Introduction to Applied

Physics
Dr. Javed Ahmed Laghari
Associate Professor
( javed@quest.edu.pk )

Department of Electrical Engineering QUEST, Nawabshah, Sindh, Pakistan.

 Books Recommended:
 Atomic Structure:
 All matter is made of atoms.
 Matter is anything that has mass and
takes up some space.
 An atom is the smallest particle of an
element that retains the characteristics
of that element.
 Every element has atoms that are
different from the atoms of all other
elements. This gives each element a
unique atomic structure.
 All atoms consist of electrons, protons,
and neutrons.
 An atom consists of a central nucleus
surrounded by orbiting electrons as
illustrated in Figure.
 Atomic Structure:
 The nucleus consists of positively charged particles
called protons and uncharged particles called neutrons.
 The basic particles of negative charge are called
electrons, which orbit the nucleus.
 All elements are arranged in the periodic table of the
elements in order according to their atomic number.
 The atomic number equals the number of
protons in the nucleus.
 Consider the Figure which shows that hydrogen
has an atomic number of 1.
 In their normal (or neutral) state, all atoms of a
given element have the same number of
electrons as protons; the positive charges
cancel the negative charges, and the atom has a
net charge of zero, making it electrically
balanced.
 Modern Electron Theory of Matter:
 Several theories have been developed to explain the nature of electricity.
 The only theory that has survived over the years to explain the nature of electricity is the
Modern Electron theory of matter.
 This theory has been the result of research work conducted by scientists like William
Crooks, J.J. Thomson, Rutherford and Neil Bohr.
 According to this theory,
 Every matter is electrical in nature. It is because every element is made up of atoms which
contains particles of electricity called electrons and protons.
 Whether a given body exhibits electricity (i.e. charge) or not depends upon the relative
number of these particles of electricity.
 If the number of protons is equal to the number of electrons in a body, the resultant charge
is zero and the body will be electrically neutral.
 If from a neutral body, some electrons are removed, there occurs a deficit of electrons in
the body. Consequently, the body attains a positive charge.
 If a neutral body is supplied with electrons, there occurs an excess of electrons. Consequently, the
body attains a negative charge.
 The Electron:
 Since electrical engineering generally deals with tiny particles called electrons, these
small particles require detailed study.
 We know that an electron is a negatively charged particle having negligible mass. Some of
the important properties of an electron are :
 Some of the important properties of an electron are:
19
 Charge on an electron, e  1.602 10 Coulomb
 Mass of an electron, m  9.0 10 31 kg
 An electron moving around the nucleus possesses two types of energies (kinetic energy
due to its motion and potential energy due to the charge on the nucleus).
 The energy of an electron increases as its distance from the nucleus increases.
 Thus, an electron in the second orbit possesses more energy than the electron in the first
orbit; electron in the third orbit has higher energy than in the second orbit.
 The Electron:
 It is clear that electrons in the last orbit possess very high energy as compared to the
electrons in the inner orbits.
 These last orbit electrons play an important role in determining the physical, chemical
and electrical properties of a material.
 The electrons in an atom revolve in different orbits or shells.
 The shells are named as K, L, M, and N.

 The number of electrons in a shell

 2 n 2
 where n is shell number 1, 2, 3, 4, etc.

 2  (1)  2
 The number of electrons 2
in First Shell, n = 1

 The number of electrons

in Second Shell, n = 2  2  (2)  2  4  8
2
 The Electron:
 The number of electrons
in Second Shell, n = 3  2  (3)  2  9  18
2

 The number of electrons

in Second Shell, n = 4  2  (4)  2 16  32
2

 The filled outermost shell should always contain a

maximum number of eight electrons.
 The outermost shell of an atom may have less than
eight electrons.
 As for example, copper has an atomic number of
29. The electrons are distributed in the K, L, M,
and N shells as 2, 8, 18, and 1 electrons,
respectively.
 The outermost shell of a copper atom has one
electron only whereas this shell could have 8
electrons.
 The Electron:
 There exists a force of attraction between the orbiting electron and the nucleus due to the
opposite charge of electron and proton.
 The electrons in the inner orbits are closely bound to the nucleus than the electrons of the
outer or outermost orbit.
 If the electron is far away from the nucleus, the
force of attraction is weak.
 The valence electrons are so weakly attached to
their nuclei that they can be easily removed or
detached. Such electrons are called free electrons.
 Those valence electrons which are very loosely
attached to the nucleus of an atom are called free
electrons.
 For example, a copper atom has only one electron
in the last orbit which otherwise could have eight
electrons.
 For instance, one cubic centimetre of copper has
about 8.5 × 1022 free electrons at room temperature.
 Valance Shell and Valance Electron:
 The outermost shell of an atom is known as valance shell.
 The electrons in the outermost orbit of an atom are known as valence electrons.
 The outermost orbit can have a maximum of 8
electrons i.e. the maximum number of
valence electrons can be 8.
 The valence electrons determine the
physical and chemical properties of a
material.

Don’t be UPSET in your life,

Always keep passion UP to SET your life.
( J. A. Laghari)
 Conductors, Insulators & Semi-conductors:
 On the basis of electrical conductivity, materials are generally classified into conductors,
insulators and semi-conductors.
 As a rough rule, one can determine the electrical behavior of a material from the number
of valence electrons as under:

 Conductors:
 When the number of valence electrons of an atom
is less than 4 (i.e. half of the maximum eight
electrons), the material is usually a metal and a
conductor.
 Examples are copper and
aluminum which have 1,
and 3 valence electrons
respectively.
 Conductors:
 Def:The conductor is the
type of metal which allows
the electrical current to
flow through it.
  Def:The insulator is the type of metal which does
 Insulators: not allow the electrical current to flow through it.
 When the number of valence electrons of an atom is more than 4, the material is usually a
non-metal and an insulator.
 Examples are nitrogen, sulphur and neon which have 5, 6 and 8 valence electrons
respectively.
 Semi-conductors:
 When the number of valence electrons of an atom is 4 (i.e. exactly one-half of the
maximum 8 electrons), the material has both metal and non-metal properties and is
usually a semi-conductor.
 Examples are carbon, silicon and germanium.
 Electric Charge ( Q ):
 Def: Electrical charge is an electrical property of matter that exists because of an excess or
deficiency of electrons.
 When an excess of electrons exists in a material, there is a net negative electrical charge.
 When a deficiency of electrons exists, there is a net positive electrical charge.
 Charge is denoted by the letter Q.
 SI unit of charge is coulomb abbreviated as C.
 One coulomb of charge is equal to the charge on 625 × 1016 electrons, i.e.
 1 coulomb = Charge on 6.25 × 1018 electrons

 Charge on electron
1 1 19
    1.6  10 C
625 1016
6.25 1018
 This example is taken from the book Principles of Electric

 Electric Charge ( Q ): Circuits by Thomas Floyd, Chapter Two, Example 1.

 The total charge Q, expressed in coulombs, for a given number of electrons is stated in the
following formula:
Number of Electrons
Q=
6.25 × 1018 electrons/C
 Example:
 How many coulombs do 93.8 x 1016 electrons represent?
 Solution:
 As we know that
Number of Electrons 93.8 × 1016 electrons
Q= = = 15×10-2 C
6.25 × 1018 electrons/C 6.25 × 1018 electrons/C
Q = 0.15 C
 Electric Current ( I ):
 Def: Electrical current is the rate of flow of charge.
 Def: The directed flow of free electrons (or charge) is called electric current.
 Electric Current is denoted by I. Mathematically,

Q The conventional symbol for current is I, which originates from the French
I phrase intensité du courant, (current intensity). The I symbol was used
t by André-Marie Ampère, after whom the unit of electric current is named.

 Where I is current in amperes (A),

 Q is charge in coulombs (C), and
 t is time in seconds (s).
 The flow of electric current can be
beautifully explained by referring to
Figure.
 The copper strip has a large number
of free electrons.
 Electric Current ( I ):
 When electric pressure or voltage is
applied, then free electrons, being
negatively charged, will start moving
towards the positive terminal around
the circuit as shown in Figure.
 This directed flow of electrons is called
electric current.
 Following points may be noted:
 The actual direction of current (i.e. flow of electrons) is from negative terminal to the
positive terminal through that part of the circuit external to the cell.
 However, prior to Electron theory, it was assumed that current flowed from positive
terminal to the negative terminal of the cell via the circuit.
 This convention is so firmly established that it is still in use. This assumed direction of
current is now called conventional current.
 Unit of Electric Current:
 The unit of current is Ampere (A).

Q C
I A
t Second
 Therefore, one ampere = couloumb/second.

 One Ampere may be defined as:

 One ampere of current is said to flow through a wire if at any cross-section one coulomb of
charge flows in one second.
 One ampere is the amount of current that flows when number of electrons having a total
charge of one coulomb move through a given cross-sectional area in one second.
 Types of Electric Current:
 The electric current may be classified into three main
classes:
 Direct Current (DC) or steady current
 varying current and
 Alternating Current (AC) or Sinusoidal Current.

 Direct Current (DC):

 When the magnitude of current does not change with
time, it is called a steady current or direct current.
 Figure shows the graph between steady current and time.
 It may be noted that value of current remains the same
as the time changes.
 The current provided by a battery is almost a steady
current (d.c.).
 Varying Current:
 When the magnitude of current changes with time, it is called a
varying current.
 Figure shows the graph between varying current and time.
 Note that value of current varies with time.

 Alternating Current (AC):

 An alternating current is one whose magnitude
changes continuously with time and direction
changes periodically.
 The current that changes its magnitude and
polarity at regular intervals of time is called an
alternating current.
 Alternating Current (AC):
 Due to technical and economical reasons, we produce
alternating currents that have sine waveform as shown in
Figure.
 In Pakistan, Alternating Current has the frequency of 50 Hz.
 The frequency in USA is 60 Hz.
 In DC System, the frequency is zero.
 Mechanism of Current Conduction in Metals:
 Every metal has a large number of free electrons which wander randomly within the body
of the conductor.
 During random motion, the free electrons collide with positive ions again and again and
after each collision, their direction of motion changes.
 When we consider all the free electrons, their random motions average to zero.
Consequently, no current is established in the conductor.
 When potential difference is applied across the ends of a conductor (say copper wire) as
shown in Figure, electric field is applied at every point of the copper wire.
 The electric field exerts force on the free electrons which start accelerating towards the
positive terminal (i.e., opposite to the direction of the field).
 Mechanism of Current Conduction in Metals:
 As the free electrons move, they collide again and again with positive ions of the metal.
 Each collision destroys the extra velocity gained by the free electrons.
 Although the free electrons are continuously accelerated by the electric field, collisions
prevent their velocity from becoming large.
 The result is that electric field provides a small constant velocity towards positive terminal
which is superimposed on the random motion of the electrons. This constant velocity is
called the drift velocity.
 The average velocity with which free electrons get drifted in a metallic conductor under the
influence of electric field is called drift velocity ( Vd ).
 Drift Velocity:
 Voltage ( V ):
 Def: Voltage is the electrical force or pressure that causes free electrons to move from one
atom to another.
 Voltage is the measure of work required to move a unit charge from one location to another,
against the electric field.
 As already discussed, a force of attraction exists between a positive and a negative charge.
 A certain amount of energy must be exerted, in the form of work, to overcome the force and
move the charges a given distance apart.
 All opposite charges possess a
certain potential energy
because of the separation
between them. The difference
in potential energy per charge
is the potential difference or
voltage.
 Electric Potential ( V ):
 Electric potential is the amount of work needed to move a unit charge from one point to
another point against an electric field.
 Mathematically,
Work Done, W
Electric Potential, V =
Charge, Q
 In other words, work done per unit charge is known as electrical potential.
 The work done is measured in joules and charge in coulombs. Therefore, the unit of
electric potential will be joules/coulomb or volt.
W joule
V Volt 
Q Coulumb
 One volt may be defined as:
 One volt is the potential difference (voltage) between two points when one joule of work or
energy is used to move one coulomb of charge from one point to the other.
 Thus, when we say that a body has an electric potential of 5 volts, it means that 5 joules of
work has been done to charge the body to 1 coulomb.
 These examples are taken from the book Principles

 Example: of Electric Circuits by Thomas Floyd, Chapter Two.

 If 50 J of energy are required to move 10 C of charge, what is the voltage?

 Solution: Work Done, W
 As we know that Electric Potential, V =
Charge, Q
W 50
V   5V
Q 10
 Example:
 How much energy is required to move 50 C from one point to
another when the voltage between the two points is 12 V ?
 Solution:
W
V OR W  V  Q  12  50  600 j
Q
 Potential Difference ( P.D ):
 Def: The difference in the potentials of two charged bodies is called potential difference.

 Electromotive force (emf):

 Electromotive force (emf) is defined as the electric potential produced by either
electrochemical cell or by changing the magnetic field.

 Concept of EMF and Potential Difference:

 Emf and potential difference (V) are both measured in volts, however they are not the same
thing.
 The e.m.f. of a device, say a battery, is the amount measure of the energy the battery gives
to each coulomb of charge.
 Thus, if a battery supplies 4 joules of energy per coulomb, we say that it has an e.m.f. of 4
volts.
 The energy given to each coulomb in a battery is due to the chemical action.
 Concept of EMF and Potential Difference:
 The potential difference between two points, say A and B, is the amount of the energy used
by one coulomb in moving from A to B.
 Thus, if potential difference between points A and B
is 2 volts, it means that each coulomb will give up
an energy of 2 joules in moving from A to B.
 Thus, potential difference causes the current to flow
while an e.m.f. maintains the potential difference.
 Resistance ( R ):
 Def:The opposition offered by a substance to the flow of electric current is called its
resistance.
 When there is current through a material, the free electrons move through the material and
occasionally collide with atoms.
 These collisions cause the electrons to lose some of their energy, and thus their movement
is restricted.
 Greater the collisions, higher will be the restriction to the flow of electrons.

 This restriction varies and is determined by the type of material. The property of a material
that restricts the flow of electrons is called resistance.
 It may be noted here that resistance is the electric friction offered by the substance and
causes production of heat with the flow of electric current.
 The moving electrons collide with atoms or molecules of the substance; each collision
resulting in the liberation of minute quantity of heat.
 Resistance ( R ): Voltage, V
 Mathematically, Resistance, R =
Current, I
 An ohm may be defined as
 A wire is said to have a resistance of 1 ohm if a potential
difference of 1 volt across its ends causes 1 ampere to flow
through it.
 Resistance is expressed in ohms, symbolized by the Greek letter omega (Ω ).
 The schematic symbol for resistance is shown in Figure whereas bottom
figure shows the schematic symbol for variable resistor.

 Resistor:
 Def:The resistor is an electrical component which offers resistance in the flow of electric
current.
 The principal applications of resistors are
 To limit current in a circuit,
 To divide voltage, and, in certain cases,
 To generate heat.
 Law of Resistance: OR
 Factors Upon Which Resistance Depends:
 The resistance R of a conductor
 (i) is directly proportional to its length i.e. R l 1
 (ii) is inversely proportional to its area of X-section i.e. R
 (iii) depends upon the nature of material. a
 (iv) depends upon temperature.
 From the first three points (leaving temperature for the time being), we have,
l l
R R  
a a
 where ρ (Greek letter ‘Rho’) is a constant and is
known as resistivity or specific resistance of
the material. Its value depends upon the nature
of the material.
 Specific Resistance OR Resistivity:
 Def:The specific resistance of a material is the resistance offered by 1 m length of wire of
material having an area of cross-section of 1 m2.
 Take a cube of the material having each side 1 m.
Considering any two opposite faces, the area of cross-
section is 1 m2 and length is 1 m as shown in Figure.
 Hence specific resistance of a material may be defined as
the resistance between the opposite faces of a metre cube of
the material.
 Unit of Resistivity:
 We know that

R  
l R a m 2
    m
a l m
 Resistivity:
 The resistivity of substances varies over a wide range. To give an idea to the reader, the
following table may be referred:
S.No. Material Nature Resistivity (Ω-m) at room temperature
1. Copper Metal 1.7x10-8
2. Iron Metal 9.68x10-8
3. Pure Silicon Semiconductor 2.5x103
4. Pure Germanium Semiconductor 0.6
5. Glass Insulator 1010 to 1014
6. Mica Insulator 1011 to 1015
 It may be noted from the above Table that resistivity of metals is very small. Therefore,
these materials are good conductors of electric current.
 On the other hand, resistivity of insulators is extremely large. As a result, these materials
hardly conduct any current.
 There is also an intermediate class of semiconductors. The resistivity of these substances
lies between conductors and insulators.
 Examples on Resistance:
 Figure shows tow conductors namely A and B of the same
length and different area of cross-section. Can you guess
which conductor will have higher resistance?
 If the length of conductor A and B is 1 meter and 5 meters
respectively. Then, which conductor will have lower resistance?
 If the area of conductor A and B is 1 m2 and 5 m2 respectively.
Then, which conductor will have lower resistance? A L=1m
Area =1m2
A B L=3m

Area =3m2
B
 Conductance ( G ):
 Def:The reciprocal of resistance of a conductor is called its conductance.
 Def: Conductance is the measure of ease with which current can pass through a material.
 Whereas resistance of a conductor is the opposition to current flow, the conductance of a
conductor is the inducement to current flow.
 If a conductor has resistance R, then its conductance G is given by;

1
G
R
 The SI unit of conductance is mho (i.e., ohm spelt
backward). These days, it is a usual practice to use
Siemen as the unit of conductance. It is denoted by
the symbol S.
 Conductivity (  ):
 Def:The reciprocal of resistivity of a conductor is called its conductivity or specific
conductance of the material.
 Def: It represents a material's ability to conduct electric current.
 It is denoted by Greek letter σ (Sigma’).
 Its value depends upon the nature of the material. 1
 If a conductor has resistivity ρ, then its conductivity is given by;

1 1 1a
G G G  G  
a
R (  l) / a  l l
G  
a Gl S m
    s/m
l a m 2

 The SI unit of electrical conductivity is Siemens per meter (S/m) or (Sm-1).

 Resistor Color Codes:
 Fixed resistors with value
tolerances of 5% or 10% are
color coded with four bands
to indicate the resistance
value and the tolerance.
 The color code is listed in
Table.
 Resistor Color Codes:
 This color-code band system is shown in
Figure. The bands are always closer to
one end.
 The color code is read as follows:
 1. Start with the band closest to one end of the resistor. The first band is the first digit of the
resistance value. If it is not clear which is the banded end, start from the end that does not
begin with a gold or silver band.
 2. The second band is the second digit of the resistance value.
 3. The third band is the number of zeros following the second
digit, or the multiplier.
 4. The fourth band indicates the percent tolerance and is
usually gold or silver. For example, a 5% tolerance
means that the actual resistance value is within ± 5% of
the color-coded value. Thus, a 100 Ω resistor with a
tolerance of ± 5% can have an acceptable range of
values from a minimum of 95 Ω to a maximum of 105 Ω.
 This example is taken from the book Principles of

 Example: Electric Circuits by Thomas Floyd, Chapter Two.

 Find the resistance value in ohms and the percent

tolerance for each of the color-coded resistors
shown in Figure (a)-(c).
 Solution:
 (a) First band is red = 2, second band is violet = 7,
third band is orange = 3 zeros,
fourth band is silver = 10% tolerance.
 R = 27,000 Ω ±10%
 (b) First band is brown = 1, second band is black = 0,
third band is brown = 1 zeros,
fourth band is silver = 10% tolerance.
 R = 100 Ω ±10%
 (c) First band is green = 5, second band is blue = 6,
third band is green = 5 zeros,
fourth band is gold = 5% tolerance.
 R = 5600000 Ω ±5%
 Effect of Temperature on Resistance:
 In general, the resistance of a material changes with the change in temperature.
 The effect of temperature upon resistance varies according to the type of material as
discussed below:
 (i) The resistance of pure metals (e.g. copper, aluminium) increases with the increase of
temperature.
 The change in resistance is fairly
regular for normal range of
temperatures so that
temperature/resistance graph is a
straight line as shown in Figure (for
copper).
 Since the resistance of metals
increases with the rise in temperature,
they have positive temperature co-
efficient of resistance.
 Effect of Temperature on Resistance:
 (ii) The resistance of electrolytes, insulators (e.g. glass, mica, rubber etc.) and
semiconductors (e.g. germanium, silicon etc.) decreases with the increase in temperature.
 Hence these materials have negative temperature co-efficient of resistance.
 (iii) The resistance of alloys increases with the rise in temperature but this increase is very
small and irregular.
 For some high resistance alloys (e.g. Eureka, manganin, constantan etc.), the change in
resistance is practically negligible over a wide range of temperatures.
 Figure shows temperature/resistance graph for
copper which is a straight line.
 If this line is extended backward, it would cut the
temperature axis at −234.5°C.
 It means that theoretically, the resistance of
copper wire is zero at −234.5°C.
 However, in actual practice, the curve departs
(point A) from the straight line path at very low
temperatures.
 Temperature Coefficient of Resistance:
 Consider a conductor having resistance R0 at 0°C and Rt at t°C.
 It has been found that in the normal range of temperatures, the increase in resistance (i.e.
Rt − R0)
 (i) is directly proportional to the initial resistance i.e. ( Rt  R0 )  R0
 (ii) is directly proportional to the rise in temperature i.e. ( Rt  R0 )  t
 (iii) depends upon the nature of material.
 Combining the first two, we get,

( Rt  R0 )  R0  t
( Rt  R0 )   0 R0  t
 Where α0 is a constant and is called temperature co-
efficient of resistance at 0°C.
 Its value depends upon the nature of material and
temperature.
 Temperature Coefficient of Resistance:
 A little reflection shows that unit of α will be ohm/ohm°C i.e./°C.
 Thus, copper has a temperature co-efficient of resistance of 0.00426/°C.
 It means that if a copper wire has a resistance of 1 Ω at 0°C, then it will increase by 0.00426
Ω for 1°C rise in temperature i.e. it will become 1.00426 Ω at 1°C.
 Similarly, if temperature is raised to 10°C, then resistance will become 1 + 10 × 0.00426 =
1.0426 ohms.
 Note: The life expectancy of electrical apparatus is limited by the temperature of its
insulation; the higher the temperature, the shorter the life.
 The useful life of electrical apparatus reduces approximately by half every time the
temperature increases by 10°C.
 This means that if a motor has a normal life expectancy of eight years at a temperature of
100°C, it will have a life expectancy of only four years at a temperature of 110°C, of two years
at a temperature of 120°C and of only one year at 130°C.
 Ohms Law:
 The relationship between voltage ( V ), the current ( I ) and resistance ( R )
in a d.c. circuit was first discovered by German scientist George Simon
*Ohm. This relationship is called Ohm’s law and may be stated as under:
 Ohm’s law states that current is directly proportional to voltage and
inversely proportional to resistance provided that physical conditions * The unit of
remains constant. resistance (i.e.
 Mathematically, 1
IV
ohm) was named
I  V V  IR in his honour.
V R
= Constant = R
I
 The ratio of potential difference ( V ) between the ends of a conductor to the current ( I )
flowing between them is constant, provided the physical conditions (e.g. temperature etc.)
do not change.
 These formulae can be applied to any part of a d.c. circuit or to a complete circuit.
 Ohms Law:
 It means that if the voltage across a resistor is increased, the current through the resistor
will also increase; and, likewise, if the voltage is decreased, the current will decrease.
 For example, if the voltage is doubled, the current will double. If the voltage is halved, the
current will also be halved.
 Similarly, if the voltage is held constant, less resistance results in more current, and, also,
more resistance results in less current.
 For example, if the resistance is
halved, the current doubles. If the
resistance is doubled, the current is
halved.
 Ohms law provides linear graph.
 Ohms Law:
 Just look at the Figure A and B, both graphs shows the
ohms law V and I relationship. Can you guess which
one is correct ?
 The correct answer is that Figure A is the correct
graph. It is because Voltage is an independent variable
which must be placed on the x-axis.
The ohms law graph given in
V.K.Mehta book consists of Figure
B. Therefore, it is not correct.
B

A Hint: The independent

variable belongs on the x-
axis (horizontal line) of
the graph and the
dependent variable
belongs on the y-axis
(vertical line).
 Non-Ohmic Conductors:
 Def:Those conductors which do not obey Ohm’s law (1 ∝ V) are called non-ohmic
conductors e.g., vacuum tubes, transistors, electrolytes, etc.
 A non-ohmic conductor may have one or more of the following properties :
 (i) The V-I graph is non-linear i.e. V/I is variable.
 (ii) The V-I graph may not pass through the origin as in case of an ohmic conductor.
 (iii) A non-ohmic conductor may conduct poorly or not at all when the p.d. is reversed.
 Water Analogy:
 A water system can be used as an analogy for a simple
circuit.
 Voltage can be considered analogous to the pressure
required to force water through the pipes.
 Current through wires can be thought of as analogous to the water moving through the
pipes.
 Resistance can be thought of as analogous to the restriction on the water flow produced by
adjusting a valve.