Electro Magnetic Field Theory

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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 LATEX 2ε 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   . . . . . . 252.2 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

    Downloaded fromhttp://www.plasma.uu.se/CED/Book   Draft version released 13th November 2000 at 22:01.

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    iii

    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 . . . . . . . . . . . . . . . . . . . . . .