Solved Problems in Electromagnetics PDF
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Undergraduate Lecture Notes in Physics Félix Salazar Bloise Rafael Medina Ferro Ana Bayón Rojo Francisco Gascón Latasa Solved Problems in Electromagnetics Undergraduate Lecture Notes in Physics Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topicsthroughoutpureandappliedphysics.Eachtitleintheseriesissuitableasabasisfor undergraduateinstruction,typicallycontainingpracticeproblems,workedexamples,chapter summaries, andsuggestions for further reading. ULNP titles mustprovide at least oneof thefollowing: (cid:129) Anexceptionally clear andconcise treatment ofastandard undergraduate subject. (cid:129) Asolidundergraduate-levelintroductiontoagraduate,advanced,ornon-standardsubject. (cid:129) Anovel perspective oranunusual approach toteaching asubject. ULNPespeciallyencouragesnew,original,andidiosyncraticapproachestophysicsteaching at theundergraduate level. ThepurposeofULNPistoprovideintriguing,absorbingbooksthatwillcontinuetobethe reader’spreferred reference throughout theiracademic career. Series editors Neil Ashby University of Colorado, Boulder, CO, USA William Brantley Department of Physics, Furman University, Greenville, SC, USA Matthew Deady Physics Program, Bard College, Annandale-on-Hudson, NY, USA Michael Fowler Department of Physics, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen Department of Physics, University of Oslo, Oslo, Norway Michael Inglis SUNY Suffolk County Community College, Long Island, NY, USA Heinz Klose Humboldt University, Oldenburg, Niedersachsen, Germany Helmy Sherif Department of Physics, University of Alberta, Edmonton, AB, Canada More information about this series at http://www.springer.com/series/8917 é F lix Salazar Bloise Rafael Medina Ferro (cid:129) ó ó Ana Bay n Rojo Francisco Gasc n Latasa (cid:129) Solved Problems in Electromagnetics 123 Félix Salazar Bloise Ana Bayón Rojo Departamento deFísica Aplicadaalos Departamento deFísica Aplicadaalos RecursosNaturales (FARN) RecursosNaturales (FARN) E.T.S.I.Minas y Energía. Universidad E.T.S.I.Minas y Energía. Universidad PolitécnicadeMadrid (UPM) PolitécnicadeMadrid (UPM) Madrid Madrid Spain Spain Rafael Medina Ferro Francisco GascónLatasa Departamento deFísica Aplicadaalos Departamento deFísica AplicadaII RecursosNaturales (FARN) E.T.S. Arquitectura. Universidad E.T.S.I.Minas y Energía. Universidad deSevilla (US) PolitécnicadeMadrid (UPM) Seville Madrid Spain Spain ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notesin Physics ISBN978-3-662-48366-4 ISBN978-3-662-48368-8 (eBook) DOI 10.1007/978-3-662-48368-8 LibraryofCongressControlNumber:2016951890 ©Springer-VerlagBerlinHeidelberg2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringer-VerlagGmbHGermany Theregisteredcompanyaddressis:HeidelbergerPlatz3,14197Berlin,Germany To our parents Preface This work has been developed by the authors after 30 years of teaching several courses of electricity and magnetism. The book contains more than three hundred solved problems, the majority of which have been proposed by the Department of Applied Physics to Natural Resources in official exams of Advance Physics, Physics II and Electromagnetism and Waves in the Mining and Energy School (ETSIME) of the Polytechnic University of Madrid (UPM). The book has been written for both beginners and advanced students in this subject.However,itmaybeusefulforphysicistsandengineers,andalsoforpeople thatworkwithrelatedtopicsandneedtheelectromagnetictheoryforunderstanding other disciplines. The objective of this book is to expose the fundamental concepts of electro- magnetism through problems. Starting with this idea each chapter is divided into two parts. The first one contains a brief theoretical introduction where the most important concepts and formulae employed in the chapter are usually presented withoutdemonstratingthem.Thesecondoneisdevotedtotheexerciseslabeledas problems A, B and C, respectively, depending on its difficulty and sometimes thematically. Problems of type A are thought for beginners in electricity and magnetismorforlecturesonGeneralPhysics,wheredefinitionsandconceptsabout this subject appear for first time. Problems of type B are a bit tougher and can be worked by students who have some basic knowledge in calculus and electromag- netics.Forclosingeachchapter,problemsoftypeCareintroduced.Thesekindsof exercises, even though they are not very difficult, have some conceptual or/and mathematical complications which make them more adequate for advanced lec- tures. According with the academic level of the student, this chapter distribution gives to the reader the possibility of using the book in a flexible way. Ourexperienceshowedusthatthemostsimplestthingsmaybeverydifficultfor the student at the beginning of learning a subject, if the explanations of the ideas involved are not clear. Sometimes, the supposition by part of the writer that one ideaorconceptisobviousmayleadtowastereader’stime.Forthisreasonwehave triedtoexplaintheproblemsin-depthwithanemphasisonphysicalconceptsrather than on the mathematical developments. vii viii Preface The book is structured in 14 chapters. It begins with an introductory chapter devotedtothebasicmathematicaltheoremsandformulaethatareneededforfurther developments. The next two chapters deal with the electric field in different situ- ations, namely in vacuum and when matter is present. An important topic when studying fields and circuits are the currents; this is the subject of Chap. 4. In the samewayascommentedfortheelectricfield,Chaps.5and6studytheoriginofthe magneticfieldandthephenomenonofthemagnetization.Untilthispartofthebook thetechniquesfor solvingtheelectric andmagnetic fields generated under specific circumstances are based on direct calculations. In Chap. 7 other more complicated methods for obtaining these fields are studied. Chapter 8 works out the important topic of the electromagnetic induction. The different causes of producing electro- motive force are explained in detail. The understanding of this phenomenon encompasses the knowledge of former chapters. For this reason it is not recom- mended to be studied without studying previously the fundamentals of the electric and magnetic fields. Chapter 9 refers to energetic aspects of the electromagnetic fieldandadidactic investigationoftheMaxwellequationsislefttoChap.10.The solution of partial differential equations may be very difficult. However, the viewpoint adopted in this chapter is more conceptual than mathematical. In fact, for systems of high symmetry it is possible to find a solution in a simpler way without solving the system of differential equations. In our opinion, numerous questions can be answered using a simple mathematical apparatus, without losing rigorandclarity.Duetotheimportanceoftheplasmas,cosmicrays,andmachines as cyclotron and betatron, among others, we have included the study of the movement of charged particles in electromagnetic field. This is the subject of Chap.11.OneofthemostimportantconsequencesoftheMaxwellequationsisthe unification of theelectricity andthe magnetism andalso thelight.In this regard in Chap.12ageneralviewoftheelectromagneticwavesisgivenandinChap.13,the phenomena of reflexion and refraction are treated. In this context, an interesting approach to the propagation of electromagnetic waves throughout anisotropic media is dealt with in the last part, Chap. 14. Wewouldliketoexpressoursinceregratitude,wholeheartedly,toourcolleague and friend Faustino Fernández López, for many years of his teaching, experience and for help wherever we needed. We are also in indebted to Prof. Aristide Dogariu and his scientific team at CREOL of the University of Central Florida. Their valuable discussions in the groupmeetingsprovidedusveryimportantinformationwhichallowedtointroduce some ideas and explanations in some parts of this book. A very special place in this book isfor our colleague Dr. James Germann, who reviewed all this work, word by word, and also corrected all equations carefully. We would like to thank him for his invaluable work, dedication, and profession- alism. Without his help this book would not have seen the light. We wish him all thebestforhisfutureandexpectthebestinhisworkwithtwo-photonmicroscopy and corneal imaging. We hope his work in this book can be appreciated in his scientific career. Preface ix ThanksarealsoduetotheEditorialSpringer,especiallyDr.ClausAscheronfor providinghelpinthepreparation ofthisbook.Hisgenerosityandpatience withus during the writing is largely appreciated. Finally, we would like to give special thanks to all our students, without whom we would have never written this book. Madrid, Spain Félix Salazar Bloise Madrid, Spain Rafael Medina Ferro Madrid, Spain Ana Bayón Rojo Seville, Spain Francisco Gascón Latasa Contents 1 A Mathematical Introduction.. .... .... .... .... .... ..... .... 1 1.1 Coordinate Systems and Transformations.... .... ..... .... 1 1.2 Differential Length, Area and Volume .. .... .... ..... .... 4 1.3 Scalar and Vector Fields. .... .... .... .... .... ..... .... 5 1.4 Concept and Definition of Regions, Curves and Surfaces. .... 6 1.5 Line, Surface and Volume Integrals. Circulation and Flux .... 9 1.6 Gradient .... .... ..... .... .... .... .... .... ..... .... 11 1.7 Curl ... .... .... ..... .... .... .... .... .... ..... .... 13 1.8 Divergence .. .... ..... .... .... .... .... .... ..... .... 14 1.9 Stokes’s Theorem and Divergence Theorem.. .... ..... .... 15 1.10 Normal Vector to a Surface .. .... .... .... .... ..... .... 16 1.10.1 Vectorial or Parametric ... .... .... .... ..... .... 16 1.10.2 Explicit .. ..... .... .... .... .... .... ..... .... 17 1.10.3 Implicit .. ..... .... .... .... .... .... ..... .... 17 1.11 Further Developments... .... .... .... .... .... ..... .... 19 1.12 Classification of Vector Fields .... .... .... .... ..... .... 21 1.13 Obtaining the Scalar Potential. .... .... .... .... ..... .... 23 1.13.1 First Method ... .... .... .... .... .... ..... .... 23 1.13.2 Second Method . .... .... .... .... .... ..... .... 24 1.13.3 Third Method*.. .... .... .... .... .... ..... .... 25 1.14 Vectorial Field from the Vector Potential.... .... ..... .... 26 1.14.1 First Method ... .... .... .... .... .... ..... .... 26 1.14.2 Second Method . .... .... .... .... .... ..... .... 28 Solved Problems.... .... ..... .... .... .... .... .... ..... .... 29 Problems A.... .... .... ..... .... .... .... .... .... ..... .... 29 Problems B.... .... .... ..... .... .... .... .... .... ..... .... 51 2 Static Electric Field in Vacuum .... .... .... .... .... ..... .... 67 2.1 Electric Charge... ..... .... .... .... .... .... ..... .... 67 2.2 Coulomb’s Law .. ..... .... .... .... .... .... ..... .... 68 2.3 Electric Field .... ..... .... .... .... .... .... ..... .... 70 xi
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