CONTENTS
1 Maxwell’s Equations 1
1.1 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Example 1.1 Macroscopic Maxwell equations . . . . . . . . . 1
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Example 1.2 Maxwell’s equations in component form . . . . . 4
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Example 1.3 The charge continuity equation . . . . . . . . . 5
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Electromagnetic Potentials and Waves 9
2.1 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Example 2.1 The Aharonov-Bohm effect . . . . . . . . . . . 9
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Example 2.2 Invent your own gauge . . . . . . . . . . . . . 11
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Example 2.3 Fourier transform of Maxwell’s equations . . . . 13
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Example 2.4 Simple dispersion relation . . . . . . . . . . . . 15
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Relativistic Electrodynamics 17
3.1 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Example 3.1 Covariance of Maxwell’s equations . . . . . . . 18
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Example 3.2 Invariant quantities constructed from the field tensor 20
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Example 3.3 Covariant formulation of common electrodynamics
formulas . . . . . . . . . . . . . . . . . . . . 21
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Example 3.4 Fields from uniformly moving charge via Lorentz
transformation . . . . . . . . . . . . . . . . . . . 23
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Lagrangian and Hamiltonian Electrodynamics 27
4.1 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Example 4.1 Canonical quantities for a particle in an EM field . 28
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Example 4.2 Gauge invariance of the Lagrangian density . . . 29
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Electromagnetic Energy, Momentum and Stress 31
5.1 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.3 Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Example 5.1 EM quantities potpourri . . . . . . . . . . . . . 32
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Example 5.2 Classical electron radius . . . . . . . . . . . . 35
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Example 5.3 Solar sailing . . . . . . . . . . . . . . . . . . 37
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Example 5.4 Magnetic pressure on the earth . . . . . . . . . 39
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Radiation from Extended Sources 41
6.1 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.3 Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Example 6.1 Instantaneous current in an infinitely long conductor 42
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Example 6.2 Multiple half-wave antenna . . . . . . . . . . . 47
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Example 6.3 Travelling wave antenna . . . . . . . . . . . . . 50
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Example 6.4 Microwave link design . . . . . . . . . . . . . 51
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7 Multipole Radiation 53
7.1 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.3 Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Example 7.1 Rotating Electric Dipole . . . . . . . . . . . . 54
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Example 7.2 Rotating multipole . . . . . . . . . . . . . . . 56
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Example 7.3 Atomic radiation . . . . . . . . . . . . . . . . 58
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Example 7.4 Classical Positronium . . . . . . . . . . . . . . 59
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8 Radiation from Moving Point Charges 63
8.1 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.2 Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.3 Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Example 8.1 Poynting vector from a charge in uniform motion 64
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Example 8.2 Synchrotron radiation perpendicular to the acceleration
. . . . . . . . . . . . . . . . . . . . . . 66
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Example 8.3 The Larmor formula . . . . . . . . . . . . . . 67
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Example 8.4 Vavilov- ˇ Cerenkov emission . . . . . . . . . . . 69
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9 Radiation from Accelerated Particles 71
9.1 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2 Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.3 Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Example 9.1 Motion of charged particles in homogeneous static
EM fields . . . . . . . . . . . . . . . . . . . . . 72
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Example 9.2 Radiative reaction force from conservation of energy 74
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Example 9.3 Radiation and particle energy in a synchrotron . . 77
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Example 9.4 Radiation loss of an accelerated charged particle . 79
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.1 The turn-on of a linear current at t 0 . . . . . . . . . . . . . . 43
6.2 Snapshots of the field . . . . . . . . . . . . . . . . . . . . . . . . 44
6.3 Multiple half-wave antenna standing current . . . . . . . . . . . . 47
9.1 Motion of a charge in an electric and a magnetic field . . . . . . . 74
PREFACE
This is a companion volume to the book Electromagnetic Field Theory by Bo Thidé.
The problems and their solutions were created by the co-authors who all have
taught this course or its predecessor.
It should be noted that this is a preliminary draft version but it is being corrected
and expanded with time.
Uppsala, Sweden B. T.
December, 1999
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