ELECTRO MAGNETIC FIELD THEORY - ¢â‚¬“main¢â‚¬â€Œ 2000/11/13 page 1 ELECTROMAGNETIC FIELD THEORY Bo Thid£©

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    ELECTRO MAGNETIC FIELD THEORY

    Υ

    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

    Υ 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 LATEX2ε 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 xi

    1 Classical Electrodynamics 1 1.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.1.1 Coulomb’s law . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 The electrostatic field . . . . . . . . . . . . . . . . . . 2

    1.2 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Ampère’s law . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 The magnetostatic field . . . . . . . . . . . . . . . . . 6

    1.3 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Equation of continuity . . . . . . . . . . . . . . . . . 9 1.3.2 Maxwell’s displacement current . . . . . . . . . . . . 9 1.3.3 Electromotive force . . . . . . . . . . . . . . . . . . . 10 1.3.4 Faraday’s law of induction . . . . . . . . . . . . . . . 11 1.3.5 Maxwell’s microscopic equations . . . . . . . . . . . 14 1.3.6 Maxwell’s macroscopic equations . . . . . . . . . . . 14

    1.4 Electromagnetic Duality . . . . . . . . . . . . . . . . . . . . 15 Example 1.1 Duality of the electromagnetodynamic equations 16 Example 1.2 Maxwell from Dirac-Maxwell equations for a

    fixed mixing angle . . . . . . . . . . . . . . . 17 Example 1.3 The complex field six-vector . . . . . . . . 18 Example 1.4 Duality expressed in the complex field six-vector 19

    Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    2 Electromagnetic Waves 23 2.1 The wave equation . . . . . . . . . . . . . . . . . . . . . . . 24

    2.1.1 The wave equation for E . . . . . . . . . . . . . . . . 24 2.1.2 The wave equation for B . . . . . . . . . . . . . . . . 24 2.1.3 The time-independent wave equation for E . . . . . . 25

    2.2 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.1 Telegrapher’s equation . . . . . . . . . . . . . . . . . 27

    i

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    ii CONTENTS

    2.2.2 Waves in conductive media . . . . . . . . . . . . . . . 29 2.3 Observables and averages . . . . . . . . . . . . . . . . . . . . 30 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    3 Electromagnetic Potentials 33 3.1 The electrostatic scalar potential . . . . . . . . . . . . . . . . 33 3.2 The magnetostatic vector potential . . . . . . . . . . . . . . . 34 3.3 The electromagnetic scalar and vector potentials . . . . . . . . 34

    3.3.1 Electromagnetic gauges . . . . . . . . . . . . . . . . 36 Lorentz equations for the electromagnetic potentials . 36 Gauge transformations . . . . . . . . . . . . . . . . . 36

    3.3.2 Solution of the Lorentz equations for the electromag- netic potentials . . . . . . . . . . . . . . . . . . . . . 38 The retarded potentials . . . . . . . . . . . . . . . . . 41

    Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

    4 The Electromagnetic Fields 43 4.1 The magnetic field . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 The electric field . . . . . . . . . . . . . . . . . . . . . . . . 47 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

    5 Relativistic Electrodynamics 51 5.1 The special theory of relativity . . . . . . . . . . . . . . . . . 51

    5.1.1 The Lorentz transformation . . . . . . . . . . . . . . 52 5.1.2 Lorentz 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

    5.1.3 Minkowski space . . . . . . . . . . . . . . . . . . . . 58 5.2 Covariant classical mechanics . . . . . . . . . . . . . . . . . 61 5.3 Covariant classical electrodynamics . . . . . . . . . . . . . . 62

    5.3.1 The four-potential . . . . . . . . . . . . . . . . . . . 62 5.3.2 The Liénard-Wiechert potentials . . . . . . . . . . . . 63 5.3.3 The electromagnetic field tensor . . . . . . . . . . . . 65

    Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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    6 Interactions of Fields and Particles 69 6.1 Charged Particles in an Electromagnetic Field . . . . . . . . . 69

    6.1.1 Covariant equations of motion . . . . . . . . . . . . . 69 Lagrange formalism . . . . . . . . . . . . . . . . . . 69 Hamiltonian formalism . . . . . . . . . . . . . . . . . 72

    6.2 Covariant Field Theory . . . . . . . . . . . . . . . . . . . . . 76 6.2.1 Lagrange-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 . . . . . . . . . . . . . . . . . . . . . . . 84

    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 7.2 Magnetisation and the magnetising field . . . . . . . . . . . . 90 7.3 Energy and momentum . . . . . . . . . . . . . . . . . . . . . 91

    7.3.1 The energy theorem in Maxwell’s theory . . . . . . . 92 7.3.2 The momentum theorem in Maxwell’s theory . . . . . 93

    Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

    8 Electromagnetic Radiation 97 8.1 The radiation fields . . . . . . . . . . . . . . . . . . . . . . . 97 8.2 Radiated energy . . . . . . . . . . . . . . . . . . . . . . . . . 99

    8.2.1 Monochromatic signals . . . . . . . . . . . . . . . . . 100 8.2.2 Finite bandwidth signals . . . . . . . . . . . . . . . . 100

    8.3 Radiation from extended sources . . . . . . . . . . . . . . . . 102 8.3.1 Linear antenna . . . . . . . . . . . . . . . . . . . . . 102

    8.4 Multipole radiation . . . . . . . . . . . . . . . . . . . . . . . 104 8.4.1 The Hertz potential . . . . . . . . . . . . . . . . . . . 104 8.4.2 Electric dipole radiation . . . . . . . . . . . . . . . . 108 8.4.3 Magnetic dipole radiation . . . . . . . . . . . . . . . 109 8.4.4 Electric quadrupole radiation . . . . . . . . . . . . . . 110

    8.5 Radiation from a localised charge in arbitrary motion . . . . . 111 8.5.1 The Liénard-Wiechert potentials . . . . . . . . . . . . 112 8.5.2 Radiation from an accelerated point charge . . . . . . 114

    Example 8.1 The fields from a uniformly moving charge . 121 Example 8.2 The convection potential and the convection

    force . . . . . . . . . . . . . . . . . . . . . 123

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    Radiation for small velocities . . . . . . . . . . . . . 125 8.5.3 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . 127

    Example 8.3 Bremsstrahlung for low speeds and short ac- celeration times . . . . . . . . . . . . . . . . 130

    8.5.4 Cyclotron and synchrotron radiation . . . . . . . . . . 132 Cyclotron radiation . . . . . . . . . . . . . . . . . . . 134 Synchrotron radiation . . . . . . . . . . . . . . . . . . 134 Radiation in the general case . . . . . . . . . . . . . . 137 Virtual photons . . . . . . . . . . . . . . . . . . . . . 137

    8.5.5 Radiation from charges moving in matter . . . . . . . 139 Vavilov-Čerenkov radiation . . . . . . . . . . . . . . 142

    Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

    F Formulae 149 F.1 The Electromagnetic Field . . . . . . . . . . . . . . . . . . . 149

    F.1.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . 149 Constitutive relations . . . . . . . . . . . . . . . . . . 149

    F.1.2 Fields and potentials . . . . . . . . . . . . . . . . . . 149 Vector and scalar potentials . . . . . . . . . . . . . . 149 Lorentz’ gauge condition in vacuum . . . . . . . . . . 150

    F.1.3 Force and energy . . . . . . . . . . . . . . . . . . . . 150 Poynting’s vector . . . . . . . . . . . . . . . . . . . . 150 Maxwell’s stress tensor . . . . . . . . . . . . . . . . . 150

    F.2 Electromagnetic Radiation . . . . . . . . . . . . . . . .