Electromagnetic
Field Theory





Electromagnetic
Field Theory
BO THI D�
Swedish Institute of Space Physics
Uppsala, Sweden
and
Department of Astronomy and Space Physics
Uppsala University, Sweden
and
LOIS Space Centre
School of Mathematics and Systems Engineering
V�xj� University, Sweden

U PS IL ON BOO KS UP PS AL A SWE DE N


Also available
ELECTROMAG NETIC FIELD TH EO RY
EXERCISES
by
Tobia Carozzi, Anders Eriksson, Bengt Lundborg,
Bo Thid� and Mattias Waldenvik
Freely downloadable from
www.plasma.uu.se/CED
This book was typeset in LATEX 2 (based on TEX 3.141592 and Web2C 7.4.4) on an HP Visu-
alize 9000/3600 workstation running HP-UX 11.11.
Copyright �1997�2008 by
Bo Thid�
Uppsala, Sweden
All rights reserved.
Electromagnetic Field Theory
ISBN X-XXX-XXXXX-X


To the memory of professor
LEV MI KHAIL OVICH ERUKHIMOV (1936�1997)
dear friend, great physicist, poet
and a truly remarkable man.




Down loade d f r om ht tp :/ /www. plas m a. uu. se /CE D/ Bo ok
Ve r si on re leas ed 8t h J une 200 8 at 23: 04.
CONTENTS
Contents
ix
List of Figures
xiii
Preface
xv
1 Classical Electrodynamics
1
1.1 Electrostatics
2
1.1.1 Coulomb�s law
2
1.1.2 The electrostatic field
3
1.2 Magnetostatics
6
1.2.1 Amp�re�s law
6
1.2.2 The magnetostatic field
7
1.3 Electrodynamics
9
1.3.1 Equation of continuity for electric charge 10
1.3.2 Maxwell�s displacement current
10
1.3.3 Electromotive force
11
1.3.4 Faraday�s law of induction
12
1.3.5 Maxwell�s microscopic equations
15
1.3.6 Maxwell�s macroscopic equations
15
1.4 Electromagnetic duality
16
1.5 Bibliography
18
1.6 Examples
20
2 Electromagnetic Waves
25
2.1 The wave equations
26
2.1.1 The wave equation for E
26
2.1.2 The wave equation for B
27
2.1.3 The time-independent wave equation for E 27
ix


Contents
2.2 Plane waves
30
2.2.1 Telegrapher�s equation
31
2.2.2 Waves in conductive media
32
2.3 Observables and averages
33
2.4 Bibliography
35
2.5 Example
36
3 Electromagnetic Potentials
39
3.1 The electrostatic scalar potential
39
3.2 The magnetostatic vector potential
40
3.3 The electrodynamic potentials
40
3.4 Gauge transformations
41
3.5 Gauge conditions
42
3.5.1 Lorenz-Lorentz gauge
43
3.5.2 Coulomb gauge
47
3.5.3 Velocity gauge
49
3.6 Bibliography
49
3.7 Examples
51
4 Electromagnetic Fields and Matter
53
4.1 Electric polarisation and displacement
53
4.1.1 Electric multipole moments
53
4.2 Magnetisation and the magnetising field
56
4.3 Energy and momentum
58
4.3.1 The energy theorem in Maxwell�s theory 58
4.3.2 The momentum theorem in Maxwell�s theory 59
4.4 Bibliography
62
4.5 Example
63
5 Electromagnetic Fields from Arbitrary Source Distributions 65
5.1 The magnetic field
67
5.2 The electric field
69
5.3 The radiation fields
71
5.4 Radiated energy
74
5.4.1 Monochromatic signals
74
5.4.2 Finite bandwidth signals
75
5.5 Bibliography
76
6 Electromagnetic Radiation and Radiating Systems
77
6.1 Radiation from an extended source volume at rest 77
6.1.1 Radiation from a one-dimensional current distribution 78
x
Ver s io n r el eas ed 8th J une 2008 at 2 3:04 . Downloa ded fr om htt p: // www.pl as ma. uu. s e/ CED/ B ook


6.1.2 Radiation from a two-dimensional current distribution 81
6.2 Radiation from a localised source volume at rest 85
6.2.1 The Hertz potential
85
6.2.2 Electric dipole radiation
90
6.2.3 Magnetic dipole radiation
91
6.2.4 Electric quadrupole radiation
93
6.3 Radiation from a localised charge in arbitrary motion 93
6.3.1 The Li�nard-Wiechert potentials
94
6.3.2 Radiation from an accelerated point charge 97
6.3.3 Bremsstrahlung
105
6.3.4 Cyclotron and synchrotron radiation
108
6.3.5 Radiation from charges moving in matter 116
6.4 Bibliography
123
6.5 Examples
124
7 Relativistic Electrodynamics
133
7.1 The special theory of relativity
133
7.1.1 The Lorentz transformation
134
7.1.2 Lorentz space
136
7.1.3 Minkowski space
141
7.2 Covariant classical mechanics
143
7.3 Covariant classical electrodynamics
145
7.3.1 The four-potential
145
7.3.2 The Li�nard-Wiechert potentials
146
7.3.3 The electromagnetic field tensor
148
7.4 Bibliography
152
8 Electromagnetic Fields and Particles
155
8.1 Charged particles in an electromagnetic field
155
8.1.1 Covariant equations of motion
155