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Preface 1 Introductory concepts 1.1 Notation, conventions, and symbology 1.2 The field concept of electromagnetics 1.2.1 Historical perspective 1.2.2 Formalization of field theory 1.3 The

Electrical Engineering Textbook Series

Richard C Dorf, Series Editor University of California, Davis

Forthcoming Titles

Applied Vector Analysis

Matiur Rahman and Issac Mulolani

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Continuous Signals and Systems with MATLAB

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Taan ElAli

Edward J Rothwell

Michigan State University

East Lansing, Michigan

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© 2001 by CRC Press LLC

No claim to original U.S Government works International Standard Book Number 0-8493-1397-X Library of Congress Card Number 00-065158 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

Printed on acid-free paper

Library of Congress Cataloging-in-Publication Data

Rothwell, Edward J.

Electromagnetics / Edward J Rothwell, Michael J Cloud.

p cm.—(Electrical engineering textbook series ; 2) Includes bibliographical references and index.

ISBN 0-8493-1397-X (alk paper)

1 Electromagnetic theory I Cloud, Michael J II Title III Series.

QC670 R693 2001

CIP

In memory of Catherine Rothwell

This book is intended as a text for a first-year graduate sequence in engineering magnetics Ideally such a sequence provides a transition period during which a studentcan solidify his or her understanding of fundamental concepts before proceeding to spe-cialized areas of research

electro-The assumed background of the reader is limited to standard undergraduate topics

in physics and mathematics Worthy of explicit mention are complex arithmetic, tor analysis, ordinary differential equations, and certain topics normally covered in a

vec-“signals and systems” course (e.g., convolution and the Fourier transform) Further alytical tools, such as contour integration, dyadic analysis, and separation of variables,are covered in a self-contained mathematical appendix

an-The organization of the book is in six chapters In Chapter 1 we present essentialbackground on the field concept, as well as information related specifically to the electro-magnetic field and its sources Chapter 2 is concerned with a presentation of Maxwell’stheory of electromagnetism Here attention is given to several useful forms of Maxwell’sequations, the nature of the four field quantities and of the postulate in general, somefundamental theorems, and the wave nature of the time-varying field The electrostaticand magnetostatic cases are treated in Chapter 3 In Chapter 4 we cover the representa-tion of the field in the frequency domains: both temporal and spatial Here the behavior

of common engineering materials is also given some attention The use of potentialfunctions is discussed in Chapter 5, along with other field decompositions including thesolenoidal–lamellar, transverse–longitudinal, and TE–TM types Finally, in Chapter 6

we present the powerful integral solution to Maxwell’s equations by the method of ton and Chu A main mathematical appendix near the end of the book contains brief butsufficient treatments of Fourier analysis, vector transport theorems, complex-plane inte-gration, dyadic analysis, and boundary value problems Several subsidiary appendicesprovide useful tables of identities, transforms, and so on

Strat-We would like to express our deep gratitude to those persons who contributed to thedevelopment of the book The reciprocity-based derivation of the Stratton–Chu formulawas provided by Prof Dennis Nyquist, as was the material on wave reflection frommultiple layers The groundwork for our discussion of the Kronig–Kramers relations wasprovided by Michael Havrilla, and material on the time-domain reflection coefficient wasdeveloped by Jungwook Suk We owe thanks to Prof Leo Kempel, Dr David Infante,and Dr Ahmet Kizilay for carefully reading large portions of the manuscript during itspreparation, and to Christopher Coleman for helping to prepare the figures We areindebted to Dr John E Ross for kindly permitting us to employ one of his computerprograms for scattering from a sphere and another for numerical Fourier transformation.Helpful comments and suggestions on the figures were provided by Beth Lannon–Cloud

Thanks to Dr C L Tondo of T & T Techworks, Inc., for assistance with the LaTeXmacros that were responsible for the layout of the book Finally, we would like to thankthe staff members of CRC Press — Evelyn Meany, Sara Seltzer, Elena Meyers, HelenaRedshaw, Jonathan Pennell, Joette Lynch, and Nora Konopka — for their guidance andsupport.

Preface

1 Introductory concepts

1.1 Notation, conventions, and symbology

1.2 The field concept of electromagnetics

1.2.1 Historical perspective

1.2.2 Formalization of field theory

1.3 The sources of the electromagnetic field

1.3.1 Macroscopic electromagnetics

1.3.2 Impressed vs secondary sources

1.3.3 Surface and line source densities

2.2 The well-posed nature of the postulate

2.2.1 Uniqueness of solutions to Maxwell’s equations

2.2.2 Constitutive relations

2.3 Maxwell’s equations in moving frames

2.3.1 Field conversions under Galilean transformation

2.3.2 Field conversions under Lorentz transformation

2.4 The Maxwell–Boffi equations

2.5 Large-scale form of Maxwell’s equations

2.5.1 Surface moving with constant velocity

2.5.2 Moving, deforming surfaces

2.5.3 Large-scale form of the Boffi equations

2.6 The nature of the four field quantities

2.7 Maxwell’s equations with magnetic sources

2.8 Boundary (jump) conditions

2.8.1 Boundary conditions across a stationary, thin source layer

2.8.2 Boundary conditions across a stationary layer of field discontinuity

2.8.3 Boundary conditions at the surface of a perfect conductor

2.8.4 Boundary conditions across a stationary layer of field discontinuity usingequivalent sources

2.8.5 Boundary conditions across a moving layer of field discontinuity

2.10.2 Wave equation for bianisotropic materials

2.10.3 Wave equation in a conducting medium

2.10.4 Scalar wave equation for a conducting medium

2.10.5 Fields determined by Maxwell’s equations vs fields determined by thewave equation

2.10.6 Transient uniform plane waves in a conducting medium

2.10.7 Propagation of cylindrical waves in a lossless medium

2.10.8 Propagation of spherical waves in a lossless medium

2.10.9 Nonradiating sources

2.11 Problems

3 The static electromagnetic field

3.1 Static fields and steady currents

3.1.1 Decoupling of the electric and magnetic fields

3.1.2 Static field equilibrium and conductors

3.1.3 Steady current

3.2 Electrostatics

3.2.1 The electrostatic potential and work

3.2.2 Boundary conditions

3.2.3 Uniqueness of the electrostatic field

3.2.4 Poisson’s and Laplace’s equations

3.2.5 Force and energy

3.2.6 Multipole expansion

3.2.7 Field produced by a permanently polarized body

3.2.8 Potential of a dipole layer

3.2.9 Behavior of electric charge density near a conducting edge

3.2.10 Solution to Laplace’s equation for bodies immersed in an impressed field

3.3 Magnetostatics

3.3.1 The magnetic vector potential

3.3.2 Multipole expansion

3.3.3 Boundary conditions for the magnetostatic field

3.3.4 Uniqueness of the magnetostatic field

3.3.5 Integral solution for the vector potential

3.3.6 Force and energy

3.3.7 Magnetic field of a permanently magnetized body

3.3.8 Bodies immersed in an impressed magnetic field: magnetostatic shielding

3.4 Static field theorems

3.4.1 Mean value theorem of electrostatics

3.4.2 Earnshaw’s theorem

3.4.3 Thomson’s theorem

3.4.4 Green’s reciprocation theorem

3.5 Problems

4 Temporal and spatial frequency domain representation

4.1 Interpretation of the temporal transform

4.2 The frequency-domain Maxwell equations

4.3 Boundary conditions on the frequency-domain fields

4.4 The constitutive and Kronig–Kramers relations

4.4.1 The complex permittivity

4.4.2 High and low frequency behavior of constitutive parameters

4.4.3 The Kronig–Kramers relations

4.5 Dissipated and stored energy in a dispersive medium

4.5.1 Dissipation in a dispersive material

4.5.2 Energy stored in a dispersive material

4.5.3 The energy theorem

4.6 Some simple models for constitutive parameters

4.6.1 Complex permittivity of a non-magnetized plasma

4.6.2 Complex dyadic permittivity of a magnetized plasma

4.6.3 Simple models of dielectrics

4.6.4 Permittivity and conductivity of a conductor

4.6.5 Permeability dyadic of a ferrite

4.7 Monochromatic fields and the phasor domain

4.7.1 The time-harmonic EM fields and constitutive relations

4.7.2 The phasor fields and Maxwell’s equations

4.7.3 Boundary conditions on the phasor fields

4.8 Poynting’s theorem for time-harmonic fields

4.8.1 General form of Poynting’s theorem

4.8.2 Poynting’s theorem for nondispersive materials

4.8.3 Lossless, lossy, and active media

4.9 The complex Poynting theorem

4.9.1 Boundary condition for the time-average Poynting vector

4.10 Fundamental theorems for time-harmonic fields

4.10.1 Uniqueness

4.10.2 Reciprocity revisited

4.10.3 Duality

4.11 The wave nature of the time-harmonic EM field

4.11.1 The frequency-domain wave equation

4.11.2 Field relationships and the wave equation for two-dimensional fields

4.11.3 Plane waves in a homogeneous, isotropic, lossy material

4.11.4 Monochromatic plane waves in a lossy medium

4.11.5 Plane waves in layered media

4.11.6 Plane-wave propagation in an anisotropic ferrite medium

4.11.7 Propagation of cylindrical waves

4.11.8 Propagation of spherical waves in a conducting medium

4.11.9 Nonradiating sources

4.12 Interpretation of the spatial transform

4.13 Spatial Fourier decomposition

4.13.1 Boundary value problems using the spatial Fourier representation

4.14 Periodic fields and Floquet’s theorem

4.14.1 Floquet’s theorem

4.14.2 Examples of periodic systems

4.15 Problems

5 Field decompositions and the EM potentials

5.1 Spatial symmetry decompositions

5.1.1 Planar field symmetry

5.2 Solenoidal–lamellar decomposition

5.2.1 Solution for potentials in an unbounded medium: the retarded potentials

5.2.2 Solution for potential functions in a bounded medium

5.3 Transverse–longitudinal decomposition

5.3.1 Transverse–longitudinal decomposition in terms of fields

5.4 TE–TM decomposition

5.4.1 TE–TM decomposition in terms of fields

5.4.2 TE–TM decomposition in terms of Hertzian potentials

5.4.3 Application: hollow-pipe waveguides

5.4.4 TE–TM decomposition in spherical coordinates

5.5 Problems

6 Integral solutions of Maxwell’s equations

6.1 Vector Kirchoff solution

6.1.1 The Stratton–Chuformula

6.1.2 The Sommerfeld radiation condition

6.1.3 Fields in the excluded region: the extinction theorem

6.2 Fields in an unbounded medium

6.2.1 The far-zone fields produced by sources in unbounded space

6.3 Fields in a bounded, source-free region

6.3.1 The vector Huygens principle

6.3.2 The Franz formula

6.3.3 Love’s equivalence principle

6.3.4 The Schelkunoff equivalence principle

6.3.5 Far-zone fields produced by equivalent sources

6.4 Problems

A Mathematical appendix

A.1 The Fourier transform

A.2 Vector transport theorems

A.3 Dyadic analysis

A.4 Boundary value problems

B Useful identities

C Some Fourier transform pairs

D Coordinate systems

E Properties of special functions

E.1 Bessel functions

E.2 Legendre functions

E.3 Spherical harmonics

References

Chapter 1

Introductory concepts

1.1 Notation, conventions, and symbology

Any book that covers a broad range of topics will likely harbor some problems withnotation and symbology This results from having the same symbol used in different areas

to represent different quantities, and also from having too many quantities to represent.Rather than invent new symbols, we choose to stay close to the standards and warn thereader about any symbol used to represent more than one distinct quantity

The basic nature of a physical quantity is indicated by typeface or by the use of a

diacritical mark Scalars are shown in ordinary typeface: q , , for example Vectors

are shown in boldface: E, Π Dyadics are shown in boldface with an overbar: ¯
, ¯A.

Frequency dependent quantities are indicated by a tilde, whereas time dependent tities are written without additional indication; thus we write ˜E(r, ω) and E(r, t) (Some

quan-quantities, such as impedance, are used in the frequency domain to interrelate Fourierspectra; although these quantities are frequency dependent they are seldom written inthe time domain, and hence we do not attach tildes to their symbols.) We often combine

diacritical marks:for example, ˜¯
denotes a frequency domain dyadic We distinguish

carefully between phasor and frequency domain quantities The variable ω is used for

the frequency variable of the Fourier spectrum, while ˇω is used to indicate the constant

frequency of a time harmonic signal We thus further separate the notion of a phasor

field from a frequency domain field by using a check to indicate a phasor field: ˇE(r).

However, there is often a simple relationship between the two, such as ˇE= ˜E( ˇω).

We designate the field and source point position vectors by r and r, respectively, and

the corresponding relative displacement or distance vector by R:

R = r − r.

A hat designates a vector as a unit vector (e.g., ˆx) The sets of coordinate variables in

rectangular, cylindrical, and spherical coordinates are denoted by

(x, y, z), (ρ, φ, z), (r, θ, φ),

respectively (In the spherical systemφ is the azimuthal angle and θ is the polar angle.)

We freely use the “del” operator notation ∇ for gradient, curl, divergence, Laplacian,and so on

The SI (MKS) system of units is employed throughout the book

1.2 The field concept of electromagnetics

Introductory treatments of electromagnetics often stress the role of the field in force

transmission:the individual fields E and B are defined via the mechanical force on a

small test charge This is certainly acceptable, but does not tell the whole story Wemight, for example, be left with the impression that the EM field always arises from

an interaction between charged objects Often coupled with this is the notion that thefield concept is meant merely as an aid to the calculation of force, a kind of notationalconvenience not placed on the same physical footing as force itself In fact, fields aremore than useful — they are fundamental Before discussing electromagnetic fields inmore detail, let us attempt to gain a better perspective on the field concept and its role

in modern physical theory Fields play a central role in any attempt to describe physicalreality They are as real as the physical substances we ascribe to everyday experience

In the words of Einstein [63],

“It seems impossible to give an obvious qualitative criterion for distinguishing betweenmatter and field or charge and field.”

We must therefore put fields and particles of matter on the same footing:both carryenergy and momentum, and both interact with the observable world

Early nineteenth century physical thought was dominated by the action at a distance

concept, formulated by Newton more than 100 years earlier in his immensely successfultheory of gravitation In this view the influence of individual bodies extends across space,instantaneously affects other bodies, and remains completely unaffected by the presence

of an intervening medium Such an idea was revolutionary; until then action by contact, in

which objects are thought to affect each other through physical contact or by contact withthe intervening medium, seemed the obvious and only means for mechanical interaction.Priestly’s experiments in 1766 and Coulomb’s torsion-bar experiments in 1785 seemed toindicate that the force between two electrically charged objects behaves in strict analogywith gravitation:both forces obey inverse square laws and act along a line joining theobjects Oersted, Ampere, Biot, and Savart soon showed that the magnetic force onsegments of current-carrying wires also obeys an inverse square law

The experiments of Faraday in the 1830s placed doubt on whether action at a distancereally describes electric and magnetic phenomena When a material (such as a dielec-tric) is placed between two charged objects, the force of interaction decreases; thus, theintervening medium does play a role in conveying the force from one object to the other

To explain this, Faraday visualized “lines of force” extending from one charged object toanother The manner in which these lines were thought to interact with materials theyintercepted along their path was crucial in understanding the forces on the objects Thisalso held for magnetic effects Of particular importance was the number of lines passing

through a certain area (the flux ), which was thought to determine the amplitude of the

effect observed in Faraday’s experiments on electromagnetic induction

Faraday’s ideas presented a new world view:electromagnetic phenomena occur in theregion surrounding charged bodies, and can be described in terms of the laws governingthe “field” of his lines of force Analogies were made to the stresses and strains in materialobjects, and it appeared that Faraday’s force lines created equivalent electromagnetic

stresses and strains in media surrounding charged objects His law of induction wasformulated not in terms of positions of bodies, but in terms of lines of magnetic force.Inspired by Faraday’s ideas, Gauss restated Coulomb’s law in terms of flux lines, andMaxwell extended the idea to time changing fields through his concept of displacementcurrent.

In the 1860s Maxwell created what Einstein called “the most important inventionsince Newton’s time”— a set of equations describing an entirely field-based theory ofelectromagnetism These equations do not model the forces acting between bodies, as doNewton’s law of gravitation and Coulomb’s law, but rather describe only the dynamic,time-evolving structure of the electromagnetic field Thus bodies are not seen to inter-act with each other, but rather with the (very real) electromagnetic field they create,

an interaction described by a supplementary equation (the Lorentz force law) To ter understand the interactions in terms of mechanical concepts, Maxwell also assignedproperties of stress and energy to the field

bet-Using constructs that we now call the electric and magnetic fields and potentials,Maxwell synthesized all known electromagnetic laws and presented them as a system ofdifferential and algebraic equations By the end of the nineteenth century, Hertz haddevised equations involving only the electric and magnetic fields, and had derived thelaws of circuit theory (Ohm’s law and Kirchoff’s laws) from the field expressions Hisexperiments with high-frequency fields verified Maxwell’s predictions of the existence ofelectromagnetic waves propagating at finite velocity, and helped solidify the link betweenelectromagnetism and optics But one problem remained:if the electromagnetic fieldspropagated by stresses and strains on a medium, how could they propagate through a

vacuum? A substance called the luminiferous aether, long thought to support the

trans-verse waves of light, was put to the task of carrying the vibrations of the electromagneticfield as well However, the pivotal experiments of Michelson and Morely showed that theaether was fictitious, and the physical existence of the field was firmly established.The essence of the field concept can be conveyed through a simple thought experiment.Consider two stationary charged particles in free space Since the charges are stationary,

we know that (1) another force is present to balance the Coulomb force between thecharges, and (2) the momentum and kinetic energy of the system are zero Now supposeone charge is quickly moved and returned to rest at its original position Action at adistance would require the second charge to react immediately (Newton’s third law),but by Hertz’s experiments it does not There appears to be no change in energy ofthe system:both particles are again at rest in their original positions However, after atime (given by the distance between the charges divided by the speed of light) we findthat the second charge does experience a change in electrical force and begins to moveaway from its state of equilibrium But by doing so it has gained net kinetic energyand momentum, and the energy and momentum of the system seem larger than at thestart This can only be reconciled through field theory If we regard the field as aphysical entity, then the nonzero work required to initiate the motion of the first chargeand return it to its initial state can be seen as increasing the energy of the field Adisturbance propagates at finite speed and, upon reaching the second charge, transfersenergy into kinetic energy of the charge Upon its acceleration this charge also sends out

a wave of field disturbance, carrying energy with it, eventually reaching the first chargeand creating a second reaction At any given time, the net energy and momentum of thesystem, composed of both the bodies and the field, remain constant We thus come toregard the electromagnetic field as a true physical entity:an entity capable of carryingenergy and momentum