Electrical Engineering Textbook Series Richard C Dorf, Series Editor University of California, Davis Forthcoming Titles Applied Vector Analysis Matiur Rahman and Issac Mulolani Optimal Control Systems Subbaram Naidu Continuous Signals and Systems with MATLAB Taan ElAli and Mohammad A Karim Discrete Signals and Systems with MATLAB Taan ElAli Edward J Rothwell Michigan State University East Lansing, Michigan Michael J Cloud Lawrence Technological University Southfield, Michigan CRC Press Boca Raton London New York Washington, D.C 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) Electromagnetic theory I Cloud, Michael J II Title III Series QC670 R693 2001 530.14′1—dc21 00-065158 CIP 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 Printed on acid-free paper In memory of Catherine Rothwell 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 we present essential background on the field concept, as well as information related specifically to the electromagnetic field and its sources Chapter 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 In Chapter 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 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 & 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 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 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 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 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 1+x (3x − 1) ln − x 1−x 1 + x Q (x) = (5x − 3x) ln − x2 + 1−x 1+x 55 35 Q (x) = (35x − 30x + 3) ln − x3 + x 16 1−x 24 Q (x) = P11 (x) = −(1 − x )1/2 = − sin θ P21 (x) = −3x(1 − x )1/2 = −3 cos θ sin θ P22 (x) = 3(1 − x ) = sin2 θ 3 P31 (x) = − (5x − 1)(1 − x )1/2 = − (5 cos2 θ − 1) sin θ 2 P32 (x) = 15x(1 − x ) = 15 cos θ sin2 θ P33 (x) = −15(1 − x )3/2 = −15 sin3 θ 5 P41 (x) = − (7x − 3x)(1 − x )1/2 = − (7 cos3 θ − cos θ) sin θ 2 15 15 2 P4 (x) = (7x − 1)(1 − x ) = (7 cos2 θ − 1) sin2 θ 2 P43 (x) = −105x(1 − x )3/2 = −105 cos θ sin3 θ P44 (x) = 105(1 − x )2 = 105 sin4 θ (E.132) (E.133) (E.134) (E.135) (E.136) (E.137) (E.138) (E.139) (E.140) (E.141) (E.142) (E.143) (E.144) Functional relationships Pnm (x) = 0, m > n, m (1−x )m/2 d n+m (x −1)n (−1) , m ≤ n 2n n! d x n+m (E.145) d n (x − 1)n dxn (E.146) Pn (x) = 2n n! Rnm (x) = (−1)m (1 − x )m/2 Pn−m (x) = (−1)m d m Rn (x) dxm (n − m)! m P (x) (n + m)! n Pn (−x) = (−1)n Pn (x) Q n (−x) = (−1)n+1 Q n (x) Pnm (−x) = (−1)n+m Pnm (x) Qm n (−x) = (−1) n+m+1 Qm n (x) (E.147) (E.148) (E.149) (E.150) (E.151) (E.152) m = 0, m > (E.153) |Pn (x)| ≤ Pn (1) = (E.154) Pnm (1) = 1, 0, Pn (0) = √ n π Pn−m (x) = (−1)m + n 2 +1 cos nπ (E.155) (n − m)! m P (x) (n + m)! n (E.156) Power series n Pn (x) = k=0 (−1)k (n + k)! (1 − x)k + (−1)n (1 + x)k (n − k)!(k!)2 2k+1 (E.157) Recursion relationships m m (n + − m)Rn+1 (x) + (n + m)Rn−1 (x) = (2n + 1)x Rnm (x) (E.158) m (1 − x )Rnm (x) = (n + 1)x Rnm (x) − (n − m + 1)Rn+1 (x) (E.159) (2n + 1)x Rn (x) = (n + 1)Rn+1 (x) + n Rn−1 (x) (x − 1)Rn (x) = (n + 1)[Rn+1 (x) − x Rn (x)] Rn+1 (x) − Rn−1 (x) = (2n + 1)Rn (x) Integral representations (E.160) (E.161) (E.162) √ π sin n + 12 u du √ π cos θ − cos u π Pn (x) = x + (x − 1)1/2 cos θ π Pn (cos θ ) = (E.163) n dθ (E.164) Addition formula Pn (cos γ ) = Pn (cos θ)Pn (cos θ ) + n (n − m)! m +2 Pn (cos θ)Pnm (cos θ ) cos m(φ − φ ), (n + m)! m=1 cos γ = cos θ cos θ + sin θ sin θ cos(φ − φ ) (E.165) (E.166) Summations = |r − r | r + r − 2rr cos γ = ∞ n=0 rn+1 Pn (cos γ ) cos γ = cos θ cos θ + sin θ sin θ cos(φ − φ ) r< = |r|, |r | , r> = max |r|, |r | (E.167) (E.168) (E.169) Integrals Pn (x) d x = −1 Pn+1 (x) − Pn−1 (x) +C 2n + x m Pn (x) d x = 0, m = max |r|, |r | (E.200) (E.201) Series expansion of a function f (θ, φ) 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of 10−10 m or less, and temporally... electrical machines using such concepts Must we therefore retreat to the macroscopic idea and ignore the discretization of charge completely? A viable alternative is to use atomic theories of matter to estimate the useful scope of macroscopic electromagnetics Remember, we are completely free to postulate a theory of nature whose scope may be limited Like continuum mechanics, which treats distributions of matter... describe the e ect 1 Attempts have been made to formulate electromagnetic theory purely in action-at-a-distance terms, but this viewpoint has not been generally adopted [69] of the supporting medium on the fields and are dependent upon the physical state of the medium The state may include macroscopic e ects, such as mechanical stress and thermodynamic temperature, as well as the microscopic, quantum-mechanical... ∂ m (r, t) = 0 ∂t (1.18) With these new sources Maxwell’s equations become appealingly symmetric Despite uncertainties about the existence and physical nature of magnetic monopoles, magnetic charge and current have become an integral part of electromagnetic theory We often use the concept of fictitious magnetic sources to make Maxwell’s equations symmetric, and then derive various equivalence theorems... and Adler [70] Importantly, all of these various forms of Maxwell’s equations produce the same values of the physical fields (at least external to the material where the fields are measurable) We must include several other constituents, besides the field equations, to make the postulate complete To form a complete field theory we need a source field, a mediating field, and a set of field differential equations... sources or material parameters are discontinuous The electromagnetic fields carry SI units as follows: E is measured in Volts per meter (V /m) , B is measured in Teslas (T), H is measured in Amperes per meter (A /m) , and D is measured in Coulombs per square meter (C /m2 ) In older texts we find the units of B given as Webers per square meter (Wb /m2 ) to reflect the role of B as a flux vector; in that case the... 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 the charges, and (2) the momentum and kinetic energy of the system are zero Now suppose one... given time, the net energy and momentum of the system, composed of both the bodies and the field, remain constant We thus come to regard the electromagnetic field as a true physical entity: an entity capable of carrying energy and momentum 1.2.2 Formalization of field theory Before we can invoke physical laws, we must find a way to describe the state of the system we intend to study We generally begin... possible We shall refer to E as the electric field, H as the magnetic field, D as the electric flux density and B as the magnetic flux density When we use the term electromagnetic field we imply the entire set of field vectors (E, D, B, H) used in Maxwell’s theory Invariance of Maxwell’s equations Maxwell’s differential equations are valid for any system in uniform relative motion with respect to the laboratory... charged particle from which the electric field diverges In contrast, experiments show that magnetic fields are created only by currents or by time changing electric fields; hence, magnetic fields have moving electric charge as their source The elemental source of magnetic field is the magnetic dipole, representing a tiny loop of electric current (or a spinning electric particle) The observation made in