.5
Conservation of momentum
6
1.6
Junction conditions
7
1.7
Potentials
10
1.8
Greens function for Poissons equation
12
1.9
Greens function for the wave equation
14
1.10 Summary
16
1.11 Problems
17
2 Electrostatics
21
2.1
Equations of electrostatics
21
2.2
Point charge
21
2.3
Dipole
22
2.4
Boundary-value problems: Greens theorem
24
2.5
Laplaces equation in spherical coordinates
28
2.6
Greens function in the absence of boundaries
30
2.7
Dirichlet Greens function for a spherical inner boundary
34
2.8
Point charge outside a grounded, spherical conductor
35
2.9
Ring of charge outside a grounded, spherical conductor
38
2.10 Multipole expansion of the electric eld
40
2.11 Multipolar elds
42
2.12 Problems
45
3 Magnetostatics
47
3.1
Equations of magnetostatics
47
3.2
Circular current loop
47
3.3
Spinning charged sphere
49
3.4
Multipole expansion of the magnetic eld
51
3.5
Problems
53
4 Electromagnetic waves in matter
55
4.1
Macroscopic form of Maxwells equations
55
4.1.1
Microscopic and macroscopic quantities
55
4.1.2
Macroscopic smoothing
56
4.1.3
Macroscopic averaging of the charge density
57
4.1.4
Macroscopic averaging of the current density
60
4.1.5
Summary macroscopic Maxwell equations
61
4.2
Maxwells equations in the frequency domain
62
4.3
Dielectric constant
63
i ii
Contents
4.4
Propagation of plane, monochromatic waves
65
4.5
Propagation of wave packets
68
4.5.1
Description of wave packets
68
4.5.2
Propagation without dispersion
69
4.5.3
Propagation with dispersion
69
4.5.4
Gaussian wave packet
71
4.6
Problems
73
5 Electromagnetic radiation from slowly moving sources
75
5.1
Equations of electrodynamics
75
5.2
Plane waves
76
5.3
Spherical waves the oscillating dipole
78
5.3.1
Plane versus spherical waves
78
5.3.2
Potentials of an oscillating dipole
79
5.3.3
Wave-zone elds of an oscillating dipole
81
5.3.4
Poynting vector of an oscillating dipole
83
5.3.5
Summary oscillating dipole
84
5.4
Electric dipole radiation
84
5.4.1
Slow-motion approximation; near and wave zones
85
5.4.2
Scalar potential
85
5.4.3
Vector potential
87
5.4.4
Wave-zone elds
88
5.4.5
Energy radiated
90
5.4.6
Summary electric dipole radiation
90
5.5
Centre-fed linear antenna
91
5.6
Classical atom
94
5.7
Magnetic-dipole and electric-quadrupole radiation
95
5.7.1
Wave-zone elds
96
5.7.2
Charge-conservation identities
98
5.7.3
Vector potential in the wave zone
99
5.7.4
Radiated power (magnetic-dipole radiation)
100
5.7.5
Radiated power (electric-quadrupole radiation)
101
5.7.6
Total radiated power
102
5.7.7
Angular integrations
103
5.8
Pulsar spin-down
103
5.9
Problems
105
6 Electrodynamics of point charges
109
6.1
Lorentz transformations
109
6.2
Fields of a uniformly moving charge
112
6.3
Fields of an arbitrarily moving charge
114
6.3.1
Potentials
115
6.3.2
Fields
116
6.3.3
Uniform motion
120
6.3.4
Summary
121
6.4
Radiation from an accelerated charge
122
6.4.1
Angular prole of radiated power
122
6.4.2
Slow motion: Larmor formula
123
6.4.3
Linear motion
124
6.4.4
Circular motion
126
6.4.5
Synchrotron radiation
127
6.4.6
Extremely relativistic motion
131
6.5
Radiation reaction
132
6.6
Problems
135 Chapter 1
Maxwells electrodynamics
1.1
Dynamical variables of electromagnetism
The classical theory of the electromagnetic eld, as formulated by Maxwell, involves
the vector elds
E
(t, x) electric eld at position x and time t
(1.1.1)
and
B
(t, x) magnetic eld at position x and time t.
(1.1.2)
We have two vectors at each position of space and at each moment of time. The
dynamical system is therefore much more complicated than in mechanics, in which
there is a nite number of degrees of freedom. Here the number of degrees of freedom
is innite.
The electric and magnetic elds are produced by charges and currents. In a
classical theory these are best described in terms of a uid picture in which the
charge and current distributions are imagined to be continuous (and not made of
pointlike charge carriers). Although this is not a true picture of reality, these con-
tinuous distributions t naturally within a classical treatment of electrodynamics.
This will be our point of view here, but we shall see that the formalism is robust
enough to also allow for a description in terms of point particles.
An element of charge is a macroscopically small (but microscopically large)
portion of matter that contains a net charge. An element of charge is located
at position x and has a volume dV . It moves with a velocity v that depends on
time and on position. The volume of a charge element must be suciently large
that it contains a macroscopic number of elementary charges, but suciently small
that the density of charge within the volume is uniform to a high degree of accuracy.
In the mathematical description, an element of charge at position x is idealized as
the point x itself. The quantities that describe the charge and current distributions
are
(t, x)
density of charge at position x and time t,
(1.1.3)
v
(t, x)
velocity of an element of charge at position x and time t,(1.1.4)
j
(t, x)
current density at position x and time t.
(1.1.5)
We shall now establish the important relation
j
= v.
(1.1.6)
The current density j is dened by the statement
j
da current crossing an element of area da,
(1.1.7)
1 2
Maxwells electrodynamics (v t )cos Figure 1.1: Point charges hitting a small surface.
so that j is the current per unit area. The current owing across a surface S is then
S
j
da.
On the other hand, current is dened as the quantity of charge crossing a surface
per unit time. Let us then calculate the current associated with a distribution of
charge with density and velocity v.
Figure 1.1 shows a number of charged particles approaching with speed v a small
surface of area a; there is an angle between the direction of the velocity vector
v
and the normal
n
to the surface. The particles within the dashed box will all hit
the surface within a time t. The volume of this box is (vt cos )a = v
n
ta.
The current crossing the surface is then calculated as
current =
(charge crossing the surface)/t
=
(total charge) volume of box
total volume
1
t
=
(density)(volume of box)/t
= (v
n
ta)/t
= v (
n
a)
= v a.
Comparing this with the expression given previously, current = j x, we nd that
indeed, v is the current density.
As was mentioned previously, the uid description still allows for the existence
of point charges. For these, the density is zero everywhere except at the position
of a charge, where it is innite. This situation can be described by a -function.
Suppose that we have a point charge q at a position r. Its charge density can be
written as
(x) = q(x r),
where (x) (x)(y)(z) is a three-dimensional -function. If the charge is moving
with velocity v, then
j
(x) = qv(x r) 1.2
Maxwells equations and the Lorentz force
3
is the current density.
More generally, let us have a collection of point charges q
A
at positions r
A
(t),
moving with velocities v
A
(t) = dr
A
/dt. Then the charge and current densities of
the charge distribution are given by
(t, x) =
A
q
A
x r
A
(t)
(1.1.8)
and
j
(t, x) =
A
q
A
v
A
(t) x r
A
(t) .
(1.1.9)
Each q
A
is the integral of (t, x) over a volume V
A
that encloses this charge but no
other:
q
A
=
V
A
(t, x) d
3
x,
(1.1.10)
where d
3
x dxdydz. The total charge of the distribution is obtained by integrating
the density over all space:
Q (t,x)d
3
x =
A
q
A
.
(1.1.11)
1.2
Maxwells equations and the Lorentz force
The four Maxwell equations determine the electromagnetic eld once the charge
and current distributions are specied. They are given by E
=
1 0
,
(1.2.1) B
=
0,
(1.2.2) × E = B
t ,
(1.2.3) × B =
0
j
+
0 0
E
t .
(1.2.4)
Here,
0
and
0
are constants, and = (
x
,
y
,
z
) is the gradient operator
of vector calculus (with the obvious notation
x
f = f /x for any function f ).
The Maxwell equations state that the electric eld is produced by charges and
time-varying magnetic elds, while the magnetic eld is produced by currents and
time-varying electric elds. Maxwells equations can also be presented in integral
form, by invoking the Gauss and Stokes theorems of vector calculus.
The Lorentz-force law determines the motion of the charges once the electro-
magnetic eld is specied. Let
f
(t, x) force density at position x and time t,
(1.2.5)
where the force density is dened to be the net force acting on an element of charge
at x divided by the volume of this charge element. The statement of the Lorentz-
force law is then
f
= E + j × B = E + v × B .
(1.2.6)
The net force F acting on a volume V of the charge distribution is the integral of
the force density over this volume:
F
(t, V ) =
V
f
(t, x) d
3
x.
(1.2.7) 4
Maxwells electrodynamics
For a single point charge we have = q(x r), j = qv(xr), and the total force
becomes
F
= q E(r) + v × B(r) .
Here the elds are evaluated at the charges position. This is the usual expression
for the Lorentz force, but the denition of Eq. (1.2.6) is more general.
Taken together, the Maxwell equations and the Lorentz-force law determine the
behaviour of the charge and current distributions, and the evolution of the electric
and magnetic elds. Those ve equations summarize the complete theory of classical
electrodynamics. Every conceivable phenomenon involving electromagnetism can be
predicted from them.
1.3
Conservation of charge
One of the most fundamental consequences of Maxwells equations is that charge is
locally conserved
: charge can move around but it cannot be destroyed nor created.
Consider a volume V bounded by a two-dimensional surface S. Charge conser-
vation means that the rate of decrease of charge within V must be equal to the
total current owing across S:
ddt
V
d
3
x =
S
j
da.
(1.3.1)
Local charge conservation