A Student’s Guide to Maxwell’s Equations
Maxwell’s Equations are four of the most inﬂuential equations in science: Gauss’s law for electric ﬁelds, Gauss’s law for magnetic ﬁelds, Faraday’s law, and the Ampere–Maxwell law. In this guide for students, each equation is the subject of an entire chapter, with detailed, plain-language explanations of thephysical meaning of each symbol in the equation, for both the integral and differential forms. The ﬁnal chapter shows how Maxwell’s Equations may be combined to produce the wave equation, the basis for the electromagnetic theory of light. This book is a wonderful resource for undergraduate and graduate courses in electromagnetism and electromagnetics. A website hosted by the author, and availablethrough www.cambridge.org/9780521877619, contains interactive solutions to every problem in the text. Entire solutions can be viewed immediately, or a series of hints can be given to guide the student to the ﬁnal answer. The website also contains audio podcasts which walk students through each chapter, pointing out important details and explaining key concepts.
da n i e l fl eis ch is AssociateProfessor in the Department of Physics at Wittenberg University, Ohio. His research interests include radar cross-section measurement, radar system analysis, and ground-penetrating radar. He is a member of the American Physical Society (APS), the American Association of Physics Teachers (AAPT), and the Institute of Electrical and Electronics Engineers (IEEE).
A Student’s Guide to Maxwell’sEquations
DANIEL FLEISCH Wittenberg University
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521877619 © D. Fleisch 2008This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008
ISBN-13 978-0-511-39308-2 ISBN-13 978-0-521-87761-9
eBook (EBL) hardback
Cambridge University Press has noresponsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
Preface Acknowledgments 1 1.1 Gauss’s law for electric ﬁelds The integral form of Gauss’s law The electric ﬁeld The dot product The unit normal vector ~ Thecomponent of E normal to a surface The surface integral The ﬂux of a vector ﬁeld The electric ﬂux through a closed surface The enclosed charge The permittivity of free space Applying Gauss’s law (integral form) The differential form of Gauss’s law Nabla – the del operator Del dot – the divergence The divergence of the electric ﬁeld Applying Gauss’s law (differential form) Gauss’s law for magneticﬁelds The integral form of Gauss’s law The magnetic ﬁeld The magnetic ﬂux through a closed surface Applying Gauss’s law (integral form) The differential form of Gauss’s law The divergence of the magnetic ﬁeld Applying Gauss’s law (differential form) v
page vii ix 1 1 3 6 7 8 9 10 13 16 18 20 29 31 32 36 38 43 43 45 48 50 53 54 55
Faraday’s law The integral form of Faraday’s law The induced electric ﬁeld The line integral The path integral of a vector ﬁeld The electric ﬁeld circulation The rate of change of ﬂux Lenz’s law Applying Faraday’s law (integral form) The differential form of Faraday’s law Del cross – the curl The curl of the electric ﬁeld Applying Faraday’s law (differential form) The Ampere–Maxwell...