Table Of ContentUndergraduate 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
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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
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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
