Electromagnetics Made Easy

660 Pages·2020·10.152 MB·English

by  S. Balaji

S. Balaji
Electromagnetics
Made Easy
Electromagnetics Made Easy
S. Balaji
Electromagnetics Made Easy
123
S. Balaji
Indira Gandhi Centrefor Atomic Research
Kalpakkam,Tamil Nadu, India
ISBN978-981-15-2657-2 ISBN978-981-15-2658-9 (eBook)
https://doi.org/10.1007/978-981-15-2658-9
©SpringerNatureSingaporePteLtd.2020
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
authors or the editors give a warranty, expressed or implied, with respect to the material contained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard
tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd.
The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,
Singapore
Preface
Engineering electromagnetics is an interesting subject in the electrical/electronics
engineeringcurriculum.However,studentsfinditdifficulttograsptheconceptsand
usually complain the paper is tedious. The common conception about electro-
magnetics is that the subject is hard to understand and is not as beautiful as other
papers. As an example, mechanics appears to be friendlier to students. Concrete
concepts and dealing with tangible objects make the learning of mechanics simple
forstudents.Studiesonprojectilethrowninspace,anobjectslidingonafloor,can
beclearlyvisualized.Onthecontrary,electromagneticsischaracterizedbyabstract
concepts and intangible fields. Along with complex mathematics, imperceptible
concepts and invisible fields confuse the students, and understanding electromag-
netics becomes a dream for the reader.
An attempt has been made in this book to change the above scenario. Real-life
examples have been used throughout the book for easy grasping of the abstract
concepts. This book is a result of a decade of teaching electromagnetics for
electrical/electronicsengineeringstudentsandphysicspostgraduates.Motivatedby
the positive feedback from students for the simple and lucid form of presentation
ofthesubject,aneedwasfelttoturnthepresentationintheformofatextbookfora
wider audience.
The book contains ten chapters. The backbone of electromagnetics is vector
analysis, and to make sure the reader becomes more familiar with vector calculus,
Chap. 1 has been devoted to the study of vectors. Various theorems in vector
calculus and coordinate systems, which are the fundamental concepts required to
understand electromagnetics, are discussed in detail in the first chapter.
Chapters 2–10 can be grouped under four major sections—electrostatics, mag-
netostatics,time-varyingfieldsandapplicationsofelectromagnetics.Chapters2and
3focusonelectrostatics.Chapter2commenceswithanintroductiontoCoulomb’s
lawandproceedstocalculateelectricfieldforvariouschargedistributions.Chapter
3 continues with the calculation of electric field and ends up with a detailed dis-
cussion on dielectrics and capacitors. Chapters 4 and 5 concentrate on magneto-
statics. Chapter 4introducesBiot–Savartlawandcontinues with thecalculationof
magnetic flux density using various methods, and the methods are compared with
v
vi Preface
their electrostatic counterpart. Chapter 5 elaborates about magnetization and mag-
netic materials. Chapter 6 introduces time-varying fields by discussing Faraday’s
lawandinduction,followedbyadetailedaccountonMaxwell’sequation.Thelast
sections of Chap. 6 describe the gauge transformations, retarded potentials and
time-harmonic fields. Chapter 7 gives a detailed account on the flow of energy in
electromagnetic wave—the Poynting theorem, wave polarization and reflections
andtransmissionofelectromagneticwaves.Chapters8–10describetheapplication
part of electromagnetics. The theory so developed in Chaps. 1–7 is applied in the
development of transmission lines, waveguides and antennas in Chaps. 8–10.
Sequentialdevelopmentofthesubjectandutilizingconcreteexamplestoexplain
abstracttheoryarethedistinctfeaturesofthebook.Thereareanumberoftopicsin
thisbookmakingituniqueinthepresentationoftheconcepts.Fewsuchtopicsare
mentioned below:
1. Real-life examples have been used to explain
(a) that “Vectors are difficult to deal with as compared to scalars” in Sect. 1.1
(b) Flux in Sect. 1.12
(c) Existence of source and sink as related to the divergence of a vector in
Sect. 1.14
(d) Non-existence of magnetic monopoles in Sect. 4.12
(e) Deficiency as related to magnetic scalar potential in the calculation of B in
Sect. 4.15.
2. Insphericalpolarcoordinates,hisallowedtovaryfrom0topandnotupto2p.
ThereasonforsuchavariationisexplainedwithsuitablefiguresinSect.1.19a.
3. For a given charge distribution all the three methods (Coulomb’s law, Gauss’s
law and potential formulation) have been utilized to calculate the electric field.
4. In Sect. 2.18, the three methods (Coulomb’s law, Gauss’s law and potential
formulation) utilized for the calculation of the electric field for a given charge
distribution have been compared.
5. A detailed discussion on reference point R in the calculation of potential is
elaborated in Sect. 2.22.
Much deeper explanations are put forward for a number of concepts, which
makes the subject easy and understandable. Here are few examples.
When the instructor states “Vectors are difficult to deal with as compared to
scalars,”atoncestudentsquestion—Whyisitso?Section1.1ofthisbookexplains
why.
Whenthestudentsask—whymagneticscalarpotentialV cannotbeusedinthe
m
calculation, the reply given will be V is a multi-valued function. At once the
m
question to shoot up in the students’ mind is—let V be a multi-valued function,
m
thenhowdoesitaffectthecalculation?Howthemulti-valuednessofV affectsthe
m
calculation is explained in Sect. 4.15.
Preface vii
Likewise, there are a number of questions in the students’ mind for which they
get partial answers which do not satisfy the students’ quest to learn, leading to
students not understanding the subject, and finally feel that electromagnetics is
difficultandcanneverbeunderstood.So,asmuchaspossibleefforthasbeenmade
inthisbooktomakesurethatallthequestionsareansweredandstudentscangrasp
the subject easily.
Understanding the concepts is a different skill, while applying the concepts to
solve the problems is a different skill. To make sure that the students apply
whatever they have learnt, in solving problems, a number of examples have been
includedthroughoutthetext.Alistofexercisesisgivenattheendofeachchapter,
anditisemphasizedthatthestudentsshouldattemptthoseexercisesontheirownto
further strengthen their problem-solving skill.
Finally, I should thank my wife E. Sumathi and my daughter B. S. Nakshathra
for their extreme patience they had during the development of this book.
Kalpakkam, India S. Balaji
Contents
1 Vector Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Graphical Representation of Vectors . . . . . . . . . . . . . . . . . . . . 6
1.3 Symbolic Representation of Vectors. . . . . . . . . . . . . . . . . . . . . 6
1.4 Vector Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Subtraction of Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Multiplication of a Vector by a Scalar . . . . . . . . . . . . . . . . . . . 8
1.7 Multiplication of Vectors: Dot Product of Two Vectors . . . . . . 8
1.8 Multiplication of Vectors—Cross—Product of Two
Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.9 Vector Components and Unit Vectors . . . . . . . . . . . . . . . . . . . 11
1.10 Triple Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.11 Line, Surface and Volume Integration . . . . . . . . . . . . . . . . . . . 20
1.12 Flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.13 Vector Differentiation: Gradient of a Scalar Function . . . . . . . . 24
1.14 Vector Differentiation: Divergence of a Vector. . . . . . . . . . . . . 27
1.15 Vector Differentiation: Curl of a Vector . . . . . . . . . . . . . . . . . . 32
1.16 Divergence Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.17 Stoke’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.18 The Gradient Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.19 Others Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.19.1 Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . . 54
1.19.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . 59
1.20 Important Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
1.21 Two and Three Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2 Electric Charges at Rest: Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.1 Coulomb’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.2 Electric Field Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
ix
x Contents
2.3 Electric Field Intensity Due to a Group of Discrete Point
Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.4 Continuous Charge Distributions . . . . . . . . . . . . . . . . . . . . . . . 75
2.5 A Note about Coulomb’s Law. . . . . . . . . . . . . . . . . . . . . . . . . 77
2.6 Calculating Electric Field E Using Coulomb’s Law . . . . . . . . . 78
2.7 Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.8 Gauss’s Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.9 Sketches of Field Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.10 Curl of E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.11 Potential of Discrete and Continuous Charge Distributions . . . . 101
2.12 Calculating Electric Field Using Gauss’s Law and Potential . . . 105
2.13 Electric Field Due to an Infinite Line Charge . . . . . . . . . . . . . . 107
2.14 Electric Field Due to the Finite Line Charge . . . . . . . . . . . . . . 113
2.15 ElectricFieldAlongtheAxisofaUniformlyChargedCircular
Disc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
2.16 Electric Field Due to an Infinite Plane Sheet of Charge. . . . . . . 121
2.17 Electric Field of a Uniformly Charged Spherical Shell . . . . . . . 128
2.18 Comparison of Coulomb’s Law, Gauss’s Law and Potential
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
2.19 Electric Field of a Dipole . . . . . . .R. . . . . . . . . . . . . . . . . . . . . 141
2.20 Calculation of Potential Using V¼ E(cid:2)dl . . . . . . . . . . . . . . . 146
2.21 The Conservative Nature of Electric Field . . . . . . . . . . . . . . . . 151
RP
2.22 The Reference Point R in the Equation V¼(cid:3) E(cid:2)dl . . . . . . .
153
R
2.23 Poisson’s and Laplace’s Equation . . . . . . . . . . . . . . . . . . . . . . 161
2.24 Conductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
2.25 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
2.26 Uniqueness Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
3 Electric Charges at Rest—Part II . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.1 Work Done . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.2 Energy in Electrostatic Fields . . . . . . . . . . . . . . . . . . . . . . . . . 183
3.3 Equipotential Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
3.4 A Note on Work Done . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
3.5 Method of Images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
3.6 Point Charge Near a Grounded Conducting Sphere. . . . . . . . . . 202
3.7 Laplace’s Equation—Separation of Variables . . . . . . . . . . . . . . 206
3.8 Separation of Variables Laplace’s Equation in Cartesian
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
3.9 Potential Between Two Grounded Semi Infinite Parallel
Electrodes Separated by a Plane Electrode Held
by a Potential V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
o
Contents xi
3.10 Potential Between Two Grounded Conducting Electrodes
Separated by Two Conducting Side Plates Maintained at V
o
Potentials V and V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
o o
3.11 Separation of Variables—Laplace’s Equation in Spherical
Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
3.12 Separation of Variables—Laplace’s Equation in Cylindrical
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
3.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
3.14 Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
3.15 Dielectric in an Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . 234
3.16 Polar and Non-Polar Molecules . . . . . . . . . . . . . . . . . . . . . . . . 234
3.17 Potential Produced by the Polarized Dielectric . . . . . . . . . . . . . 236
3.18 Bound Charges rpandqp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
3.19 Electric Displacement Vector and Gauss Law
in Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
3.20 Linear Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
3.21 Dielectric Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
3.22 Boundary Conditions in the Presence of Dielectrics . . . . . . . . . 243
3.23 Capacitance and Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
3.24 Principle of a Capacitor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
3.24.1 Capacity of a Parallel Plate Capacitor . . . . . . . . . . . . . 254
3.24.2 Capacity of a Parallel Plate Capacitor with Two
Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
3.25 Capacitance of a Spherical Capacitor . . . . . . . . . . . . . . . . . . . . 257
3.26 Capacitance of a Cylindrical Capacitor. . . . . . . . . . . . . . . . . . . 258
3.27 Capacitors in Parallel and Series . . . . . . . . . . . . . . . . . . . . . . . 260
3.28 Energy Stored in a Capacitor. . . . . . . . . . . . . . . . . . . . . . . . . . 263
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
4 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
4.2 Lorentz Force Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
4.3 Applications of Lorentz Force—Hall Effect . . . . . . . . . . . . . . . 277
4.4 Sources of Magnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
4.5 Magnetic Force Between Two Current Elements. . . . . . . . . . . . 280
4.6 Biot–Savart Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
4.7 Current Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
4.8 Magnetic Flux Density Due to a Steady Current
in a Infinitely Long Straight Wire . . . . . . . . . . . . . . . . . . . . . . 287
4.9 Ampere’s Circuital Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
4.10 Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
4.11 The Divergence of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
4.12 Magnetic Monopoles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299