Introduction to Electrodynamics

Contents
Preface
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1 Vector Analysis
	1.1 Vector Algebra
		1.1.1 Vector Operations
		1.1.2 Vector Algebra: Component Form
		1.1.3 Triple Products
		1.1.4 Position, Displacement, and Separation Vectors
		1.1.5 How Vectors Transform
	1.2 Differential Calculus
		1.2.1 “Ordinary” Derivatives
		1.2.2 Gradient
		1.2.3 The Del Operator
		1.2.4 The Divergence
		1.2.5 The Curl
		1.2.6 Product Rules
		1.2.7 Second Derivatives
	1.3 Integral Calculus
		1.3.1 Line, Surface, and Volume Integrals
		1.3.2 The Fundamental Theorem of Calculus
		1.3.3 The Fundamental Theorem for Gradients
		1.3.4 The Fundamental Theorem for Divergences
		1.3.5 The Fundamental Theorem for Curls
		1.3.6 Integration by Parts
	1.4 Curvilinear Coordinates
		1.4.1 Spherical Coordinates
		1.4.2 Cylindrical Coordinates
	1.5 The Dirac Delta Function
		1.5.1 The Divergence of ˆr/r2
		1.5.2 The One-Dimensional Dirac Delta Function
		1.5.3 The Three-Dimensional Delta Function
	1.6 The Theory of Vector Fields
		1.6.1 The Helmholtz Theorem
		1.6.2 Potentials
2 Electrostatics
	2.1 The Electric Field
		2.1.1 Introduction
		2.1.2 Coulomb’s Law
		2.1.3 The Electric Field
		2.1.4 Continuous Charge Distributions
	2.2 Divergence and Curl of Electrostatic Fields
		2.2.1 Field Lines, Flux, and Gauss’s Law
		2.2.2 The Divergence of E
		2.2.3 Applications of Gauss’s Law
		2.2.4 The Curl of E
	2.3 Electric Potential
		2.3.1 Introduction to Potential
		2.3.2 Comments on Potential
		2.3.3 Poisson’s Equation and Laplace’s Equation
		2.3.4 The Potential of a Localized Charge Distribution
		2.3.5 Boundary Conditions
	2.4 Work and Energy in Electrostatics
		2.4.1 The Work It Takes to Move a Charge
		2.4.2 The Energy of a Point Charge Distribution
		2.4.3 The Energy of a Continuous Charge Distribution
		2.4.4 Comments on Electrostatic Energy
	2.5 Conductors
		2.5.1 Basic Properties
		2.5.2 Induced Charges
		2.5.3 Surface Charge and the Force on a Conductor
		2.5.4 Capacitors
3 Potentials
	3.1 Laplace’s Equation
		3.1.1 Introduction
		3.1.2 Laplace’s Equation in One Dimension
		3.1.3 Laplace’s Equation in Two Dimensions
		3.1.4 Laplace’s Equation in Three Dimensions
		3.1.5 Boundary Conditions and Uniqueness Theorems
		3.1.6 Conductors and the Second Uniqueness Theorem
	3.2 The Method of Images
		3.2.1 The Classic Image Problem
		3.2.2 Induced Surface Charge
		3.2.3 Force and Energy
		3.2.4 Other Image Problems
	3.3 Separation of Variables
		3.3.1 Cartesian Coordinates
		3.3.2 Spherical Coordinates
	3.4 Multipole Expansion
		3.4.1 Approximate Potentials at Large Distances
		3.4.2 The Monopole and Dipole Terms
		3.4.3 Origin of Coordinates in Multipole Expansions
		3.4.4 The Electric Field of a Dipole
4 Electric Fields in Matter
	4.1 Polarization
		4.1.1 Dielectrics
		4.1.2 Induced Dipoles
		4.1.3 Alignment of Polar Molecules
		4.1.4 Polarization
	4.2 The Field of a Polarized Object
		4.2.1 Bound Charges
		4.2.2 Physical Interpretation of Bound Charges
		4.2.3 The Field Inside a Dielectric
	4.3 The Electric Displacement
		4.3.1 Gauss’s Law in the Presence of Dielectrics
		4.3.2 A Deceptive Parallel
		4.3.3 Boundary Conditions
	4.4 Linear Dielectrics
		4.4.1 Susceptibility, Permittivity, Dielectric Constant
		4.4.2 Boundary Value Problems with Linear Dielectrics
		4.4.3 Energy in Dielectric Systems
		4.4.4 Forces on Dielectrics
5 Magnetostatics
	5.1 The Lorentz Force Law 5.1.1 Magnetic Fields
	5.1.2 Magnetic Forces
	5.1.3 Currents
	5.2 The Biot-Savart Law
		5.2.1 Steady Currents
		5.2.2 The Magnetic Field of a Steady Current
	5.3 The Divergence and Curl of B
		5.3.1 Straight-Line Currents
		5.3.2 The Divergence and Curl of B
		5.3.3 Ampère’s Law
		5.3.4 Comparison of Magnetostatics and Electrostatics
	5.4 Magnetic Vector Potential
		5.4.1 The Vector Potential
		5.4.2 Boundary Conditions
		5.4.3 Multipole Expansion of the Vector Potential
6 Magnetic Fields in Matter
	6.1 Magnetization
		6.1.1 Diamagnets, Paramagnets, Ferromagnets
		6.1.2 Torques and Forces on Magnetic Dipoles
		6.1.3 Effect of a Magnetic Field on Atomic Orbits
		6.1.4 Magnetization
	6.2 The Field of a Magnetized Object
		6.2.1 Bound Currents
		6.2.2 Physical Interpretation of Bound Currents
		6.2.3 The Magnetic Field Inside Matter
	6.3 The Auxiliary Field H
		6.3.1 Ampère’s Law in Magnetized Materials
		6.3.2 A Deceptive Parallel
		6.3.3 Boundary Conditions
	6.4 Linear and Nonlinear Media
		6.4.1 Magnetic Susceptibility and Permeability
		6.4.2 Ferromagnetism
7 Electrodynamics
	7.1 Electromotive Force
		7.1.1 Ohm’s Law
		7.1.2 Electromotive Force
		7.1.3 Motional emf
	7.2 Electromagnetic Induction
		7.2.1 Faraday’s Law
		7.2.2 The Induced Electric Field
		7.2.3 Inductance
		7.2.4 Energy in Magnetic Fields
	7.3 Maxwell’s Equations
		7.3.1 Electrodynamics Before Maxwell
		7.3.2 How Maxwell Fixed Ampère’s Law
		7.3.3 Maxwell’s Equations
		7.3.4 Magnetic Charge
		7.3.5 Maxwell’s Equations in Matter
		7.3.6 Boundary Conditions
8 Conservation Laws
	8.1 Charge and Energy
		8.1.1 The Continuity Equation
		8.1.2 Poynting’s Theorem
	8.2 Momentum
		8.2.1 Newton’s Third Law in Electrodynamics
		8.2.2 Maxwell’s Stress Tensor
		8.2.3 Conservation of Momentum
		8.2.4 Angular Momentum
	8.3 Magnetic Forces Do No Work
9 Electromagnetic Waves
	9.1 Waves in One Dimension
		9.1.1 The Wave Equation
		9.1.2 Sinusoidal Waves
		9.1.3 Boundary Conditions: Reflection and Transmission
		9.1.4 Polarization
	9.2 Electromagnetic Waves in Vacuum
		9.2.1 The Wave Equation for E and B
		9.2.2 Monochromatic Plane Waves
		9.2.3 Energy and Momentum in Electromagnetic Waves
	9.3 Electromagnetic Waves in Matter
		9.3.1 Propagation in Linear Media
		9.3.2 Reflection and Transmission at Normal Incidence
		9.3.3 Reflection and Transmission at Oblique Incidence
	9.4 Absorption and Dispersion
		9.4.1 Electromagnetic Waves in Conductors
		9.4.2 Reflection at a Conducting Surface
		9.4.3 The Frequency Dependence of Permittivity
	9.5 Guided Waves
		9.5.1 Wave Guides
		9.5.2 TE Waves in a Rectangular Wave Guide
		9.5.3 The Coaxial Transmission Line
10 Potentials and Fields
	10.1 The Potential Formulation
		10.1.1 Scalar and Vector Potentials
		10.1.2 Gauge Transformations
		10.1.3 Coulomb Gauge and Lorenz Gauge
		10.1.4 Lorentz Force Law in Potential Form
	10.2 Continuous Distributions
		10.2.1 Retarded Potentials
		10.2.2 Jefimenko’s Equations
	10.3 Point Charges
		10.3.1 Liénard-Wiechert Potentials
		10.3.2 The Fields of a Moving Point Charge
11 Radiation
	11.1 Dipole Radiation
		11.1.1 What is Radiation?
		11.1.2 Electric Dipole Radiation
		11.1.3 Magnetic Dipole Radiation
		11.1.4 Radiation from an Arbitrary Source
	11.2 Point Charges
		11.2.1 Power Radiated by a Point Charge
		11.2.2 Radiation Reaction
		11.2.3 The Mechanism Responsible for the Radiation Reaction
12 Electrodynamics and Relativity
	12.1 The Special Theory of Relativity
		12.1.1 Einstein’s Postulates
		12.1.2 The Geometry of Relativity
		12.1.3 The Lorentz Transformations
		12.1.4 The Structure of Spacetime
	12.2 Relativistic Mechanics
		12.2.1 Proper Time and Proper Velocity
		12.2.2 Relativistic Energy and Momentum
		12.2.3 Relativistic Kinematics
		12.2.4 Relativistic Dynamics
	12.3 Relativistic Electrodynamics
		12.3.1 Magnetism as a Relativistic Phenomenon
		12.3.2 How the Fields Transform
		12.3.3 The Field Tensor
		12.3.4 Electrodynamics in Tensor Notation
		12.3.5 Relativistic Potentials
A Vector Calculus in Curvilinear Coordinates
	A.1 Introduction
	A.2 Notation
	A.3 Gradient
	A.4 Divergence
	A.5 Curl
	A.6 Laplacian
B The Helmholtz Theorem
C Units
Index