The Theory of Electromagnetism

The Theory of the Electomagnetism covers the behavior of electromagnetic fields and those parts of applied mathematics necessary to discover this behavior. This book is composed of 11 chapters that emphasize the Maxwell's equations. The first chapter is concerned with the general properties of solutions of Maxwell's equations in matter, which has certain macroscopic properties. The succeeding chapters consider specific problems in electromagnetism, including the determination of the field produced by a variable charge, first in isolation and then in the surface distributions of an antenna. The next two chapters are concerned with the effects of surrounding the medium by a perfectly conducting boundary as in a cavity resonator and as in a waveguide. Other chapters are devoted to discussions on the effect of a plane interface where the properties of the medium change discontinuously; the propagation along cylindrical surfaces; the study of the waves scattered by objects both with and without edges. This book further reviews the harmonic waves and the difficulties involved in going from harmonic waves to those with a more general time dependence. The final chapter provides some information about the classical theory of electrons, magneto-hydrodynamics and waves in a plasma. This book will prove useful to physicists, and physics teachers and students.

Preface

1. The Representation of the Electromagnetic Field

Maxwell's Equations

1.1 The Field Equations

1.2 The Equation of Continuity

Macroscopic Properties of Matter

1.3 Dielectric Constant and Permeability

1.4 Permanent Magnetism

1.5 The Characteristics of a Ferrite

1.6 Physical Properties of Conductors

The Electromagnetic Potentials

1.7 The Scalar and Vector Potentials

1.8 The Potentials in a Homogeneous Conductor

1.9 The Hertz Vector

1.10 The Representation in Terms of Two Scalare

Orthogonal Curvilinear Coordinates

1.11 Curvilinear Coordinates

1.12 The Differential Operators

1.13 Particular Coordinate Systems

Integral Representations

1.14 The Scalar Potential of a Point Charge

1.15 Generalized Functions

1.16 Retarded Potentials

1.17 Kirchhoff’s Solution of the Wave Equation

1.18 Volterra's Solution in Two Dimensions

1.19 An Integral Formula for the Electromagnetic Field

1.20 Uniqueness

Boundary Conditions

1.21 Discontinuities in the Field

Stress and Energy

1.22 The Stress Tensor in Free Space

1.23 Force in a General Medium

1.24 Electromagnetic Momentum

1.25 Poynting's Theorem

Harmonic Waves

1.26 Helmholtz's Theorem

1.27 Boundary Condition and Uniqueness

1.28 The Complex Poynting Vector

1.29 Green's Functions

1.30 Green's Tensor

1.31 The Exterior Problem

1.32 Reciprocity Theorem

1.33 Special Functions

1.34 Formulae for Two Dimensional Fields

Exercises

2. The Special Theory of Relativity

Tensor Calculus

2.1 Coordinate Transformation

2.1 Tensors

2.3 Contraction

2.4 Fundamental Tensors

2.5 Derivatives

Tensor in Three-Dimensional Space

2.6 Pseudo-Tensors

2.7 Cartesian Tensors

2.8 The Derivatives of Cartesian Tensors

2.9 The Divergence and Stokes' Theorem

The Lorentz Transformation

2.10 The Michelson-Morley Experiment

2.11 The Lorentz Transformation

2.12 The Lorentz-Fitzgerald Contraction

2.13 The Clock Paradox

Relativistic Mechanics

2.14 The Transformation of Velocity

2.15 The Variation of Mass with Velocity

2.16 The Conservation of Momentum and Energy

Electrodynamics in Free Space

2.17 The Invariant Form of Maxwell's Equations

2.18 The Lorentz Force

2.19 The Doppler Effect

2.20 Electromagnetic Stress and Momentum

Electrodynamics in Moving Media

2.21 The Field Equations

2.22 Boundary Conditions

2.23 The Convection of Light

2.24 The Convection of Charge by a Moving Medium

Exercises

3. Radiation

The Field of a Moving Point Charge

3.1 The Liénard-Wiechert Potentials

3.2 The Radiated Energy

3.3 The Self-Force of an Electron

3.4 The Field of a Moving Charge in a Dielectric and Cerenkov Radiation

The Field of a Variable Charge

3.5 The Electric Dipole

3.6 The Magnetic Dipole

3.7 The Harmonic Dipole

3.8 Two-Dimensional Dipoles

The Characteristics of Linear Antenna Systems

3.9 The Radiation from a Thin Wire

3.10 Linear Arrays

3.11 Schelkunoff’s Method for Linear Arrays

3.12 Beam Synthesis

3.13 The Helical Antenna

The Antenna Boundary Value Problem

3.14 The Integral Equation of the Perfectly Conducting Antenna

3.15 Pocklington's Theory

3.16 The Murray-Pidduck Theory

3.17 Hallén's Method of Iteration

3.18 The Theory of Albert and Synge

3.19 Comparison of Theories

Exercises

4. Cavity Resonators

The Theory of Eigenfunctions

4.1 The One-Dimensional Field

4.2 Fourier Series and Gibbs' Phenomenon

4.3 The Sturm-Liouville Equation

4.4 The Variational Method of Calculating Eigenvalues

4.5 The Equivalent Integral Equation

Linear Operators

4.6 The Lebesgue Integral

4.7 The Space L2

4.8 Hilbert Space

4.9 Symmetric and Completely Continuous Operators

4.10 The Determination of Eigenvalues

The Eigenvalues of Differential Operators

4.11 The Boundary Condition u = 0

4.12 The Boundary Condition әu/әv + σu = 0

Cavity Resonators

4.13 The Eigenvalues of a Cavity Resonator

4.14 Typical Eigenfunctions

4.15 The Eigenvalues of Maxwell's Equations

Perturbation Theory

4.16 The Effect of Conductivity

4.17 Boundary Perturbation

4.18 The Effect of an Aperture

Exercises

5. The Theory of Waveguides

5.1 Boundary Conditions

5.2 The Modal Expansion of the Field

5.3 Energy Flow

5.4 The Attenuation Due To Surface Loss

5.5 Typical Waveguides

Junctions

5.6 General Waveguide Junction

5.7 The Scattering Matrix

5.8 T-Junctions

5.9 Directional Couplers

Determination of Matrix Elements

5.10 The Source Method for the Inductive Post

5.11 The Capacitive Iris

5.12 General Theory

5.13 Approximation to the Kernel

5.14 The Equivalent Static Method

5.15 The Wiener-Hopf Method

5.16 Equivalence Theorems

5.17 Waveguides Containing Dielectric

Ferrites in Waveguides

5.18 Waves in a Gyromagnetic Medium

5.19 Waveguide Modes

5.20 Circular Waveguide with a Ferrite Core

Radiation from Waveguides and Horns

5.21 The Sectoral Horn

5.22 The Conical Horn

5.23 Radiation Properties

Exercises

6. Refraction

The Homogeneous Isotropie Medium

6.1 The Plane Wave

6.2 Harmonic Plane Waves

6.3 Polarization

6.4 The Effect of Conductivity

6.5 Refraction at a Plane Interface

6.6 Dielectric Media

6.7 Conducting Media

6.8 The Plane Slab

6.9 The Sandwich

The Homogeneous Anisotropie Medium

6.10 The Plane Wave

6.11 Refraction in a Crystal

The Inhomogeneous Isotropie Medium

6.12 General Considerations

6.13 The Rayleigh-Gans Approximation

6.14 The High Frequency Approximation

6.15 Geometrical Optics

6.16 Fermat's Principle

6.17 Focusing Properties of a Pencil

6.18 Horizontally Stratified Medium

6.19 The Wave Equation in a Stratified Medium

6.20 Laminated Media

6.21 The WKB Method

6.22 Langer's Method

6.23 Uniformly Valid Asymptotic Expansions for Bessel Functions

Propagation over a Plane Earth

6.24 The Earth's Atmosphere

6.25 Propagation in a Homogeneous Atmosphere

6.26 Asymptotic Evaluation of the Field

6.27 The Impedance Boundary Condition and Wave Tilt

6.28 Propagation in a Quasi-Homogeneous and Standard Atmosphere

6.29 Various Approximations

6.30 Ray Theory for a Standard Atmosphere

6.31 The Stratified Atmosphere

6.32 Ducts

6.33 The Ionosphere

6.34 The Influence of the Earth's Magnetic Field

6.35 Scattering by Atmospheric Irregularities

Exercises

7. Surface Waves

Propagation along a Cylindrical Surface

7.1 General Considerations

7.2 The Conducting Circular Cylinder

7.3 The Dielectric Circular Rod

7.4 Several Conductors

7.5 Transmission Lines

7.6 More General Cylindrical Structures

Propagation along a Plane Surface

7.7 General Remarks

7.8 Launching Efficiency

The Polyrod Antenna

7.9 The Radiation Pattern

Exercises

8. Scattering by Objects without Edges

Asymptotic Evaluation of Integrals

8.1 Watson's Lemma

8.2 The Method of Steepest Descent

8.3 Some Examples

8.4 Finite Range of Integration

8.5 The Method of Stationary Phase

Two-Dimensional Scattering Problems

8.6 The Circular Cylinder

8.7 The High Frequency behavior of the Circular Cylinder

8.8 The Diffracted Field behind the Cylinder

8.9 The Line Source

8.10 The Parabolic Cylinder

8.11 The Parabolic Cylinder—General Incidence

8.12 The Current Distributions on the Parabolic Cylinder

8.13 The Elliptic Cylinder

8.14 Inhomogeneous Cylinders

Three-Dimensional Scattering Problems

8.15 Scattering by Infinitely Long Cylinders

8.16 Spherical Waves

8.17 The Series Expansion of the Field

8.18 The Expansions of Various Fields

8.19 Scattering by a Sphere

8.20 General Discussion of the Scattered Field

8.21 The Effect of Conductivity

8.22 Comparison with Scalar Field Theory

8.23 The Scattering Coefficient

8.24 Alternative Expressions for the Scattering Coefficient

8.25 Rayleigh Scattering

8.26 Rayleigh-Gans Scattering by a Diaphanous Sphere

8.27 High Frequency Scattering by a Perfectly Conducting Sphere

8.28 Propagation near a Spherical Earth

8.29 The Effect of Refraction

8.30 The Prolate Spheroid

8.31 The Oblate Spheroid

Arbitrary Curved Obstacles

8.32 Rayleigh Scattering

8.33 Rayleigh-Gans Scattering by Diaphanous Objects

8.34 High Frequency Scattering by a Diaphanous Object

8.35 High Frequency Scattering

8.36 Behavior near a Focal Line

Assemblages of Particles

8.37 General Theory for Widely Spaced Objects

8.38 Independent Scattering

8.39 The Grating

Exercises

9. Diffraction by Obstacles With Edges

General Results

9.1 Uniqueness

9.2 The Edge Conditions

9.3 Babinet's Principle

9.4 The Scattering Coefficient

Transform Techniques

9.5 The Laplace Transform

9.6 The Mellin Inverse

9.7 The Bilateral Laplace Transform

9.8 The Semi-Infinite Plane

9.9 The Diffraction of a Spherical Wave by a Semi-Infinite Plane

9.10 The Radiation from a Semi-Infinite Circular Pipe

9.11 The 'Split' Functions

9.12 The Perfectly Conducting Strip

9.13 The Kontorowich-Lebedev Transform

9.14 Application to the Wedge

9.15 The Cone

Separation of Variables

9.16 The Metal Strip

9.17 The Circular Disc

Approximate Methods

9.18 The Narrow Strip

9.19 The Small Disc

9.20 Kirchhoff’s Theory

9.21 Macdonald's Theory

9.22 Keller's Method

9.23 The Variational Method

Exercises

10. Aperiodic Phenomena

10.1 Reflection at a Plane Interface

Methods for Partial Differential Equations

10.2 Characteristics

10.3 Transport Equations

10.4 Uniqueness

10.5 The Initial Value Problem

10.6 Particular Cases

Propagation of Waves

10.7 The Distant Radiation from a Point Source

10.8 The Integration with Respect to ω

10.9 Asymptotic behavior when a Saddle-Point is Near a Pole

10.10 Dispersive Media

Exercises

11. Miscellaneous topics

The Theory of Electrons

11.1 The Expansion in Multipoles

11.2 The Average Equations in a Body

11.3 Polarization

11.4 Dispersion

11.5 Time Variations

The Theory of Fluid Motion

11.6 The Equations of Motion

11.7 Thermodynamic Considerations

11.8 Thermal Flux and Stress

11.9 Various Properties of Flows

11.10 Sound waves

11.11 Simple Waves

11.12 Shock waves

11.13 Boundary Conditions

Magneto-Hydrodynamics

11.14 The Equations of Motion

11.15 Boundary Conditions

11.16 Magneto-Hydrostatics in a Perfectly Conducting Fluid

11.17 An Energy Equation

11.18 Frozen-In Fields

11.19 Small Amplitude Waves

11.20 Waves of Finite Amplitude

11.21 One-Dimensional Waves

11.22 Shock Waves

11.23 The Effects of Dissipation

11.24 Steady Flow of a Viscous Fluid

11.25 Other Problems

Plasma Dynamics

11.26 Boltzmann's Equation

11.27 Average Properties

11.28 The Maxwellian Distribution

11.29 The Equations for a Plasma

11.30 Average Properties of a Plasma

11.31 Some Magnetic Effects

11.32 The Current Flow

11.33 Plasma Waves

Exercises

Tables

Author Index

Subject Index

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