Covariant Electrodynamics

by Charap, John M.

• ISBN13: 9781421400150
• ISBN10: 1421400154
• Format: Paperback
• Copyright: 2011-04-26
• Publisher: Johns Hopkins Univ Pr

Summary

A notoriously difficult subject, covariant electrodynamics is nonetheless vital for understanding relativistic field theory. John M. Charap's classroom-tested introduction to the mathematical foundations of the topic presents the material in an approachable manner. Charap begins with a historical overview of electrodynamics and a discussion of the preliminary mathematics one needs in order to grasp the advanced and abstract concepts underlying the theory. He walks the reader through Maxwell's four equations, explaining how they were developed and demonstrating how they are applied. From there, Charap moves through the other components of electrodynamics, such as Lorentz Transformations, tensors, and charged particle behavior. At each point, he carefully works through the mathematics, applies the concepts to simple physical systems, and provides historical context that makes clear the connections among the theories and the mathematicians responsible for developing them. A concluding chapter reviews the history of electrodynamics and points the way for independent testing of the theory. Thorough, evenly paced, and intuitive, this friendly introduction to high-level covariant electrodynamics is a handy and helpful addition to any physicist's toolkit.

Author Biography

John M. Charap is an emeritus professor of theoretical physics at the University of London's Queen Mary College. He is the editor of Geometry of Constrained Dynamical Systems and the author of Explaining the Universe: The New Age of Physics.

Table of Contents

Preface	p. ix
Introduction	p. 1
Mathematical Preliminaries	p. 11
A Reminder of Vector Calculus	p. 11
Special Relativity	p. 13
Four-Vectors	p. 14
Covariant and Contravariant Vectors	p. 15
Tensors	p. 17
Time Dilation and the Lorentz-FitzGerald Contraction	p. 19
The Four-Velocity	p. 21
Energy and Momentum	p. 22
Plane Waves	p. 22
Exercises for Chapter 2	p. 22
Maxwell's Equations	p. 25
Our Starting Point	p. 25
The Experimental Background	p. 27
Coulomb's Law	p. 27
Absence of Magnetic Monopoles	p. 28
Ørsted and Ampère	p. 28
The Law of Biot and Savart	p. 29
The Displacement Current	p. 29
Faraday's Law of Induction	p. 30
The Lorentz Force	p. 30
Capacitors and Solenoids	p. 31
Energy	p. 32
Electromagnetic Waves	p. 33
Polarization	p. 34
Electromagnetism and Light	p. 34
Exercises for Chapter 3	p. 36
Behavior under Lorentz Transformations	p. 37
The Charge-Current Density Four-Vector	p. 37
The Lorentz Force	p. 38
The Potential Four-Vector	p. 39
Gauge Transformations	p. 40
The Field-Strength Tensor	p. 43
The Dual Field-Strength Tensor	p. 45
Exercises for Chapter 4	p. 46
Lagrangian and Hamiltonian	p. 49
Lagrange's Equations	p. 49
The Lagrangian for a Charged Particle	p. 51
The Hamiltonian for a Charged Particle	p. 53
The Lagrangian for the Electromagnetic Field	p. 54
The Hamiltonian for the Electromagnetic Field	p. 57
Noether's Theorem	p. 58
Exercises for Chapter 5	p. 60
Stress, Energy, and Momentum	p. 61
The Canonical Stress Tensor	p. 61
The Symmetrical Stress Tensor	p. 63
The Conservation Laws with Sources	p. 66
The Field as an Ensemble of Oscillators	p. 68
Exercises for Chapter 6	p. 69
Motion of a Charged Particle	p. 71
Fields from an Unaccelerated Particle	p. 71
Motion of a Particle in an External Field	p. 72
Uniform Static Magnetic Field	p. 72
Crossed E and B Fields	p. 73
Nonuniform Static B-Field	p. 74
Curved Magnetic Field Lines	p. 75
Exercises for Chapter 7	p. 76
Fields from Sources	p. 79
Introducing the Green's Function	p. 79
The Delta Function	p. 80
The Green's Function	p. 81
The Covariant Form for the Green's Function	p. 84
Exercises for Chapter 8	p. 85
Radiation	p. 87
Potentials from a Moving Charged Particle	p. 88
The Liénard-Wiechert Potentials	p. 89
Fields from an Unaccelerated Particle	p. 91
Fields from a Charged Oscillator	p. 94
The General Case	p. 96
The Multipole Expansion	p. 98
Electric Dipole Radiation	p. 100
Magnetic Dipole and Higher-Order Terms	p. 102
Motion in a Circle	p. 103
Radiation from Linear Accelerators	p. 105
Radiation from an Antenna	p. 106
Exercises for Chapter 9	p. 107
Media	p. 109
Dispersion	p. 110
Newton on the ôPhænomena of Coloursö	p. 111
Refraction	p. 111
The Boundary Conditions at the Interface	p. 113
Cerenkov Radiation	p. 116
Exercises for Chapter 10	p. 119
Scattering	p. 121
Scattering from a Small Scatterer	p. 122
Many Scatterers	p. 123
Scattering from the Sky	p. 124
The Born Approximation	p. 126
Rayleigh's Explanation for the Blue Sky	p. 128
Critical Opalescence	p. 133
Dispersion	p. 137
The Oscillator Model	p. 137
The High-Frequency Limit	p. 140
The Drude Model	p. 140
Dispersion Relations	p. 142
The Optical Theorem	p. 145
Epilogue	p. 149
Index	p. 161
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