electrodynamics / Electromagnetic Field Theory - Bo Thide
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ELECTRO
MAGNETIC
FIELD
THEORY
Bo Thidé
U P S I L O N M E D I A
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Bo Thidé
ELECTROMAGNETIC FIELD THEORY
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Also available
ELECTROMAGNETIC FIELD THEORY
EXERCISES
by
Tobia Carozzi, Anders Eriksson, Bengt Lundborg,
Bo Thidé and Mattias Waldenvik
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ELECTROMAGNETIC
FIELD THEORY
Bo Thidé
Swedish Institute of Space Physics
and
Department of Astronomy and Space Physics
Uppsala University, Sweden
U P S I L O N M E D I A U P P S A L A S W E D E N
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This book was typeset in LATEX 2" on an HP9000/700 series workstation
and printed on an HP LaserJet 5000GN printer.
Copyright ©1997, 1998, 1999 and 2000 by Bo Thidé
Uppsala, Sweden All rights reserved.
Electromagnetic Field Theory
ISBN X-XXX-XXXXX-X
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Contents
Preface |
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xi |
1 Classical Electrodynamics |
1 |
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1.1 |
Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . |
1 |
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1.1.1 |
Coulomb's law . . . . . . . . . . . . . . . . . . . . . |
1 |
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1.1.2 The electrostatic field . . . . . . . . . . . . . . . . . . |
2 |
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1.2 |
Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . |
4 |
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1.2.1 Ampère's law . . . . . . . . . . . . . . . . . . . . . . |
4 |
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1.2.2 The magnetostatic field . . . . . . . . . . . . . . . . . |
6 |
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1.3 |
Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . |
8 |
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1.3.1 |
Equation of continuity . . . . . . . . . . . . . . . . . |
9 |
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1.3.2 |
Maxwell's displacement current . . . . . . . . . . . . |
9 |
1.3.3Electromotive force . . . . . . . . . . . . . . . . . . . 10
1.3.4Faraday's law of induction . . . . . . . . . . . . . . . 11
1.3.5 Maxwell's microscopic equations . . . . . . . . . . . 14
1.3.6Maxwell's macroscopic equations . . . . . . . . . . . 14
1.4 |
Electromagnetic Duality . . . . . . . . . . . . . . . . . . . . |
15 |
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Example 1.1 Duality of the electromagnetodynamic equations |
16 |
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Example 1.2 |
Maxwell from Dirac-Maxwell equations for a |
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fixed mixing angle . . . . . . . . . . . . . . . |
17 |
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Example 1.3 |
The complex field six-vector . . . . . . . . |
18 |
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Example 1.4 |
Duality expressed in the complex field six-vector |
19 |
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
20 |
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2 Electromagnetic Waves |
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23 |
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2.1 |
The wave equation . . . . . . . . . . . . . . . . . . . . . . . |
24 |
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2.1.1The wave equation for E . . . . . . . . . . . . . . . . 24
2.1.2The wave equation for B . . . . . . . . . . . . . . . . 24
2.1.3 The time-independent wave equation for E . . . . . . 25
2.2Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.1 Telegrapher's equation . . . . . . . . . . . . . . . . . 27
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CONTENTS |
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2.2.2 Waves in conductive media . . . . . . . . . . . . . . . 29
2.3Observables and averages . . . . . . . . . . . . . . . . . . . . 30
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
31 |
3 Electromagnetic Potentials |
33 |
3.1 The electrostatic scalar potential . . . . . . . . . . . . . . . . |
33 |
3.2The magnetostatic vector potential . . . . . . . . . . . . . . . 34
3.3The electromagnetic scalar and vector potentials . . . . . . . . 34
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3.3.1 Electromagnetic gauges . . . . . . . . . . . . . . . . |
36 |
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Lorentz equations for the electromagnetic potentials . |
36 |
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Gauge transformations . . . . . . . . . . . . . . . . . |
36 |
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3.3.2 Solution of the Lorentz equations for the electromag- |
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netic potentials . . . . . . . . . . . . . . . . . . . . . |
38 |
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The retarded potentials . . . . . . . . . . . . . . . . . |
41 |
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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
41 |
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The Electromagnetic Fields |
43 |
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4.1 |
The magnetic field . . . . . . . . . . . . . . . . . . . . . . . |
45 |
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4.2 |
The electric field . . . . . . . . . . . . . . . . . . . . . . . . |
47 |
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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
49 |
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5 |
Relativistic Electrodynamics |
51 |
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5.1The special theory of relativity . . . . . . . . . . . . . . . . . 51
5.1.1 The Lorentz transformation . . . . . . . . . . . . . . 52
5.1.2Lorentz space . . . . . . . . . . . . . . . . . . . . . . 53
Metric tensor . . . . . . . . . . . . . . . . . . . . . . |
54 |
Radius four-vector in contravariant and covariant form |
54 |
Scalar product and norm . . . . . . . . . . . . . . . . |
55 |
Invariant line element and proper time . . . . . . . . . |
56 |
Four-vector fields . . . . . . . . . . . . . . . . . . . . |
57 |
The Lorentz transformation matrix . . . . . . . . . . . |
57 |
The Lorentz group . . . . . . . . . . . . . . . . . . . |
58 |
5.1.3Minkowski space . . . . . . . . . . . . . . . . . . . . 58
5.2 |
Covariant classical mechanics . . . . . . . . . . . . . . . . . |
61 |
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5.3 |
Covariant classical electrodynamics . . . . . . . . . . . . . . |
62 |
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5.3.1 |
The four-potential . . . . . . . . . . . . . . . . . . . |
62 |
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5.3.2 |
The Liénard-Wiechert potentials . . . . . . . . . . . . |
63 |
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5.3.3 |
The electromagnetic field tensor . . . . . . . . . . . . |
65 |
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
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6 Interactions of Fields and Particles |
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6.1Charged Particles in an Electromagnetic Field . . . . . . . . . 69 6.1.1 Covariant equations of motion . . . . . . . . . . . . . 69
Lagrange formalism . . . . . . . . . . . . . . . . . . |
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Hamiltonian formalism . . . . . . . . . . . . . . . . . |
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6.2Covariant Field Theory . . . . . . . . . . . . . . . . . . . . . 76
6.2.1Lagrange-Hamilton formalism for fields and interactions 77 The electromagnetic field . . . . . . . . . . . . . . . . 80
Example 6.1 Field energy difference expressed in the field tensor . . . . . . . . . . . . . . . . . . . . . 81
Other fields . . . . . . . . . . . . . . . . . . . . . . . |
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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
85 |
7 Interactions of Fields and Matter |
87 |
7.1 Electric polarisation and the electric displacement vector . . . |
87 |
7.1.1 Electric multipole moments . . . . . . . . . . . . . . |
87 |
7.2Magnetisation and the magnetising field . . . . . . . . . . . . 90
7.3Energy and momentum . . . . . . . . . . . . . . . . . . . . . 91
7.3.1 The energy theorem in Maxwell's theory . . . . . . . 92
7.3.2The momentum theorem in Maxwell's theory . . . . . 93
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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8 Electromagnetic Radiation |
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8.1 The radiation fields . . . . . . . . . . . . . . . . . . . . . . . |
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8.2Radiated energy . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.2.1 Monochromatic signals . . . . . . . . . . . . . . . . . 100
8.2.2 Finite bandwidth signals . . . . . . . . . . . . . . . . 100
8.3Radiation from extended sources . . . . . . . . . . . . . . . . 102
8.3.1 Linear antenna . . . . . . . . . . . . . . . . . . . . . |
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8.4 Multipole radiation . . . . . . . . . . . . . . . . . . . . . . . |
104 |
8.4.1The Hertz potential . . . . . . . . . . . . . . . . . . . 104
8.4.2 |
Electric dipole radiation |
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108 |
8.4.3 |
Magnetic dipole radiation |
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109 |
8.4.4Electric quadrupole radiation . . . . . . . . . . . . . . 110
8.5 Radiation from a localised charge in arbitrary motion . . . . . 111
8.5.1The Liénard-Wiechert potentials . . . . . . . . . . . . 112
8.5.2 Radiation from an accelerated point charge . . . . . . |
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Example 8.1 The fields from a uniformly moving charge . |
121 |
Example 8.2 The convection potential and the convection |
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force . . . . . . . . . . . . . . . . . . . . . |
123 |
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CONTENTS |
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Radiation for small velocities . . . . . . . . . . . . . 125
8.5.3Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . 127
Example 8.3 Bremsstrahlung for low speeds and short acceleration times . . . . . . . . . . . . . . . . 130
8.5.4Cyclotron and synchrotron radiation . . . . . . . . . . 132 Cyclotron radiation . . . . . . . . . . . . . . . . . . . 134 Synchrotron radiation . . . . . . . . . . . . . . . . . . 134
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Radiation in the general case . . . . . . . . . . . . . . |
137 |
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Virtual photons . . . . . . . . . . . . . . . . . . . . . |
137 |
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8.5.5 |
Radiation from charges moving in matter . . . . . . . |
139 |
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142 |
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Vavilov-Cerenkov radiation . . . . . . . . . . . . . . |
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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
147 |
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F Formulae |
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149 |
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F.1 |
The Electromagnetic Field . . . . . . . . . . . . . . . . . . . |
149 |
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F.1.1 |
Maxwell's equations . . . . . . . . . . . . . . . . . . |
149 |
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Constitutive relations . . . . . . . . . . . . . . . . . . |
149 |
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F.1.2 |
Fields and potentials . . . . . . . . . . . . . . . . . . |
149 |
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Vector and scalar potentials . . . . . . . . . . . . . . |
149 |
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Lorentz' gauge condition in vacuum . . . . . . . . . . |
150 |
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F.1.3 |
Force and energy . . . . . . . . . . . . . . . . . . . . |
150 |
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Poynting's vector . . . . . . . . . . . . . . . . . . . . |
150 |
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Maxwell's stress tensor . . . . . . . . . . . . . . . . . |
150 |
F.2 |
Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . |
150 |
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F.2.1 Relationship between the field vectors in a plane wave |
150 |
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F.2.2 |
The far fields from an extended source distribution . . |
150 |
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F.2.3 |
The far fields from an electric dipole . . . . . . . . . . |
150 |
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F.2.4 |
The far fields from a magnetic dipole . . . . . . . . . |
151 |
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F.2.5 |
The far fields from an electric quadrupole . . . . . . . |
151 |
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F.2.6 The fields from a point charge in arbitrary motion . . . 151 |
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F.2.7 |
The fields from a point charge in uniform motion . . . |
151 |
F.3 |
Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . |
152 |
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F.3.1 |
Metric tensor . . . . . . . . . . . . . . . . . . . . . . |
152 |
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F.3.2 |
Covariant and contravariant four-vectors . . . . . . . . |
152 |
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F.3.3 |
Lorentz transformation of a four-vector . . . . . . . . |
152 |
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F.3.4 Invariant line element . . . . . . . . . . . . . . . . . . |
152 |
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F.3.5 |
Four-velocity . . . . . . . . . . . . . . . . . . . . . . |
152 |
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F.3.6 |
Four-momentum . . . . . . . . . . . . . . . . . . . . |
153 |
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F.3.7 |
Four-current density . . . . . . . . . . . . . . . . . . |
153 |
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F.3.8 Four-potential . . . . . . . . . . . . . . . . . . . . . . |
153 |
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Downloaded from http://www.plasma.uu.se/CED/Book |
Draft version released 13th November 2000 at 22:01. |
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