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Electromagnetic
Field Theory
Bo Thidé
Swedish Institute of Space Physics
and
Department of Astronomy and Space Physics
Uppsala University, Sweden



Contents
Preface xi
1 Classical Electrodynamics 1
1.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Coulomb’s law . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 The electrostatic field . . . . . . . . . . . . . . . . . . 2
1.2 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Ampère’s law . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 The magnetostatic field . . . . . . . . . . . . . . . . . 6
1.3 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Equation of continuity for electric charge . . . . . . . 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 Faraday’s law as a consequence of conservation
of magnetic charge . . . . . . . . . . . . 16
Example 1.2 Duality of the electromagnetodynamic equations 18
Example 1.3 Dirac’s symmetrised Maxwell equations for a
fixed mixing angle . . . . . . . . . . . . . . . 18
Example 1.4 The complex field six-vector . . . . . . . . 20
Example 1.5 Duality expressed in the complex field six-vector 20
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 ElectromagneticWaves 23
2.1 The Wave Equations . . . . . . . . . . . . . . . . . . . . . . 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
Example 2.1 Wave equations in electromagnetodynamics . 26
2.2 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Telegrapher’s equation . . . . . . . . . . . . . . . . . 29
2.2.2 Waves in conductive media . . . . . . . . . . . . . . . 30
2.3 Observables and Averages . . . . . . . . . . . . . . . . . . . 32
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Electromagnetic Potentials 35
3.1 The Electrostatic Scalar Potential . . . . . . . . . . . . . . . . 35
3.2 The Magnetostatic Vector Potential . . . . . . . . . . . . . . . 36
3.3 The Electrodynamic Potentials . . . . . . . . . . . . . . . . . 36
3.3.1 Electrodynamic gauges . . . . . . . . . . . . . . . . . 38
Lorentz equations for the electrodynamic potentials . . 38
Gauge transformations . . . . . . . . . . . . . . . . . 39
3.3.2 Solution of the Lorentz equations for the electromagnetic
potentials . . . . . . . . . . . . . . . . . . . . . 40
The retarded potentials . . . . . . . . . . . . . . . . . 43
Example 3.1 Electromagnetodynamic potentials . . . . . 44
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Relativistic Electrodynamics 47
4.1 The Special Theory of Relativity . . . . . . . . . . . . . . . . 47
4.1.1 The Lorentz transformation . . . . . . . . . . . . . . 48
4.1.2 Lorentz space . . . . . . . . . . . . . . . . . . . . . . 49
Radius four-vector in contravariant and covariant form 50
Scalar product and norm . . . . . . . . . . . . . . . . 50
Metric tensor . . . . . . . . . . . . . . . . . . . . . . 51
Invariant line element and proper time . . . . . . . . . 52
Four-vector fields . . . . . . . . . . . . . . . . . . . . 54
The Lorentz transformation matrix . . . . . . . . . . . 54
The Lorentz group . . . . . . . . . . . . . . . . . . . 54
4.1.3 Minkowski space . . . . . . . . . . . . . . . . . . . . 55
4.2 Covariant Classical Mechanics . . . . . . . . . . . . . . . . . 57
4.3 Covariant Classical Electrodynamics . . . . . . . . . . . . . . 59
4.3.1 The four-potential . . . . . . . . . . . . . . . . . . . 59
4.3.2 The Liénard-Wiechert potentials . . . . . . . . . . . . 60
4.3.3 The electromagnetic field tensor . . . . . . . . . . . . 62
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 Electromagnetic Fields and Particles 67
5.1 Charged Particles in an Electromagnetic Field . . . . . . . . . 67
5.1.1 Covariant equations of motion . . . . . . . . . . . . . 67
Lagrange formalism . . . . . . . . . . . . . . . . . . 67
Hamiltonian formalism . . . . . . . . . . . . . . . . . 70
5.2 Covariant Field Theory . . . . . . . . . . . . . . . . . . . . . 74
5.2.1 Lagrange-Hamilton formalism for fields and interactions 74
The electromagnetic field . . . . . . . . . . . . . . . . 78
Example 5.1 Field energy difference expressed in the field
tensor . . . . . . . . . . . . . . . . . . . . . 79
Other fields . . . . . . . . . . . . . . . . . . . . . . . 82
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 Electromagnetic Fields and Matter 85
6.1 Electric Polaris__TEXT IS TOO BIG. IT WAS TRUNCATED TO 5000 SYMBOLS

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