ELECTROMAGNETICS Edward J. Rothwell Michigan State University East Lansing, Michigan Michael J. Cloud Lawrence Technological University Southfield,Michigan CRC CRC Press Boca Raton London New York Washington, D.C
Edward J. Rothwell Michigan State University East Lansing, Michigan Michael J. Cloud Lawrence Technological University Southfield, Michigan Boca Raton London New York Washington, D.C. CRC Press
Library of Congress Cataloging-in-Publication Data Electromagnetics/Edward J. Rothwell, Michael J Cloud. -Electrical engineering textbook serie Includes bibliographical references and index. ISBN 0-8493-1397-X(alk. paper) ectromagnetic theory. I. Cloud, Michael J. Il. Title. Ill. Series QC670R6932001 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable fforts have been made to publish reliable data and information, but the author and the publisher cannot me responsibility for the validity of all materials or for the consequences of their us Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N w. Corporate Blvd., Boca Raton, Florida 33431, or visit ourweBsiteatwww.crcpress.com Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe Visitourwebsiteatwww.crcpress.com. o 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 456789 0 Printed on acid-free paper
This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431, or visit our Web site at www.crcpress.com Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit our website at www.crcpress.com. © 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 530.14′1—dc21 00-065158 CIP
Preface This book is intended as a text for a first-year graduate sequence in engineering electro- magnetics. Ideally such a sequence provides a transition period during which a student can solidify his or her understanding of fundamental concepts before proceeding to spe. cialized areas of research The assumed background of the reader is limited to standard undergraduate topics in physics and mathematics. Worthy of explicit mention are complex arithmetic, vec- tor analysis, ordinary differential equations, and certain topics normally covered in a signals and systems"course(e. g, convolution and the Fourier transform). Further ar alytical tools, such as contour integration, dyadic analysis, and separation of variables are covered in a self-contained mathematical appendix. The organization of the book is in six chapters. In Chapter 1 we present essential background 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 Maxwells theory of electromagnetism. Here attention is given to several useful forms of Maxwells equations, the nature of the four field quantities and of the postulate in general, some fundamental theorems, and the wave nature of the time-varying field. The electrostatic and 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 potential functions is discussed in Chapter 5, along with other field decompositions including the solenoidal-lamellar, transverse-longitudinal, and TE-TM types. Finally, in Chapter 6 we present the powerful integral solution to Maxwells equations by the method of Strat on and Chu. A main mathematical appendix near the end of the book contains brief but ufficient treatments of Fourier analysis, vector transport theorems, complex-plane inte- gration, dyadic analysis, and boundary value problems. Several subsidiary appendices provide useful tables of identities, transforms, and so on We would like to express our deep gratitude to those persons who contributed to the development of the book. The reciprocity-based derivation of the Stratton-Chu formula was provided by Prof. Dennis Nyquist, as was the material on wave reflection from multiple layers. The groundwork for our discussion of the Kronig-Kramers relations was provided by Michael Havrilla, and material on the time-domain reflection coefficient was developed 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 its preparation, and to Christopher Coleman for helping to prepare the figures. We are indebted to Dr. John E. Ross for kindly permitting us to employ one of his computer programs for scattering from a sphere and another for numerical Fourier transformation Helpful comments and suggestions on the figures were provided by Beth Lannon-Cloud 0 2001 by CRC Press LLC
Preface This book is intended as a text for a first-year graduate sequence in engineering electromagnetics. Ideally such a sequence provides a transition period during which a student can solidify his or her understanding of fundamental concepts before proceeding to specialized areas of research. The assumed background of the reader is limited to standard undergraduate topics in physics and mathematics. Worthy of explicit mention are complex arithmetic, vector analysis, ordinary differential equations, and certain topics normally covered in a “signals and systems” course (e.g., convolution and the Fourier transform). Further analytical tools, such as contour integration, dyadic analysis, and separation of variables, are covered in a self-contained mathematical appendix. The organization of the book is in six chapters. In Chapter 1 we present essential background on the field concept, as well as information related specifically to the electromagnetic field and its sources. Chapter 2 is concerned with a presentation of Maxwell’s theory of electromagnetism. Here attention is given to several useful forms of Maxwell’s equations, the nature of the four field quantities and of the postulate in general, some fundamental theorems, and the wave nature of the time-varying field. The electrostatic and magnetostatic cases are treated in Chapter 3. In Chapter 4 we cover the representation 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 potential functions is discussed in Chapter 5, along with other field decompositions including the solenoidal–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 Stratton and Chu. A main mathematical appendix near the end of the book contains brief but sufficient treatments of Fourier analysis, vector transport theorems, complex-plane integration, dyadic analysis, and boundary value problems. Several subsidiary appendices provide useful tables of identities, transforms, and so on. We would like to express our deep gratitude to those persons who contributed to the development of the book. The reciprocity-based derivation of the Stratton–Chu formula was provided by Prof. Dennis Nyquist, as was the material on wave reflection from multiple layers. The groundwork for our discussion of the Kronig–Kramers relations was provided by Michael Havrilla, and material on the time-domain reflection coefficient was developed 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 its preparation, and to Christopher Coleman for helping to prepare the figures. We are indebted to Dr. John E. Ross for kindly permitting us to employ one of his computer programs 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 Techworks. Inc.. for assistance with the LaTeX macros that were responsible for the layout of the book. Finally, we would like to thank the staff members of CRC Press- Evelyn Meany, Sara Seltzer, Elena Meyers, Helena Redshaw, Jonathan Pennell, Joette Lynch, and Nora Konopka- for their guidance and support 0 2001 by CRC Press LLC
Thanks to Dr. C. L. Tondo of T & T Techworks, Inc., for assistance with the LaTeX macros that were responsible for the layout of the book. Finally, we would like to thank the staff members of CRC Press — Evelyn Meany, Sara Seltzer, Elena Meyers, Helena Redshaw, Jonathan Pennell, Joette Lynch, and Nora Konopka — for their guidance and support
Contents 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 1.3.4 Charge conservation 1.3.5 Magnetic charge 1.4 Problems 2 Maxwells theory of electromagnetism 2.1 The postulate 2.1.1 The Maxwell-Minkowski equation 2.1.2 Connection to mechanics 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 Maxwells 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 0 2001 by CRC Press LLC
Contents 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 1.3.4 Charge conservation 1.3.5 Magnetic charge 1.4 Problems 2 Maxwell’s theory of electromagnetism 2.1 The postulate 2.1.1 The Maxwell–Minkowski equations 2.1.2 Connection to mechanics 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 using equivalent sources 2.8.5 Boundary conditions across a moving layer of field discontinuity 2.9 Fundamental theorems 2.9.1 Linearity 2.9.2 Dualit 2.9.3 Reciprocity 2. 9.4 Similitude 2.9.5 Conservation theorems 2.10 The wave nature of the electromagnetic field omagnetic waves 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 the wave 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 wor 3.2.2 Boundary conditions 3.2.3 Uniqueness of the electrostatic field 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 bod 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 0 2001 by CRC Press LLC
2.8.4 Boundary conditions across a stationary layer of field discontinuity using equivalent sources 2.8.5 Boundary conditions across a moving layer of field discontinuity 2.9 Fundamental theorems 2.9.1 Linearity 2.9.2 Duality 2.9.3 Reciprocity 2.9.4 Similitude 2.9.5 Conservation theorems 2.10 The wave nature of the electromagnetic field 2.10.1 Electromagnetic waves 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 the wave 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 Greens reciprocation theorem 3.5 Problems 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 permittivit 4.4.2 High and low frequency behavior of constitutive parameters 4.4.3 The Kronig-Kramers relations 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 conducto 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 Maxwells 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 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 0 2001 by CRC Press LLC
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 decomposition 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 Maxwells equations 6.1 Vector Kirchoff solution 6.1.1 The Stratton-Chu formula 1.2 The Sommerfeld radiation condition 6 6.1.3 Fields in the excluded region: the extinction theorem 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 3.2 The franz formula 6.3.3 Love's equivalence principle 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 A2 Vector transport theorems 1.3 analysis ary value probl B Useful identities c Some Fourier transform pairs 0 2001 by CRC Press LLC
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 E1 Bessel functions E2 Legendre functions E3 Spherical harmonics References 0 2001 by CRC Press LLC
D Coordinate systems E Properties of special functions E.1 Bessel functions E.2 Legendre functions E.3 Spherical harmonics References
