A student guide for maxwell equations

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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 available throughwww.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, 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 2008 Thispublication 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 no responsibilityfor 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 magnetic fieldsThe 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


2 2.1




3 3.1


4 4.14.2


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...
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