Covariant Electrodynamics - A Concise Guide

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.Table of ContentPreface1. Introduction2. Mathematical Preliminaries2.1. A Reminder of Vector Calculus2.2. Special Relativity2.3. Four-Vectors2.4. Covariant and Contravariant Vectors2.5. Tensors2.6. Time Dilation and the Lorentz-FitzGerald Contraction2.7. The Four-Velocity2.8. Energy and Momentum2.9. Plane Waves2.10. Exercises for Chapter 23. Maxwell's Equations3.1. Our Starting Point3.2. The Experimental Background3.2.1. Coulomb's Law3.2.2. Absence of Magnetic Monopoles3.2.3. Ørsted and Ampere3.2.4. The Law of Biot and Savart3.2.5. The Displacement Current3.2.6. Faraday's Law of Induction3.2.7. The Lorentz Force3.3. Capacitors and Solenoids3.3.1. Energy3.4. Electromagnetic Waves3.4.1. Polarization3.4.2. Electromagnetism and Light3.5. Exercises for Chapter 34. Behavior under Lorentz Transformations4.1. The Charge-Current Density Four-Vector4.2. The Lorentz Force4.3. The Potential Four-Vector4.4. Gauge Transformations4.5. The Field-Strength Tensor4.6. The Dual Field-Strength Tensor4.7. Exercises for Chapter 45. Lagrangian and Hamiltonian5.1. Lagrange's Equations5.2. The Lagrangian for a Charged Particle5.3. The Hamiltonian for a Charged Particle5.4. The Lagrangian for the Electromagnetic Field5.5. The Hamiltonian for the Electromagnetic Field5.6. Noether's Theorem5.7. Exercises for Chapter 56. Stress, Energy, and Momentum6.1. The Canonical Stress Tensor6.2. The Symmetrical Stress Tensor6.3. The Conservation Laws with Sources6.4. The Field as an Ensemble of Oscillators6.5. Exercises for Chapter 67. Motion of a Charged Particle7.1. Fields from an Unaccelerated Particle7.2. Motion of a Particle in an External Field7.2.1. Uniform Static Magnetic Field7.2.2. Crossed E and B Fields7.2.3. Nonuniform Static B-Field7.2.4. Curved Magnetic Field Lines7.3. Exercises for Chapter 78. Fields from Sources8.1. Introducing the Green's Function8.2. The Delta Function8.3. The Green's Function8.4. The Covariant Form for the Green's Function8.5. Exercises for Chapter 89. Radiation9.1. Potentials from a Moving Charged Particle9.2. The Lienard-Wiechert Potentials9.2.1. Fields from an Unaccelerated Particle9.2.2. Fields from a Charged Oscillator9.3. The General Case9.4. The Multipole Expansion9.4.1. Electric Dipole Radiation9.4.2. Magnetic Dipole and Higher-Order Terms9.5. Motion in a Circle9.6. Radiation from Linear Accelerators9.7. Radiation from an Antenna9.8. Exercises for Chapter 910. Media10.1. Dispersion10.1.1. Newton on the "Phænomena of Colours"10.2. Refraction10.2.1. The Boundary Conditions at the Interface10.3. Cerenkov Radiation10.4. Exercises for Chapter 1011. Scattering11.1. Scattering from a Small Scatterer11.2. Many Scatterers11.3. Scattering from the Sky11.3.1. The Born Approximation11.3.2. Rayleigh's Explanation for the Blue Sky11.4. Critical Opalescence12. Dispersion12.1. The Oscillator Model12.1.1. The High-Frequency Limit12.1.2. The Drude Model12.2. Dispersion Relations12.3. The Optical TheoremEpilogueIndexReview Quote"John Charap succeeds well in making electrodynamics manifestly covariant, providing historical background and applications of far-reaching importance. The diligent reader, armed with pen and ample scratch paper for filling in the intermediate steps, will see covariant electrodynamics emerge coherently." (Dwight E. Neuenschwander, author of Emmy Noether's Wonderful Theorem)"Biographical NoteJohn 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.John Charap succeeds well in making electrodynamics manifestly covariant, providing historical background and applications of far-reaching importance. The diligent reader, armed with pen and ample scratch paper for filling in the intermediate steps, will see covariant electrodynamics emerge coherently. -- Dwight E. Neuenschwander, author of Emmy Noether's Wonderful TheoremJohn Charap succeeds well in making electrodynamics manifestly covariant, providing historical background and applications of far-reaching importance. The diligent reader, armed with pen and ample scratch paper for filling in the intermediate steps, will see covariant electrodynamics emerge coherently.—Dwight E. Neuenschwander, author of Emmy Noether's Wonderful Theorem