Introduction to Advanced Electrodynamics

Description

This book summarizes the basics of electricity and magnetism prior to covariant formulation of Maxwell's equations. The book works out the basics of special relativity and then applies the covariant formalism to understand radiation, both in vacuum and in material medium. The emphasis is on cleaner mathematical formalism based on experimental facts. The book contains many problems/exercises which will help the students to understand the basics of the subject. The difference between the present book with existing books of this level lies in the presentation of the topics and the subjects chosen. Instead of resenting a lot of material related to electromagnetism, it presents some very important but selected problems of advanced electromagnetism to students who are learning it for the first time. This book is aimed at graduate/advanced graduate students who have done at least one basic level course in electricity and magnetism.​ Preface Contents 1 Basic Laws of Electrodynamics 1.1 Electric Field and the Principle of Superposition 1.1.1 Electric Scalar Potential 1.1.2 Poisson and Laplace Equations 1.1.3 Electrostatic Potential Energy for Discrete and Continuous Charge Distribution 1.2 Magnetic Field and the Principle of Superposition 1.2.1 Biot-Savart Law of Magnetostatics 1.2.2 General Derivation of Ampere's Law 1.2.3 Vector Potential 1.3 Faraday's Law of Electromagnetic Induction 1.3.1 Energy Stored in Magnetic Field 1.4 Maxwell's Correction to Ampere's Law: The Displacement Current 1.5 Maxwell Equations; Scalar and Vector Potentials 1.6 A Short Note on Units and Dimensions 1.7 Problems 2 Boundary Value Problems in Electrostatics 2.1 Boundary Conditions in Electrostatics 2.2 Green's Theorem 2.3 Uniqueness of Solutions: Dirichlet and Neumann Boundary Conditions 2.4 Formal Solution of the Poisson Equation Using the Green Function 2.5 Method of Images 2.5.1 Point Charge Near a Conducting Sphere 2.5.2 Conducting Sphere in Uniform Electric Field 2.6 Construction of the Green Function from Images 2.6.1 Application of the Green Function to a Pair of Conducting Hemispherical Shells at Different Fixed Potentials 2.7 Laplace Equation as Boundary Value Problem 2.8 Laplace Equation in Spherical Coordinates 2.8.1 Pair of Hemispherical Shells at Different Potentials 2.9 Laplace Equation in Cylindrical Coordinates 2.9.1 Non-zero Potential only at the Top Surface 2.10 Problems 3 Electrodynamics of Material Media 3.1 Electric Multipole Expansion 3.1.1 The Electric Dipole Moment 3.2 Electrostatic Interaction of an Extended Charge Distribution with External Electric Field 3.3 Macroscopic Field Equations of Electrostatics 3.4 Boundary Conditions at the Interface of Two Linear Media 3.5 Boundary Value Problems in Presence of Dielectrics 3.5.1 Image Problem Involving Dielectrics 3.5.2 Dielectric Sphere in Uniform External Field 3.5.3 Dielectric with Spherical Cavity in Presence of Uniform Electric Field 3.6 Multipole Magnetic Moments 3.7 Magnetized Matter 3.8 Boundary Conditions on B and H at the Interface Between Two Linear Isotropic Media 3.9 Boundary Value Problems in Magnetostatics 3.9.1 A Uniformly Magnetized Sphere 3.10 Generalized Ampere's Law in Material Media 3.11 Energy and Momentum Conservation: Poynting Theorem 3.11.1 Conservation of Linear Momentum 3.12 Problems 4 Initiation to Electromagnetic Radiation 4.1 Plane-Wave Solutions 4.1.1 Plane Electromagnetic Waves in Dispersive Medium 4.2 Vectorial Properties of Plane Electromagnetic Waves 4.2.1 Linear and Circular Polarization of Plane Electromagnetic Waves 4.2.2 General Polarization Basis 4.2.3 Stokes Parameters 4.3 Radiation from Localized Charge and Current Distributions 4.3.1 Green Function for Inhomogeneous Maxwell's Equation in Lorenz Gauge 4.3.2 Radiation from Localized Charges 4.3.3 The Vector Potential in the Electric Dipole Approximation 4.3.4 The Magnetic Field and the Electric Field in the Electric Dipole Approximation 4.3.5 The Electric Monopole Term 4.3.6 Power Radiated in the Electric Dipole Approximation 4.3.7 Short, Center-Fed, Linear Antenna 4.3.8 Magnetic Dipole Radiation 4.3.9 Electric Quadrupole Radiation 4.4 Problems 5 Long Wavelength Scattering 5.1 General Formulation of the Dipole Scattering Problem 5.1.1 Scattering by a Dielectric Sphere 5.1.2 Scattering by a Perfectly Conducting Sphere 5.1.3 Scattering by a Collection of Scatterers 5.2 Perturbation Theory of Scattering by Liquids and Gases 5.2.1 Born Approximation 5.2.2 Sunset and Blue Sky: Rayleigh Scattering Law 5.3 Problems 6 Special Relativity and Fourier Transform Theory for Electrodynamics 6.1 Lorentz Boosts 6.2 More Formal Way to Define 4-Vectors and Other Tensors 6.2.1 Contravariant Vectors 6.2.2 Covariant Vectors 6.2.3 A Note on the Einstein Summation Convention 6.3 Tensors 6.3.1 Contravariant Tensors 6.3.2 Covariant Tensors 6.3.3 The Mixed Tensor 6.3.4 Quotient Theorem and the Metric Tensor 6.4 Inner Product and Raising and Lowering of Tensor Indices 6.5 Lorentz Transformations 6.5.1 Lorentz Boost with Arbitrary Velocity Direction 6.6 Why Tensors in Physics? 6.7 Brief Description of 4-Velocity, 4-Momentum and 4-Force for a Massive Particle 6.8 Relevant Portions of Fourier Transform Theory 6.8.1 The Product of Fourier Transforms: Convolution 6.9 Problems 7 Covariant form of Maxwell's Equations in the Absence of Bound Charges and Bound Currents 7.1 Maxwell's Equations Using only 3-Vectors 7.2 The Field Strength Tensor and Its Dual 7.3 Maxwell's Equations in Covariant form 7.4 Transformation of Fµν Under a Lorentz Transformation 7.4.1 The Electromagnetic Field of a Point Charge Moving with Uniform Velocity 7.4.2 Covariant form of the Lorentz Force Law 7.5 A Brief Discussion of Parity Symmetry 7.6 Problems 8 Gauge Invariance of Electrodynamics 8.1 The Lorenz Gauge 8.2 The Coulomb Gauge 8.3 Problems 9 Action Principle in Electrodynamics 9.1 Classical Particle Mechanics 9.1.1 The Lagrangian of a Relativistic Free Particle 9.1.2 Lorentz Force Law from Action Principle 9.2 Lagrangian Formalism for Electromagnetic Fields 9.2.1 Introduction to the Lagrangian Formalism 9.2.2 The Maxwell Equations 9.2.3 Difficulty with the Hamiltonian Formulation 9.3 Continuous Symmetries and Noether's Theorem 9.3.1 Pure Electromagnetic Field and Continuous Symmetries 9.3.2 Noether's Theorem 9.4 Drawbacks of the Energy-Momentum Tensor Tµν 9.4.1 Constructing the Proper Energy-Momentum Tensor 9.4.2 Energy-Momentum Conservation 9.4.3 Conserved Tensors Corresponding to Lorentz Transformations 9.5 Energy-Momentum Conservation in the Presence of Sources 9.6 Problems 10 Electromagnetic Field Produced by an Arbitrarily Moving Point Charge 10.1 Evaluation of the Green Function 10.1.1 A Different Pole Prescription and the Advanced Green Function 10.1.2 Green Functions Under Lorentz Transformations 10.2 The 4-Current for a Point Charge Moving Arbitrarily 10.3 Liénard-Wiechert Potentials 10.4 The Field Strength Tensor Fµν Obtained from the Liénard-Wiechert Potentials 10.4.1 Setting up the Stage 10.4.2 The Calculation for the Electric Field 10.4.3 The Calculation for the Magnetic Field 10.4.4 The Nature of the Fields 10.4.5 Electric and Magnetic Fields from an Uniformly Moving Charged Particle in the Liénard-Wiechert Formalism 10.5 Power Radiated by an Accelerated Charge … 10.5.1 Power Radiated by a System of Accelerated Charged Particles Moving Non-relativistically 10.6 Relativistic Generalization of Larmor's Formula 10.6.1 Radiation from a Linearly Accelerated Charge 10.6.2 Radiation from a Charged Particle in Circular Motion 10.7 Angular Distribution of Radiation Emitted by the Accelerated Charge 10.8 The Thomson Scattering Cross-Section 10.9 Problems 11 Radiation Reaction in Brief 11.1 Abraham-Lorentz Equation of Motion 11.2 The Integro-Differential Equation from Abraham-Lorentz Equation 11.3 Problems 12 Cherenkov Radiation 12.1 Electric and Magnetic Field Due to a Uniformly Moving Charged Particle Inside Dielectric Medium 12.2 Energy Loss by the Charged Particle Moving Inside the Dielectric Medium 12.2.1 Calculation of the Fields 12.2.2 The Energy Loss Mechanism 12.3 Properties of Cherenkov Radiation 12.4 Problems Appendix Index Index