Part 1: Maxwell's Equations: PHYS370 - Advanced Electromagnetism

PHYS370 Advanced Electromagnetism
Part 1: Maxwells Equations

Introduction
Contents:
1. Introduction - Maxwells equations (3 lectures)
2. Electromagnetic waves in dielectric media (3)
3. Electromagnetic waves in conducting media (2)
4. Waves incident on boundaries between two media (4)
5. Electromagnetic cavities and waveguides (4)
6. Transmission lines (3)
7. Electromagnetic potentials (2)
8. Sources of electromagnetic radiation (5)
9. Electromagnetism and special relativity (4)
Advanced Electromagnetism 1 Part 1: Maxwells Equations
Introduction
We will learn how a vast range of physical phenomena follow
from Maxwells equations...
Some Resources
Recommended text:
I.S.Grant and W.R. Phillips, Electromagnetism
Wiley, 2nd Edition, 1990
Free to download:
B. Thide, Electromagnetic Field Theory
http://www.plasma.uu.se/CED/Book/index.html
Comprehensive reference for the very ambitious:
J.D.Jackson, Classical Electrodynamics
Wiley, 3rd Edition, 1998
Notation
Scalar quantities are italicised, like this:
x, E
x
,
0
Vector quantities are written with an arrow:
E,

B
The components of a vector are written with a subscript, e.g.:
A = (A
x
, A
y
, A
z
)
The magnitude of a vector

A is written in italics:
= A
A derivative with respect to time is indicated by a dot:
B =

B
t
Vector Calculus
In electromagnetism, we use vector calculus all the time. Make
sure you are familiar with the notation, and the algebra!
The four basic operators of vector calculus are (in Cartesian
coordinates):
grad (scalar) = vector f =
_
f
x
,
f
y
,
f
z
_
div (vector) = scalar

A =
A
x
x
+
A
y
y
+
A
z
z
curl (vector) = vector

A =
_
A
z
y

A
y
z
,
A
x
z

A
z
x
,
A
y
x

A
x
y
_
laplacian (scalar) = scalar
2
f =

2
f
x
2
+

2
f
y
2
+

2
f
z
2
laplacian (vector) = vector
2

A =
_
2
A
x
,
2
A
y
,
2
A
z
_
Useful Mathematical Theorems
The mathematical identities (for any vector eld

F) are useful:

F (

F)
2
F (1)

F 0 (2)
The grad (), div () and curl () operators acting on sin
functions have the following eect:
sin(
k r) =

k cos(
k r) (3)

Asin(
k r) =

k

Acos(
k r) (4)

Asin(
k r) =

k

Acos(
k r) (5)
where r = (x, y, z) is a position vector, and

k and

A are
constant vectors.
In the above equations, we can interchange sin and cos, with a
minus sign on the right hand side.
Useful Mathematical Theorems: Gauss and Stokes
Gauss theorem for any vector eld

A:
_
V

AdV =
_
S
A

dS (6)
where S is the closed surface bounding the volume V , and the
surface area element

dS is directed out of the volume V .
Stokes theorem for any vector eld

A:
_
S

A

dS =
_
C
A

dl (7)
where C is the closed line bounding the area S.
Units and Physical Constants
Units:
We use the SI (International System) of units, in which there
are seven base units:
mass kilograms kg
length meters m
time seconds s
electric current amperes A
thermodynamic temperature kelvin K
amount of substance mole mol
luminous intensity candela cd
Some useful physical constants:
Speed of light in a vacuum c 2.998 10
8
ms
1
Impedance of free space Z
0
376.7
Permittivity of free space
0
8.854 10
12
Fm
1
Permeability of free space
0
4 10
7
Hm
1
Charge on a positron e 1.602 10
19
C
James Clerk Maxwell, 1831-1879
In 1865, Maxwell published a set of equations that describe
completely the behaviour of electromagnetic elds. These
equations are used in a huge range of applications, from the
properties of materials, to properties of radiation (radio waves
to gamma rays).
The theory of electromagnetism has been extensively tested
and is hugely successful. It provides a model for a wide variety
of eld theories.
Field Theories
In general, a eld theory describes interactions between
dierent objects.
The electrostatic interaction between two point-like objects in
a vacuum can be written:
F
1
=
1
4
0
q
1
q
2
r
21
|r
21
|
3
(8)
F
2
=
1
4
0
q
1
q
2
r
12
|r
12
|
3
(9)
where

F
1
and

F
2
are the forces on the objects which carry
charges q
1
and q
2
respectively; r
21
and r
12
are vectors giving
the relative positions of the objects; and
0
is a fundamental
physical constant that expresses the strength of the interaction
per unit charge.
Field Theories
The interaction can be written in terms of an electric eld that
is created by a charged object. For example, we can dene the
electric eld at location r
12
from an isolated point charge q
1
to
be:
E(r
12
) =
1
4
0
q
1
r
12
|r
12
|
3
(10)
In terms of this electric eld, the force F
2
on a second point
charge, q
2
, in the eld

E(r
12
) is given by:
F
2
= q
2
E(r
12
) (11)
Equation (10) relates the eld to its source. Equation (11)
tells us the eect of the eld on an object in the eld. These
are the essential ingredients of a eld theory.
Electromagnetism as a Field Theory
The full set of equations that relate electric elds (
E and

D)
and magnetic elds (
B and

H) to their sources (charge density
and current density

J) can be written, in dierential form, as
follows:

D =

B = 0
(12)

E =
B
t

H =

J +

D
t
To nd explicit expressions for the elds at a given place and
time, we have to solve these dierential equations with the
boundary conditions imposed by the sources of the elds.
The eects of the elds on a point-like object moving in an
electromagnetic eld can be written (the Lorentz force
equation):
F = q
_
E +v

B
_
(13)
where v is the velocity of the object.
Fields and Potentials
Sometimes, the eld equations take a simpler form when they
are expressed in terms of potentials rather than directly in
terms of the elds.
A potential is a mathematical scalar or vector function (of
space and time) whose derivative gives the eld.
For example, the magnetic eld can be expressed in terms of
the magnetic vector potential

A:
B =

A (14)
and the electric eld can be expressed in terms of the magnetic
vector and electric scalar potential :
E =

A
t
(15)
Note that elds are associated with forces, whereas potentials
are associated with energy.
In electrostatics:
the eld is the force per unit charge;
the potential is the potential energy per unit charge;
the eld is the gradient of the potential (since the force is
the gradient of the potential energy).
In terms of the potentials, the electromagnetic eld equations
become second-order dierential equations. But they take a
nice, symmetric form:
A
t
2
=
J (16)
t
2
=
(17)
where and are quantities that characterise the strength of
the electric and magnetic interactions in the material in which
the elds exist.
Note that for a static, point-like charge q, Equation (17) has
the familiar solution:
=
1
4
q
r
(18)
where r is the distance from the charge.
How many types of eld are there?
We believe that there are only four fundamental types of eld
(sometimes called simply forces) in nature. The familiar ones
from everyday experience are gravity (the rst eld to be
described mathematically) and electromagnetism.
The other fundamental forces are the weak nuclear and strong
nuclear forces. They dier from gravity and electromagnetism
in a number of respects: for example, they act only over very
short distances, whereas gravity and electromagnetism are both
capable of very long range interactions.
Long-Range and Short-Range Fields
The mathematical description of a short-range interaction is
very similar to that of a long-range interaction. Take Equation
(17) above, and simply add another term:
t
2

l
2
=
(19)
where l is a constant. For a static, point-like source, Equation
(19) has the solution:
=
1
4
q
e
r
l
r
(20)
For small l, the potential falls o much more quickly than 1/r...
Long-Range and Short-Range Fields
The constant l characterises the range of the force. For
gravity and electromagnetism we believe (from experiments)
that 1/l = 0.
Classical and Quantum Fields
All the equations on the previous slides are classical equations:
they take no account of quantum eects, and make no
reference to Plancks constant h.
However, we believe that when particles interact, they do so by
exchanging discrete amounts of energy, called quanta. When
we develop a classical eld theory to include quantum eects,
we construct a quantum eld theory.
In general, quantum eld theories are much more complicated
than classical eld theories. But some of the consequences of
quantisation can be understood using simplied models.
In quantum eld theory, an interaction between two particles is
understood in terms of an exchange of a third particle: the type
of particle exchanged is determined by the type of interaction.
Using the physical constants h and c, we can dene a mass m
associated with the length scale l of the interaction:
m =
h
c
1
l
(21)
In quantum eld theory, the mass m is identied with the mass
of the exchange particle. Note that as l , m 0.
This simple model explains why gravity and electromagnetism
are long-range forces (since the graviton and photon have zero
mass) while the weak interaction has only a very short-range
(since the W and Z bosons have mass of around 90 GeV/c
2
).
But it doesnt explain why the strong interaction is also
short-range, despite the fact that the gluon has zero mass.
When a eld theory is quantised, many new features can
appear that are not expected from relatively simple classical
eld theories. The quantum eld theory of the strong
interaction is especially complicated.
The electromagnetic eld was the rst to be understood as a
quantum theory. We are still searching for a quantum theory of
gravity.
The Electromagnetic Field
The classical electromagnetic eld is a good case study for
other eld theories, including quantum eld theories.
Our main goal in this course will be to understand a variety of
electromagnetic phenomena in terms of solutions to the eld
equations (Maxwells equations). The phenomena we shall
study will include:
Electromagnetic waves in various media
Electromagnetic waves at boundaries between dierent media
Propagation of electromagnetic waves in waveguides
Sources of electromagnetic radiation
We shall begin by reviewing the various quantities associated
with electromagnetic elds, and the relationships between
them...
The Electric Field
The electric eld

E at a particular point in space is the force
per unit static electric charge located at that point:
E =
F
q
(22)
In free space, the electric eld is very simple. Things get more
complicated when we need to describe electric elds within
materials...
Electric Polarisation
When a dielectric (non-conducting material) is placed in an
electric eld, the molecules within the material each acquire an
electric dipole moment. The dipole moment measures the
displacement of the electric charges within the molecule, in
response to the external electric eld.
The electric dipole moment p is dened as the magnitude of
the charge multiplied by the separation:
p = qx (23)
Note that the dipole moment is a vector quantity.
Electric Polarisation
The electric polarisation

P of a dielectric is dened as the
dipole moment p per unit volume. Thus, if there are N
molecules per unit volume, each with electric dipole moment p,
the polarisation of the dielectric will be:
P = N p = Nqx (24)
Electric Polarisation and Susceptibility
In general, the response of a material to an external electric
eld is very complicated. Even in relatively simple materials, we
need a good understanding of the quantum mechanics of the
atoms and molecules within the material if we want to
calculate the polarisation from rst principles.
However, for many materials we can make the approximation
that the polarisation is proportional to the external electric
eld. The constant of proportionality is the product of the
permittivity of free space,
0
, and the electric susceptibility,
e
:
P =
e
E (25)
Note that the susceptibility
e
is a dimensionless quantity.
Equation (25) is a good approximation for materials that are
homogeneous, isotropic and linear. There are various ways in
which the susceptibility of a given material can be measured.
Electric Displacement
The electric displacement

D is a measure of the electric eld
within a material, taking into account the polarisation

P:
D =
0
E +

P (26)
Note that the polarisation generated by an external electric
eld will tend to reduce the strength of the eld.
The electric susceptibility describes how the polarisation
depends on the external electric eld. Equation (25) tells us
that for homogeneous, isotropic, linear dielectrics:
P =
e
E
Combining equations (26) and (25), we nd:
D =
0
E +
e
E (27)
= (1 +
e
)
0
E (28)
Relative Permittivity
The electric displacement is the measure of the electric eld in
a material, taking into account the response of the material to
the eld. The magnitude of the response of a material to an
external electric eld can be measured by the electric
susceptibility
e
, or by the relative permittivity
r
.
The relative permittivity
r
is dened by:
D =
r
E (29)
Combining equations (28) and (29), we nd:
r
= 1 +
e
(30)
Note that, like the susceptibility, the relative permittivity is
dimensionless.
Permittivity
A material that has no response to an external electric eld
(like a vacuum) will have susceptibility zero, and relative
permittivity equal to 1.
A material that has a strong response to an external electric
eld (by acquiring a large polarisation) will have a susceptibiliy
much larger than zero, and a relative permittivity much larger
than 1.
We sometimes use the permittivity , instead of the relative
permittivity
r
. The permittivity is dened by:
=
r
0
(31)
Summary: The Electric Field
electric eld

E newtons/coulomb (NC
1
)
electric displacement

D coulombs/meter
2
(Cm
2
)
permittivity farads/meter (Fm
1
)
The electric displacement

D and the electric eld

E are related
by the permittivity :
D =
E (32)
The permittivity is a property of materials. The vacuum also
has a permittivity, with value
0
:
0
= 8.854 10
12
Fm
1
(33)
The ratio of the permittivity of a material to the permittivity of
the vacuum is known as the relative permittivity. The relative
permittivity measures the electric polarisation induced in a
material by an external electric eld.
The Magnetic Field
When a charged particle moves through a magnetic eld, it
experiences a force proportional to the size of the eld, the
speed of the particle, and the sine of the angle between the
eld and the velocity. The direction of the force is
perpendicular to both the eld and the velocity:
The magnetic force on the particle can be written:
F = qv

B (34)
Just as we had to account for the response of materials to
external electric elds, we have to account for the response of
materials to external magnetic elds.
The Magnetic Field in Materials
The magnetic eld around an individual atom or molecule can
modelled as a current I owing in a loop enclosing an area

A:
Note that

A is a vector, with magnitude equal to the area
enclosed by the loop, and direction perpendicular to the current
loop.
The magnetic dipole moment m of the current loop is dened
by:
m = I

A (35)
Magnetisation
The magnetisation of a material is dened as the magnetic
dipole moment per unit volume. Thus, if there are N magnetic
dipoles per unit volume with average dipole moment m, the
magnetisation

M is given by:
M = N m (36)
In a magnetic material, the magnetic moments of atoms and
molecules within the material can change in response to an
applied external magnetic eld. The response of the material
to an external magnetic eld

B is measured by the magnetic
intensity

H:
B =
0
_
H +

M
_
(37)
where
0
is a fundamental physical constant, the permeability
of free space.
Magnetic Susceptibility
In certain kinds of materials (diamagnets and paramagnets),
the magnetisation is approximately proportional to the
magnetic intensity:
M =
m
H (38)
Note that this relationship does not hold for ferromagnets,
which are more complicated. In particular, ferromagnets display
hysteresis, in which the magnetisation depends not only on the
magnetic intensity present at a given time, but on the magnetic
intensity that was present in the material at earlier times. This
means that there cannot be a simple one-to-one relationship
between magnetic intensity and magnetisation in ferromagnetic
materials.
Magnetic Permeability
In cases where equation (38) holds, we can write the
relationship (37) between the magnetic eld and the magnetic
intensity as:
B =
0
_
H +
m
H
_
(39)
= (1 +
m
)
0
H (40)
We dene the relative permeability
r
as:
r
= 1 +
m
(41)
so that for diamagnetic and paramagnetic materials:
B =
r
H (42)
We can also dene the magnetic permeability as:
=
r
0
(43)
Summary: The Magnetic Field
magnetic intensity

H amperes/meter (Am
1
)
magnetic eld

B tesla (T)
permeability henrys/meter (Hm
1
)
The magnetic intensity

H and the magnetic eld

B in a
diamagnetic or paramagnetic material are related by the
permeability :
B =
H (44)
The permeability is a property of the material. The vacuum
also has a permeability, with value
0
:
0
= 4 10
7
Hm
1
(45)
The ratio of the permeability of a material to the permeability
of free space is the relative permeability of the material. The
relative permeability measures the response of the material to
an external magnetic eld.
Permittivity and Permeability
In static cases, sources of the electromagnetic eld determine
the electric displacement

D and the magnetic intensity

H:

D =

H =

J (46)
The permeability and permittivity describe the magnetisation
and polarisation of a material in response to external elds. In
most materials, the electric eld

E is reduced by the induced
electric dipole moment; and the magnetic eld

B is enhanced
by the induced magnetic moment in the material:
E =
B =
r
H (47)
Permittivity and Permeability of the Vacuum
As we shall see later, the speed of light in a vacuum is given by:
c =
1
0
(48)
It turns out that we can choose one of the constants
0
and
0
for our own convenience; the other is then xed by the speed
of light in a vacuum, from equation (48). Dierent systems of
units make dierent choices for either
0
or
0
. In SI units, the
permeability of free space
0
is dened to be:
0
= 4 10
7
Hm
1
(49)
We then nd, using c = 2.998 10
8
ms
1
, that:
0
= 8.854 10
12
Fm
1
(50)
Charge, Charge Density, Current Density and Conductivity
Electric charge is represented by the symbol q, and is measured
in coulombs (C). The electric charge density (charge per unit
volume) is represented by the symbol , and is measured in
C/m
3
.
In an electrical conductor, an electric eld

E will cause a ow
of electric charge. The ow of charge is given by the current
density

J (which has units of amperes/meter
2
, A/m
2
).
In an ohmic conductor with conductivity , the current density
is given by:
J =
E (51)
This is equivalent to Ohms law, I = V/R.
Maxwells Equations
Maxwells equations determine the electric and magnetic elds
in the presence of sources (charge and current densities), and
in materials of given properties.

D = (52)

B = 0 (53)

E =
B
t
(54)

H =

J +

D
t
(55)
The physical signicance of Maxwells equations is most easily
understood by converting them from dierential equations into
integral equations...
Maxwells Equations
Dierential Integral

D =
_
S

D

dS =
_
V
dV

B = 0
_
S

B

dS = 0

E =
B
t
_
C

E

dl =
t
_
S

B

dS

H =

J +

D
t
_
C

H

dl =
_
S

J

dS +

t
_
S

D

dS
We can show that the dierential forms of Maxwells equations
are equivalent to the integral forms using Gauss theorem and
Stokes theorem...
Gauss Theorem and Stokes Theorem
Gauss theorem for any vector eld

A:
_
V

AdV =
_
S
A

dS (56)
where S is the closed surface bounding the volume V , and the
surface area element

dS is directed out of the volume V .
Stokes theorem for any vector eld

A:
_
S

A

dS =
_
C
A

dl (57)
where C is the closed line bounding the area S.
Maxwells Equations: Physical Interpretation (1)
Gauss theorem and Stokes theorem can be used to transform
between the dierential and integral forms of Maxwells
equations.
For example, consider the rst of Maxwells equations:

D = (58)
Apply Gauss theorem:
_
V

DdV =
_

D

dS (59)
to get the integral form:
_

D

dS =
_
V
dV (60)
This tells us that the ux of

D through a closed surface equals
the enclosed charge.
As an example, consider the eld around a point charge, q.
The eld is spherically symmetric, and at a distance r from a
point charge, passes through a sphere of surface area 4r
2
.
Since the system is spherically symmetric, the integral of the
electric displacement over the surface of a sphere centered on
the point charge is simply equal to the magnitude of the electric
diplacement vector multiplied by the surface area of the sphere:
_
S
D

dS = 4r
2
D (61)
The integral of the charge density over the volume inside the
sphere is simply equal to the charge:
_
V
dV = q (62)
Thus, from equation (60):
_

D

dS =
_
V
dV
we get:
4r
2
D = q (63)
and nally, from

D =
E, we get Coulombs law:

E =
q
4r
2
(64)
Consider the second of Maxwells equations:

B = 0 (65)
Apply Gauss theorem:
_
V

BdV =
_

B

dS (66)
_

B

dS = 0 (67)
This tells us that the ux of

B through a closed surface equals
zero. In other words, as much magnetic eld ows into a
closed surface as ows out of the surface.
The ux of

B through a closed surface equals zero. As a
consequence, there can be no sources or sinks of lines of
magnetic ux: the lines must be continuous, and have no
beginning or end.
Equation (65) is a statement that there are no magnetic
monopoles.
Consider the third of Maxwells equations:

E =
B
t
(68)
Apply Stokes theorem:
_
S

E

dS =
_
C
E

dl (69)
_
C
E

dl =
t
_
S
B

dS =
t
(70)
This tells us that the circulation of

E around any closed curve
is equal to the rate of change of magnetic ux through any
surface spanning the curve.
Electromotive force (emf) is dened by:
emf =
_
C
E

dl (71)
We have seen that applying Stokes theorem (69) to Maxwells
equation (68):

E =
B
t
gives:
_
C
E

dl =
t
_
S
B

dS =
t
(72)
So Maxwells third equation is just a statement of Faradays
(and Lenzs) law:
emf =
_
C
E

dl =
t
(73)
Consider the fourth of Maxwells equations:

H =

J +

D
t
(74)
Apply Stokes theorem:
_
S

H

dS =
_
C
H

dl (75)
_
C
H

dl =
_
S
J

dS +

t
_
S
D

dS (76)
This tells us that the circulation of

H around a closed curve is
equal to the ux of current density through any surface
spanning that curve (Amperes law) plus the rate of change of
electric displacement through any surface spanning the curve
(Maxwells extension to Amperes law).
Consider the magnetic eld around a long, straight wire
carrying an electric current I. The magnetic eld lines form
closed loops perpendicular to, and centered on, the wire.
Consider a disc of radius r perpendicular to, and centered on,
the wire. We integrate Maxwells equation:

H =

J +

D
t
(77)
across the surface of the disc (noting that

D = 0):
_
S

H

dS =
_
S
J

dS = I (78)
Applying Stokes theorem, we get:
_
C
H

dl = 2rH = I (79)
where H is the magnitude of the magnetic intensity at
perpendicular distance r from the wire:
H =
I
2r
(80)
Displacement Current
Maxwells extension to Amperes law introduces the
displacement current:
displacement current =

t
_
S
D

dS (81)
The presence of the displacement current in Maxwells
equations tells us that a changing electric eld gives rise to a
magnetic eld.
But there is another, very important consequence of this term:
it tells us that electric charge is conserved...
Displacement Current and Charge Conservation
Maxwells fourth equation is:

H =

J +

D
t
(82)
Using the vector identity (2) we nd:

H =

D +

J = 0 (83)
Maxwells rst equation is:

D = (84)
which (dierentiating with respect to time, t) gives:
D = =

t
(85)
Combining equations (83) and (85) gives:

J =
t
(86)
This equation is called the continuity equation for electric
charge. It tells us that there is local conservation of electric
charge.
Continuity Equations
The continuity equation (86) follows directly from Maxwells
equations, and tells us that electric charge is conserved locally.
In dierential form, the continuity equation is:

J =
t
(87)
Using Gauss theorem, we can convert to integral form, to
make the physical interpretation clearer:
_

J

dS =
t
_
V
dV (88)
Summary of Part 1
You should be able to:
Explain the features of a eld theory.
Explain the quantities used to describe electromagnetic elds in free
space and in materials (including: electric eld; electric displacement;
magnetic eld; magnetic intensity; electric permittivity; magnetic
permeability) and give the relationships between them.
Write down the eld equations for classical electromagnetism
(Maxwells equations), and the Lorentz force equation.
Write down Gauss theorem and Stokes theorem, and use these
theorems to convert Maxwells equations from dierential to integral
form.
Derive, from Maxwells equations, expressions for: the electric eld
around a point charge; the magnetic eld around a straight wire; the
emf in a wire loop in a time-dependent magnetic eld.
Derive, from Maxwells equations, the continuity equation expressing
the local conservation of electric charge.

Part 1: Maxwell's Equations: PHYS370 - Advanced Electromagnetism

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PHYS370 Advanced Electromagnetism

Part 1: Maxwells Equations

E, we get Coulombs law:

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