Table Of Content

Yeon Ho Lee Introduction Engineering to Electro- magnetics 123 Introduction to Engineering Electromagnetics Yeon Ho Lee Introduction to Engineering Electromagnetics ABC Author Prof.YeonHoLee SchoolofInformationandCommunicationEngineering SungkyunkwanUniversity Kyongkido RepublicofKorea ISBN978-3-642-36117-3 e-ISBN978-3-642-36118-0 DOI10.1007/978-3-642-36118-0 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012955857 (cid:2)c Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub- lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect tothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To my parents Preface This is a text book on engineering electromagnetics, designed for an undergraduate course at the sophomore or junior level. The book can be covered in two semesters. The first part begins with vector algebra and coordinate systems covered in Chapter 1, and vector calculus covered in Chapter 2. The two chapters can take about half a semester for a full coverage. Although the instructor may skip some materials in those chapters, the students may use them as references. Chapter 3 discusses electrostatics, and Chapter 4 reviews currents. Chapter 5 deals with magnetostatics. The second part of the book begins with time-varying fields and Maxwell’s equations covered in Chapter 6, and wave motion in general is covered in Chapter 7. In the two chapters, students learn about the interrelationship between time-varying electric and magnetic fields, and the concept of plane waves. Chapter 8 discusses the propagation of electromagnetic waves in material media. Chapter 9 discusses transmission lines, and Chapter 10 is on waveguides. Although electromagnetics is one of the most fundamental subjects in electrical engineering and it attracts many students to the discipline, some students feel that it is not easy to master electromagnetics. Electromagnetics covers a wide range of topics that not only deal with various physical concepts, but also involve many different mathematical concepts, such as vector functions, coordinate systems, integrals, derivatives, complex numbers, and phasors. Confusion arises not from the amount of mathematical theorems and formulas, but from the lack of a thorough knowledge of the mathematical rules and lack of a rigorous application of the rules to electromagnetic problems. Moreover, the confusion becomes worse with a lack of consistency in the notations that are used to denote various physical quantities and constants in electromagnetics. The main objective of the book is to present electromagnetic concepts in a more consistent and rigorous manner. This is achieved through elaborate reasoning and the strict application of mathematical concepts. This does not necessarily mean lengthy mathematical steps. On the contrary, I encourage students to obtain the solutions to electromagnetic problems in an intuitive way by considering the symmetry of configurations and applying the uniqueness theorem. The book contains detailed accounts of the following: 1. Students run into difficulties with the concept of vector fields at the beginning of the class, since they have been familiar only with vectors representing, for example, the force acting on a rigid body. Such vectors are closely related to the displacement of the body. However, a vector in a vector field does not necessarily VIII Preface imply a displacement of an object in space; it is a quantity specific to a point in space, and in most cases, is not allowed to move to another point in space. 2. Cylindrical and spherical coordinate systems are meaningful only if the geometry under consideration has cylindrical or spherical symmetry. When a position vector is expressed as Ra in spherical coordinates, the unit vector in R the radial direction a is treated in different ways: as a constant in the presence of R spherical symmetry, or otherwise as a function of position. Base vectors in those coordinates are generally functions of position, and are therefore differentiable and integrable. 3. Symmetry is an integral part of Gauss’s law and Ampere’s law. The final form of electric flux density or magnetic field intensity of a given problem should be predicted from symmetry configurations so that a Gaussian surface or an Amperian path may be constructed. Typical symmetries in electromagnetics are discussed in detail in the text, including cylindrical, spherical, translational, and two-fold rotational symmetries. Symmetry considerations are useful for intuitively solving boundary value problems and problems of solenoidal and toroidal coils. 4. The inconsistency in notation among different books is a less attractive aspect of electromagnetics. For some authors, the meaning of the notation V is the 12 potential difference between point 1 and point 2 (or V −V ), yet for others, it 1 2 signifies the work done in moving a unit charge from point 1 to point 2(or V −V ). This is very confusing for expressing the electric force acting on a 2 1 charge q due to a charge q as F along the direction of a unit vector a 2 1 21 12 pointing from q to q such that F = F a . This book adopts a new notation 1 2 21 21 12 to avoid the confusion. In our notation, the potential difference is denoted as V =V −V , in which the subscript 1-2 mimics the subtraction on the right-hand 1−2 1 2 side, while the hyphen implies a sense of backward direction, such as “from 2 to 1,” or the effect at point 1 due to a cause at point 2. Accordingly, the electric force on q due to q that is in the direction of a unit vector pointing from q to q is 1 2 2 1 expressed as F = F a in our notation. Subscript 12 always represents 1−2 1−2 1−2 something “from 1 to 2.” For example, Ψ represents the magnetic flux through 12 loop 2 due to the current in loop 1. 5. An electromagnetic quantity may take on different forms. Static field quantities are denoted by a boldface letter, such as E for a static electric field, while time- varying fields are denoted by a script letter, such as E for a time-varying electric field. Scalars are denoted by a regular letter, such as E for the magnitude of electric field intensity. Complex quantities are denoted by a caret on top, such as Eˆ for the complex amplitude of electric field intensity. Since an electric field o phasor is independent of time, it is also denoted as E. Preface IX The book contains 300 figures in which real data are drawn to scale; many figures provide three-dimensional views. Each subsection includes a number of examples that are elaborately worked out by putting aforementioned concepts and relations into use and illustrating rigorous approaches in steps. Each subsection ends with exercises and answers that can be solved in a few simple steps and used to check if students correctly understood the concepts. At the end of each section, several review questions are provided to point out key concepts and relations discussed in the section. Since it has been found that open-ended questions are simply ignored by many students, the review questions are given with hints referring to related equations and figures. The book contains a total of 280 end-of-chapter problems. I would like to thank the professors and students who provided valuable comments and suggestions, and corrected errors in the examples and exercises. I also wish to thank my wife, Hyunjoo, for her patience, inspiration, and confidence in me in the course of writing this book. Contents 1 Vector Algebra and Coordinate Systems ........................................................ 1 1.1 Vectors and Vector Field............................................................................. 2 1.2 Vector Algebra ............................................................................................ 4 1.2.1 Vector Addition and Subtraction ...................................................... 4 1.2.2 Vector Scaling .................................................................................. 6 1.2.3 Scalar or Dot Product ....................................................................... 7 1.2.4 Vector or Cross Product ................................................................. 11 1.2.5 Scalar and Vector Triple Products .................................................. 14 1.3 Orthogonal Coordinate Systems ................................................................ 16 1.3.1 Cartesian Coordinate System ......................................................... 17 1.3.2 Cylindrical Coordinate System ....................................................... 27 1.3.3 Spherical Coordinate System ......................................................... 36 1.4 Coordinate Transformation ....................................................................... 45 1.4.1 Cartesian-Cartesian Transformation ............................................... 45 1.4.2 Cylindrical-Cartesian Transformation ............................................ 48 1.4.3 Spherical-Cartesian Transformation ............................................... 50 2 Vector Calculus ............................................................................................... 61 2.1 Line and Surface Integrals ......................................................................... 62 2.1.1 Curves ............................................................................................ 62 2.1.2 Line Integral ................................................................................... 65 2.1.3 Surface Integral .............................................................................. 71 2.2 Directional Derivative and Gradient ......................................................... 74 2.3 Flux and Flux Density ............................................................................... 82 2.4 Divergence and Divergence Theorem ....................................................... 84 2.4.1 Divergence of a Flux Density ......................................................... 85 2.4.2 Divergence Theorem ...................................................................... 89 2.5 Curl and Stokes’s Theorem ....................................................................... 94 2.5.1 Curl of a Vector Field ..................................................................... 94 2.5.2 Stokes’s Theorem ......................................................................... 101 2.6 Dual Operations of ∇ .............................................................................. 104 2.7 Helmholtz’s Theorem .............................................................................. 107 3 Electrostatics ................................................................................................. 117 3.1 Coulomb’s Law ....................................................................................... 118 3.2 Electric Field Intensity ............................................................................ 122 3.2.1 Electric Field due to Discrete Charges ......................................... 123 XII Contents 3.2.2 Electric Field due to a Continuous Charge Distribution ............... 126 3.3 Electric Flux Density and Gauss’s Law .................................................. 133 3.3.1 Electric Flux Density .................................................................... 133 3.3.2 Gauss’s Law ................................................................................. 136 3.4 Electric Potential ..................................................................................... 144 3.4.1 Work Done in Moving a Charge .................................................. 144 3.4.2 Electric Potential due to a Charge Distribution ............................ 145 3.4.3 Conservative Field ........................................................................ 150 3.4.4 E as the Negative Gradient of V ................................................... 152 3.5 Dielectric in a Static Electric Field .......................................................... 156 3.5.1 Electric Polarization ..................................................................... 157 3.5.2 Dielectric Constant ....................................................................... 160 3.5.3 Boundary Conditions at a Dielectric Interface ............................. 164 3.6 Perfect Conductor in a Static Electric Field ............................................ 168 3.7 Electrostatic Potential Energy ................................................................. 172 3.8 Electrostatic Boundary Value Problems .................................................. 176 3.8.1 Poisson’s and Laplace’s Equations .............................................. 176 3.8.2 Uniqueness Theorem .................................................................... 178 3.8.3 Examples of Boundary Values Problems ..................................... 180 3.8.4 Method of Images ......................................................................... 185 3.9 Capacitance and Capacitors .................................................................... 191 3.9.1 Parallel-Plate Capacitor ................................................................ 193 3.9.2 Examples of Capacitors ................................................................ 194 4 Steady Electric Current ............................................................................... 211 4.1 Convection Current ................................................................................. 212 4.2 Conduction Current and Ohm’s Law ...................................................... 215 4.3 Resistance ................................................................................................ 218 4.4 Equation of Continuity ............................................................................ 221 4.4.1 Relaxation Time Constant ............................................................ 223 4.5 Power Dissipation and Joules’s Law ....................................................... 225 4.6 Steady Currents at an Interface ............................................................... 227 4.7 Analogy between D and J ....................................................................... 230 5 Magnetostatics .............................................................................................. 237 5.1 Lorentz Force Equation ........................................................................... 238 5.2 The Biot-Savart Law ............................................................................... 240 5.3 Ampere’s Circuital Law .......................................................................... 247 5.4 Magnetic Flux Density ............................................................................ 253 5.5 Vector Magnetic Potential ....................................................................... 257 5.5.1 Ampere’s Circuital Law from the Biot-Savart Law ..................... 259 5.6 The Magnetic Dipole ............................................................................... 265 5.7 Magnetic Materials .................................................................................. 269 5.7.1 Magnetization and Equivalent Current Densities ......................... 270 5.7.2 Permeability ................................................................................. 274 5.7.3 Hysteresis of a Ferromagnetic Material........................................ 281