 233
BOYLE, R ., 123 friction, 103
BRIDGMAN, P. W., 73 heat transfer, 137!
Bushel, 277 conversion table, 260
INDEX
internal friction, 114 Cycle per second, 93

linear absorption, 151
m obility, 234
recombination, 233f Day,
resistance, 103f solar, 36
self-induction, 180 stellar, 36
shear, 112 Debye, 221
surface tension, 116f Decibel, 146, 239f
temperature, 140 Decilog, 239f
van der Waals equation, 140f Decimetre, 80
volume electronic ionization, cubic, 82
233 Decrement, logarithmic damping,
Concentration, 117f 108f
conversion table, 259 Degree, 40, 128
normal, 118 (angle), 83
Conductance, 175, 191 square, 85
Conductivity, 175, 191 Baume, 241
thermal, 136f, 284 electrical, 94
conversion table, 260 Engler, 115
Constant, Density, 109f
Boltzmann’s, 125, 245 current, 174
decay, 222 energy, 102
dielectric, 169, 173, 189 mass flow, 107
Dirac, 244 measurement, 241
disintegration, 222 particle flux, 232
electric, 169 radiant energy, 204
Faraday’s, 244 radiant flux, 202, 230
fine-structure, 237, 244 spectral, 205f
gravitational, 27, 31, 49, 243 sound energy, 144
inertial, 27f, 31, 50 surface,
magnetic, 168 charge, 170
Planck, 220, 244 conversion table, 271
Rydberg, 228, 244f boat flow, 133f
in Stefan-Boltzmann law, 245 volume charge, conversion
universal, 32f table, 271
gas, 245 volumetric flow rate, 95
in Wien displacement law, 245 Diffusion, thermal, 138f
Coulomb, 184 D iffusivity, thermal, 138f
per kilogram, 230 Dimension(s),
per square metre, 189 analysis, 65ff
Coulomb-metre, 189 formulas, 42fl
COULOMB, C.A., 153, 156 derived units, 43f
Criteria, sim ilarity, 76ff use, 61, 65ff
Cross section, effective, 223ff Dioptre, 215
conversion table, 276 Dipole, 172
reduced, 225 Disintegration, 231
Curie, 232 per second, 232
Curvature, 86f Displacement, electric, 171, 185ff
Gaussian, 88 conversion table, 272
mean, 87f Distribution functions, 119
surface, 87f normalized, 119f
290 INDEX
velocity, 119 Faraday, 244

Dose, Fermi, 223
absorbed radiation, 230 Flexibility, 104
rate, 231 Flow,
D uctility, 114 energy, 202
Dyne, 24, 96f heat, 133, 136f
specific, 133f
mass, 107
Efficiency, 126 rate, volumetric, 94f
luminous, 212ff sound energy, 144
absolute, 212, 275 Fluidity, 116
monochromatic, 213 Flux,
relative, 213, 275 displacement, conversion tab
total, 213 le, 272
Einstein, 231 electric, 171f, 189
EINSTEIN, A., 234 heat, 133
Electron-volt, 225 linkage, 176f
Eman, 232 luminous, 209
Emittance, magnetic, 176, 192
luminous, 210 conversion table, 274
radiant, 203 particle, 229
Energy, 52, 102, 227 quantum, 229
t | conversion table, 257 radiant, 202, 230
free, 116 F number, 216
Foot, 52, 81
internal, 131
1 kinetic, 46, 48 square, 82
■sound, 143 Force, 23, 49f, 96f
* thermal, 131 coercive, 181f, 196
Entropy, 134f conversion table, 256
Eotvos, 278 electromotive (e.m .f.), 172,
Equivalent, 184
man roentgen, 231 gravitational unit, 28
mechanical, of light, 214 internal friction, 114
physical roentgen, 231 magnetomotive (m .m .f.),
Erg, 101 177f, 194
per second, 102 conversion table, 274
Exposure, radiation, 230 Formula,
Eye sensitivity, 212 k Ampere’s, 160
£ | Boltzanm n’s, 125
Laplace’s, 158
Planck’s, 207
Factor, Franklin, 278
absorption, 217 Frequency, 93f
acoustic absorption, 150f angular, 94
acoustic reflection, 150 Fresnel, 94
compressibility, 112 Frigorie, 133
damping, 108
diffusion, 217
reflection, 217 Gal, 92, 278
transmission, 217 GALILEO, G., 92
Farad, 169 190 Gallon, 277
INDEX 291
Gamma, metric, 103

(magnetic field intensity), 193 U .S., 103
(mass), 96 Horsepower-hour, 257
Gauss, 176 Hour, 41, 91
GAY-LUSSAC, J. L., 122
Gilbert, 178 Illum ination, 210
GILBERT, W., 153 radiant, 203
GIORGI, G., 37 Impact parameter, 224
Gon, 83 Impulse, 47, 52, 97f
Gradient, moment of force, 106
electric, 170 Inch, 41, 81
pressure, 99f cubic, 82
temperature, 133 square, 82
velocity, 95 Index, refractive, 217
Grain, 278 Inductance, 179, 194f
Gram, 24, 96 circuit, 180
(force), 97 conversion table, 274
Gram-molecule, 96 mutual, 179, 181, 194f
conversion table, 274
Induction,
Half-life of particle, 222f electric, 171, 185
Hardness, 113f magnetic, 175, 191
Brinell, 113f conversion table, 273
scales, 113, 277 residual, 181, 196
Breithaupt, 113, 277 Inerta, 9, 54, 96, 278
Mohs, 113, 277 Intensity,
HARTREE, D ., 236 current, 155, 174
Heat, conversion table, 273
capacity, 135 electric field, 155, 157, 170f,
specific, 135 185 .
mechanical equivalent, 101, conversion table, 271
132 luminous flux, 210
molar, 135 magnetic field, 156f, 176, 192
molecular, 135 circulation, 177f, 194
quantity, 130f conversion table, 274
specific, 135, 284 magnetization, 181, 195
conversion table, 259 radiant, 203
volumetric, 135f sound, 144
transformation, 136 Interaction,
transition, 136 current, 160f
Heating value, 136 electromagnetic, 157f
volumetric, 136 electrostatic, 154f
HEAVISIDE, 0 ., 167 permanent magnet, 156f
Hectare, 82 Interval,
Hectopiezo, 257 frequency, conversion table,
Hectowatt, 102 260
Hectowatt-hour, 103 musical, 147
Henry, 168, 195 Irradiance, 202
Hertz, 93, 142
Horsepower, 103 Joule, 101, 131
British, 103 per second, 102
292 INDEX
Kayser, 278 LENZ, E. Ch., 192

KELVIN, W. T., 122 LEONTOVICH, M. A., 8, 188
KHVOLSON, 0 . D., 7 Level,
Kilocalorie, 101, 132 sound intensity, 146f
per hour, 103, 133 sound pressure, 146f
per minute, 133 Lifetime of particle, 222f
per second, 133 mean, 223
Kilogram, 23, 96 Light year, 81
(force), 97 Litre, 82
prototype, 35, 36 Litre-atmosphere, 101
Kilogram-metre, 101 Loudness of sound, 149f
per second, 102 Lumen, 209
Kilogram-molecule, 96 per square centimetre, 210
Kilograv, 36 per square metre, 210
Kilohertz, 94 Lumen-second, 209
Kilometre, 80 Luminance, 210f
per hour, 91 conversion table, 275
Kilomole, 96, 111 Lux, 210
Kiloparsec, 81
Kilopond, 36, 278
K ilovolt, per centimetre, 185 Mache, 232
Kilowatt, 102 Magn, 278
Kilowatt-hour, 103 Magnetization, 181, 195
KLYATSKIN, I. G., 188 Magneton,
Knot, 91 B o h r, 221, 245
nuclear, 221
Magnitude, stellar, 238
Lambert, 211, 275 MALIKOV, M. F., 9, 54
Law, MARIOTTE, E., 123
Biot, Savart and Laplace, Mass, 47, 51, 96, 218
158, 168, 193 atomic unit, 111, 218f
Coulomb’s, 155, 156 chemical, 218f
Faraday’s, 179 physical, 219
Hooke’s, 104, 111 conversion table, 256
Kepler’s third, 30f electron, 243
Lambert, 204 magnetic, 156f, 163
Newton’s second, 23, 27, 47 molecular, 111, 219
Ohm’s, 174 neutron, 244
Planck radiation, 245 proton, 243
Stefan-Boltzmanri, 125, 204 relationship to energy, 228
thermodynamics, Maxwell, 176
first, 131 MAXWELL, J. C., 164
second, 126 Measurement, HIT
universal gravitation, 27, 57 angular, 17
Weber-Fechner, 238 direct, 16f
Length, 81 f indirect, 16
astronomical unit (AU), 81 linear, 17
conversion table, 252 methods, 12
focal, principal, 215 single-valued, 13
non-metric units, 80f Megahertz, 94
Lenz, 192, 278 Megaparsec, 81
INDEX 293
Megawatt, 102 Moment(s),

Megawatt-hour, 103 dipole, 172f, 189, 221
MENDELEEV, D. IM 123 dynamic, 105
Mes, 91, 278 of inertia,
Metre, 23, 80 axial, 90
cubic, 82 of body, 105f
inverse, 88 conversion table, 259
prototype, 35, 36, 38 polar, 90
reciprocal, 202 magnetic, 177, 193f, 220
round, 26 of momentum, 106f, 219f
per second, 91 of plane figures, 89f
per second per second, 92 statical, 89
square, 20, 81 Momentum, 47, 52, 97f
of open window, 151 angular, 106f, 219
Mho, 191, 278 Mug, 54
Microbar, 143
Microcoulomb per kilogram, 231
Microfarad, 190 Nanometre, 80
Microgram, 96 Nanosecond, 91
Micrometre, 80 Neper, 147, 239
Micron, 80 Newton, 23, 96
reciprocal, 202 per coulomb, 185
Microsecond, 91 Nit, 21 Of, 275
Microwatt, 102 Number,
Mie plates, 187f Avogadro’s, 96, 243
Mil, 81, 278 quantum, 220
Mile, 81, 278 Reynolds, 77f
international nautical, 81 spin, 220
square, 82 wave, 201f
Milliangstrom, 80
Milligram, 96
Millimetre, 80 Octave, 147, 260
of mercury, 41, 99 Oersted, 176, 192
of water, 99 OERSTED, II. C., 153, 157
Millimicron, see Nanometre Ohm, 190
Millioctave, 147, 260 acoustic, 145
Millirocntgcn, 231 international, 199
Millisecond, 91 mechanical, 145
M illiwatt, 102 Oscillation(s),
Minute, 41, 91 acoustic, 142
(angle), 83 dynamic characteristics, 108f
metric, 83 infrasonic, 142
Mobility, 234 sound, 142
reduced, 235 ultrasonic, 142
Modulus, Ounce, 278
elasticity, 112, 283 Overload, 92
shear, 112
conversion table, 259 Par, 54
Young’s, 112, 283 Parsec, 81
Mole, 96, 111 Particle, 117f
294 INDEX
Pascal, 257, 278 of electricity, 170, 184

Penetrance, acoustic, 151 of heat, 130f
Period, 93 of illum ination, 211
Permeability, radiant, 203
absolute, 196 of light, 209
magnetic, in vacuum, 169 of magnetism, 156
relative, 163, 181, 196 Quart, 278
Permeance, 179 Quintal, 96
Perm ittivity, 169
absolute, 189
relative, 163, 173, 189 Rad, 230, 231
Phase, 94 Radian, 83, 86
pH index, 242 per second, 92
Phon, 150 Radiance, 204
Phot, 210 Radius,
Picofarad, 190 classical electron, 223
Pieze, 99 curvature, 86
Pitch of sound, 147 first Bohr orbit, 223
PLANCK, M„ 59, 236 Radlux, 210
Point(s), Radphot, 210
compass, 278 Relationship, defining, 22, 25ff
temperature, 279 Relative value, absolute magni
Poise, 115 tude, 13
reciprocal, 116 Reluctance, 179, 194
Polarization, 221f Rem, 231
dielectric, 173, 189 Rcmanencc, 181, 196
Pood, 40 Rep, 231
Potential, 172, 184 Resistance, 174f, 190f
conversion table, 272 acoustic, 144
Pound, 52, 278 conversion table, 273
Poundal, 278 magnetic, 179, 194
Power, 56, 102 mechanical, 145
absorbed radiation dose, 230 wave, 197
apparent, 184 of vacuum, 196fi
conversion table, 258 R esistivity, 175, 191, 285
lens, 215 acoustic, 145, 284
total absorbing, 151 conversion table, 273
Poynting vector, 202 Reverberation, 151f
Pressure, 48, 98f Revolution, 83
conversion table, 257 REYNOLDS, O., 77
reduced, 124, Rhe, 116, 278
conversion table, 259 Roentgen, 230
sound, 142f Rutherford, 232
Rydberg, 228f
Quality of oscillating system, 109
Quantities,
basic, 21 f SABINE, W., 152
derived, 21 f Savart, 147, 260
symbols, 19, 281 ff SAY ART, F.,157
Quantity, Scale(s),
of cold, 133 hardness, 277
INDEX 295
Brvi Ili;i 111 >1, 277 Planck’s, 236

Mohs, 277 symmetrical, 39, 159, 165,
just, I uS 169
musical, 148 technical, 34
natural, 148
tom pored, 148
wind, Beaufort, 13, 277 Temperature, 39, 121 ff
Second, 23, 35, 36, 38f, 91, 93 absolute, 123, 125
(angle), 83 fixed points, 130, 279
metric, 83 scale,
Englor, 115 absolute, 122, 128
solar, 36 Fahrenheit, 128
stellar, 36 Kelvin, 128
SEDOV, L. I., 73 Rankine, 128
Siemens, 191 Reaumur, 128
per metre, 191 thermodynamic, 127
SLEPYAN, L. B., 188 Tesla, 191
Slug, metric, 54 Theorem,
Source, Lambert, 204 Gauss, 171
Standard, Viole, 208 ji-, 73f
Steradian, 85f Therm, 132
Sthene, 97 Timbre of sound, 148f
Stilb, 211, 275 Time, 91
Stokes, 116 conversion table, 255
Strength, impact, 114 reverberation, 152
Stress, 48 TME, 54
Susceptibility, Ton, 40, 98-
dielectric, 173f, 190 (force), 97
magnetic, 182, 196 long, 278
System of units, short, 278
absolute, 34, 182f Toroid, 178
Blondel’s, 166 Torr, 99
cgs, 36f, 39, 79, 165, 169ff TORRICELLI, E., 99
cgse, 37, 156, 182 TOWNSEND, F., 233
cgsl, 209ff Transparency, 217
cgsm, 37, 157, 182
construction, 24ff, 154ff
electromagnetic, 37, 157 Unit(s),
electrostatic, 37, 156 absolute, 200
emu, 157 acoustic, 142ff
esu, 156 atomic mass, 111
Gaussian, 39, 159, 169 basic, 18
Giorgi’s, 166 number, 28ff, 164
Hartree’s, 236 selection, 25, 34ff, 76, 158f
International (SI), 7, 34, 40, British, 277f
79, 165, 182ff British thermal, 277
M axwell’s, 166 conversion, 14f, 52f
metre-ton-second, 40 decimal m ultiples and sub-
metric, 14, 35 m ultiples, 41, 280
mk(force)s, 37, 79 derived, 18, 21f
natural, 235ff designation, 24
296 INDEX
Unit(s), relative, 115

dimension formulas, 42ff specific, 115
dimensionless, 86 Volt, 184
dynamic, 96ff per centimetre, 185
electrical, 153ff, 262ff international, 199
geometrical, 79ff, 248ff per metre, 185
illum ination engineering, Volt-ampere, 184
207f£ Voltage, 172, 184
international, 198ff Volume, 82
kinematic, 91 ff conversion table, 254
local, 14 gas, standard, 245
logarithmic, 238ff, molecular, 111
magnetic, 153ff, 262ff sound, 149
mechanical, 79, 248ff specific, 110
of mechanical properties,
109ff
of molecular properties, 109ff Water, triple point, 127f
non-system, 40f Watt, 102
optical instrument, 214ff international, 199
of optical properties, 217 Watt-hour, 103
radiation, 201 ff Wavelength(s), 201
radioactivity, 231 Compton, 244
static, 96ff Weber, 192
supplementary, 86 Weight,
symbols, 279f atomic, 219
systems, see Systems of units molecular, 111, 219
thermal, 121 ff specific, 11 Of
uranium, 278 W IEN, W ., 125
U.S., 277f Work, 50, 52, lOOff
V elocity, 22, 46, 91
angular, 92ff
conversion table, 256 X-unit, 80
non-system units, 93
electron, 227
light, 164f, 242 Yard, 81, 278
mass flow, 107 square, 82
volumetric, 143 Year, tropical, 39
Viscosimeters, 115
Viscosity, 114f, 283
dynamic, 116 Zero, absolute, 123
kinematic, 116

Units of Physical Quantities and Their Dimensions (Mir, 1972)

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Units of Physical Quantities and Their Dimensions (Mir, 1972)

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L A SENA

EflHHHIJbl OH3HHECKHX BEJIHUHII

Translated from the Russian

MIR PUBLISHERS * MOSCOW • 1972

The present book sets out in detail the prin­

Mir Publishers w ill welcome your com­

Printed in the Union of Soviet Socialist Republics

Chapter One. General Concepts on Systems of Basic and Derived

Chapter Two. Conversion of Units and Dimension Formulas 42

Chapter Three. Analysis ofD im ensions....................................... 65

Chapter Four. Units of Geometrical and Mechanical Quantities 79

Chapter Five. Thermal U n it s .................... 121

Chapter Six. Acoustic U n it s ............................................ 142

Chapter Seven. Electrical and Magnetic U n i t s ............................ 153

Chapter Eight. Units of R ad iation ............................................ 201

Chapter Nine. Selected Units of Atomic P h y sic s........................ 218

Procrustean bed of a single system will hardly satisfy

* In the English edition use has been made wherever possible of

GENERAL CONCEPTS ON SYSTEMS

L I. Physical Quantities and Their Units

The part played by measurements has especially grown

than the length of a pencil, or whether the volume of a cup

The existence of a great number of diverse units natu­

in its numerical value being equal to au then we can write

If, when measuring the same magnitude A using the unit

1.2. Direct and Indirect Measurements

intermediate step between the current to be measured and

length. Actually this simply means that since all phenomena

1.3. Basic and Derived Units

If by the symbols A x, A 2, lx and l2 we understand the rele­

where the factor C depends, on one hand, on the shape of

It is obvious that the accepted relationship between

ties”—should by no means be understood in the sense that

For the particular case of uniform motion expression (1.5)

where, as previously, C is a factor depending on the units

which for uniformly accelerated motion takes the form

Here the difference v2 — vx denotes the change in velocity

and Newton’s second law can be written in the following

If we take the centimetre (cm) as the unit of length and

« > ( . - £ ) < i 7 b >

In writing the formulas of physical relationships, such

1.4. Constructing a System of Units

derived units of all the other quantities we are interested

area. Let uscall this unit a “round metre” (rd m).

Regardless of how the unit of area has been determined,

We shall now show that it is possible to select essentially

where r is the distance between the points being attracted,

defining relationship and, assuming Cg = 1, determine

The supporters of the opposite viewpoint, which the

of different physical quantities, however, can be reduced

The meaning of this equation consists in that the acceleration

move along ellipses leads to the same expression, the only

to reduce the number of basic units, it will be easy to see

This arbitrariness is such only theoretically, however.

1.5. Selection of Basic Units

system, however, the use of certain other systems, and also

Besides the fact that changing of the latter would in

* A number of other names were proposed for this unit, such as

The mk(force)s system, whose basic units are the unit

The present book sets out in detail the prin

Mir Publishers w ill welcome your com

The existence of a great number of diverse units natu

If by the symbols A x, A 2, lx and l2 we understand the rele

then the dimension of C is equal to the product of the dimen

then, when measuring the mass in kilograms and the dis