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ElectrodynamicsProblems and solutions

ElectrodynamicsProblems and solutions

Carolina C Ilie and Zachariah S SchrecengostState University of New York at Oswego

Morgan & Claypool Publishers

Copyright ª 2018 Morgan & Claypool Publishers

All rights reserved. No part of this publication may be reproduced, stored in a retrieval systemor transmitted in any form or by any means, electronic, mechanical, photocopying, recordingor otherwise, without the prior permission of the publisher, or as expressly permitted by law orunder terms agreed with the appropriate rights organization. Multiple copying is permitted inaccordance with the terms of licences issued by the Copyright Licensing Agency, the CopyrightClearance Centre and other reproduction rights organisations.

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ISBN 978-1-6817-4931-0 (ebook)ISBN 978-1-6817-4928-0 (print)ISBN 978-1-6817-4929-7 (mobi)

DOI 10.1088/978-1-6817-4931-0

Version: 20180501

IOP Concise PhysicsISSN 2053-2571 (online)ISSN 2054-7307 (print)

A Morgan & Claypool publication as part of IOP Concise PhysicsPublished by Morgan & Claypool Publishers, 1210 Fifth Avenue, Suite 250, San Rafael, CA,94901, USA

IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

In the memory of my father.To the people I still learn from every day: my family, mentors, students, and

friends—CCI.

To my friends, family and mentors—ZSS.

Contents

Preface ix

Acknowledgements xi

Author biographies xii

Illustrations xiii

1 Maxwell’s equations—Electrodynamics 1-1

1.1 Theory 1-1

1.1.1 Ohm’s law 1-1

1.1.2 Joule heating law 1-1

1.1.3 Flux rule for motional electromotive force 1-2

1.1.4 Magnetic energy 1-2

1.1.5 Maxwell’s equations 1-2

1.2 Problems and solutions 1-3

Bibliography 1-28

2 Conservation laws 2-1

2.1 Theory 2-1

2.1.1 Electromagnetic energy density 2-1

2.1.2 Poynting’s theorem 2-1

2.1.3 Maxwell’s stress tensor 2-2

2.1.4 Electromagnetic force on a charge in volumeV 2-2

2.1.5 Electromagnetic momentum density 2-2

2.1.6 Continuity equation 2-2

2.2 Problems and solutions 2-3

Bibliography 2-24

3 Electromagnetic waves 3-1

3.1 Theory 3-1

3.1.1 Intensity 3-1

3.1.2 Radiation pressure 3-1

3.1.3 Monochromatic plane wave 3-2

3.1.4 Electromagnetic waves in matter 3-2

3.1.5 Coaxial transmission line 3-2

3.2 Problems and solutions 3-3

Bibliography 3-30

vii

4 Potentials and fields 4-1

4.1 Theory 4-1

4.1.1 Electric field 4-1

4.1.2 Gauge transformations 4-1

4.1.3 Retarded potentials 4-2

4.1.4 Jefimenko’s equations 4-2

4.1.5 Liénard–Wiechert potentials 4-2

4.1.6 Moving point charge 4-3

4.2 Problems and solutions 4-3

Bibliography 4-25

5 Radiation 5-1

5.1 Theory 5-1

5.1.1 Power radiated 5-1

5.1.2 Electric dipole radiation 5-2

5.1.3 Magnetic dipole radiation 5-2

5.1.4 Larmor formula 5-2

5.1.5 Radiation fields 5-2

5.2 Problems and solutions 5-3

Bibliography 5-24

Electrodynamics

viii

Preface

We wrote this book of problems and solutions as a natural continuation of our firstbook Electromagnetism: Problems and Solutions, published in November 2016 byInstitute of Physics Concise Physics, a Morgan & Claypool publication. As with thefirst book, we wrote this book for junior or senior undergraduate students, and forgraduate students who may have not studied electrodynamics yet and who may wantto work on more problems and have an immediate feedback while studying. Theauthors strongly recommend the textbook of David J Griffiths, Introduction toElectrodynamics as a first source manual, since it is recognized as one of the bestbooks on electrodynamics at the undergraduate level. We consider this book ofproblems and solutions a companion for the student who would like to workelectrodynamics problems more independently in order to deepen their under-standing and the problems solving skills and perhaps prepare for graduate school.We add brief theoretical notes and formulae, and for a complete theoreticalapproach we suggest Griffiths’s book. Every chapter is organized as follows: concisetheoretical notes, then the problem text with the solution. Each chapter ends with abrief bibliographical note.

We discuss herein the main concepts and techniques related to Maxwell’sequations, conservation laws, electromagnetic waves, potentials and fields, andradiation. For students interested in relativistic electrodynamics we recommend,Introduction to Electrodynamics, by David Griffiths, Classical Electrodynamics, byJohn David Jackson, Modern Electrodynamics, by Andrew Zangwill, andRelativistic Electrodynamics and Differential Geometry, by Stephen Parrot.

We follow here the same notation employed in our first book, Electromagnetism:Problems and solutions and introduced by David Griffiths in Introduction toElectrodynamics. Therefore, we use the r for the vector from a source point ′r tothe field point r . Please note that ˆ = = −

∣ − ∣

′

′r r

r rr r

rand, as you see, this notation

simplifies greatly already complex equations, but you need to be careful, especially ifyou only use cursive or typed letters. Also, we use the same notation s for thedistance to z axis in cylindrical coordinates, as is used in Griffiths’ book.

The chosen units are SI units—the international system. The reader should beaware that other books may employ either the Gaussian system (CGS) or theHeaviside–Lorentz (HL). Here is the Coulomb force in each of the systems:

In the SI system:

πε = ˆF

q q14 o

1 22rr

In CGS (Gaussian):

= ˆFq q1 2

2rr

And in Heaviside–Lorentz (HL)

ix

π = ˆF

q q14

1 22rr

Some of the problems are applications of the Maxwell’s equations for varioussystems which can be discussed with the available mathematical methods. Severalproblems which are presented here may appear in a variety of undergraduatetextbooks on electrodynamics, as they are classic solvable examples. However, weconsider it important to discuss these problems as they are fundamental to the studyof electrodynamics. We also present some problems of a more general nature, whichmay be a bit more challenging. We tried to keep a balance between the two types ofproblems, and we hope that the readers will enjoy this variation and have as muchthrill and excitement as we had while creating and solving these problems.

Electrodynamics

x

Acknowledgements

We want to thank our illustrator, Julia D’Rozario for making all of the figures. Wethank Dr Ildar Sabirianov for giving useful suggestions. We appreciate the supportof the administration at SUNY Oswego in granting Dr Ilie the sabbatical semester.Dr Ilie is grateful to the Electrodynamics II students Shelby Davis, Vincent DeBiase,Michael Kolacki and Stephen Porter for sharing the enthusiasm for this field. Wealso thank Victor Sabirianov for contributing to the last minute editing. Dr Ilie isgrateful to Dr Peter Dowben, from University of Nebraska at Lincoln for being awonderful mentor. And many thanks to our editors, Joel Claypool, Publisher atMorgan & Claypool Publishers, Jeanine Burke, Consulting Editor at Morgan &Claypool for the IOP Concise Physics ebook program, and Chris Benson,Production Editor at IOP Publishing. We are also grateful to Brent Beckley, theDirect Marketing Manager of Morgan & Claypool Publishers.

Last, but not the least, we thank our families and friends for their love, joy,encouragement, and sense of humor—their constant presence made this journeyexciting and rewarding.

xi

Author biographies

Carolina C Ilie

Carolina C Ilie is an Associate Professor with tenure at the StateUniversity of New York at Oswego. She taught ElectromagneticTheory for almost ten years and she designed various problems forher students’ exams, group work and quizzes and was excited towork on the two books on electromagnetism and electrodynamicsduring the last two years. Dr Ilie received her PhD in Physics andAstronomy at the University of Nebraska at Lincoln, an MSc in

Physics at the Ohio State University and another MSc in Physics at the University ofBucharest, Romania. She received the President’s Award for Teaching Excellence in2016 and the Provost Award for Mentoring in Scholarly and Creative Activity in2013. She lives in Central New York with her spouse, also a physicist, and their twosons.

Zachariah S Schrecengost

Zachariah S Schrecengost is a State University of New Yorkalumnus. He graduated with a BS degree having completed majorsin Physics, Software Engineering, and Applied Mathematics. Hetook the Advanced Electromagnetic Theory course with Dr Ilie andloved to be involved in this project. He brings to the project boththe fresh perspective of the student taking electrodynamics, aswell as the enthusiasm and talent of an alumnus who is an

electrodynamics and upper level mathematics aficionado. Mr Schrecengost workedas a software engineer in Syracuse and is now working on his PhD in Physics.

xii

Illustrations

Julia R D’Rozario

Julia R D’Rozario is an alumna of the State University of NewYork at Oswego. She graduated in December 2016 with a BS inPhysics and a BA in Cinema and Screen Studies and finished aminor in Astronomy by May 2016. She completed the AdvancedElectromagnetic Theory course with Dr Ilie and has muchexperience in the arts through her cinema studies. Ms D’Rozariocontributes her knowledge of electrodynamics and talents in

drawing in Inkscape software. She is currently working on her PhD in MicrosystemsEngineering at the Rochester Institute of Technology. She works for the NanoPowerResearch Laboratory under the advisement of Dr Seth Hubbard on III–Vcompound solar cells.

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