Contents
Preface xii
1 Preliminary Concepts 1
1.1 What is Electromagnetics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Electromagnetic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Fundamentals of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Guided and Unguided Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Electric and Magnetic Fields 17
2.1 What is a Field? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Electric Field Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Electric Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Magnetic Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 Magnetic Field Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 Electromagnetic Properties of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Transmission Lines 30
3.1 Introduction to Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Types of Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Transmission Lines as Two-Port Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Lumped-Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Telegrapher’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6 Wave Equation for a TEM Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7 Characteristic Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.8 Wave Propagation on a TEM Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.9 Lossless and Low-Loss Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.10 Coaxial Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.11 Microstrip Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.12 Voltage Reflection Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.13 Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.14 Standing Wave Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.15 Input Impedance of a Terminated Lossless Transmission Line . . . . . . . . . . . . . . . . . . . 52
3.16 Input Impedance for Open- and Short-Circuit Terminations . . . . . . . . . . . . . . . . . . . . 54
3.17 Applications of Open- and Short-Circuited Transmission Line Stubs . . . . . . . . . . . . . . . 55
3.18 Measurement of Transmission Line Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 56
3.19 Quarter-Wavelength Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.20 Power Flow on Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.21 Impedance Matching: General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.22 Single-Reactance Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.23 Single-Stub Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Vector Analysis 70
4.1 Vector Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
viii CONTENTS
4.3 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.7 Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.8 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.9 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.10 The Laplacian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Electrostatics 93
5.1 Coulomb’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Electric Field Due to Point Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 Charge Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4 Electric Field Due to a Continuous Distribution of Charge . . . . . . . . . . . . . . . . . . . . . 96
5.5 Gauss’ Law: Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.6 Electric Field Due to an Infinite Line Charge using Gauss’ Law . . . . . . . . . . . . . . . . . . 101
5.7 Gauss’ Law: Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.8 Force, Energy, and Potential Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.9 Independence of Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.10 Kirchoff’s Voltage Law for Electrostatics: Integral Form . . . . . . . . . . . . . . . . . . . . . 108
5.11 Kirchoff’s Voltage Law for Electrostatics: Differential Form . . . . . . . . . . . . . . . . . . . 109
5.12 Electric Potential Field Due to Point Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.13 Electric Potential Field due to a Continuous Distribution of Charge . . . . . . . . . . . . . . . . 112
5.14 Electric Field as the Gradient of Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.15 Poisson’s and Laplace’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.16 Potential Field Within a Parallel Plate Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.17 Boundary Conditions on the Electric Field Intensity (E) . . . . . . . . . . . . . . . . . . . . . . 118
5.18 Boundary Conditions on the Electric Flux Density (D) . . . . . . . . . . . . . . . . . . . . . . 120
5.19 Charge and Electric Field for a Perfectly Conducting Region . . . . . . . . . . . . . . . . . . . 122
5.20 Dielectric Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.21 Dielectric Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.22 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.23 The Thin Parallel Plate Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.24 Capacitance of a Coaxial Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.25 Electrostatic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 Steady Current and Conductivity 134
6.1 Convection and Conduction Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2 Current Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.4 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.5 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.6 Power Dissipation in Conducting Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7 Magnetostatics 146
7.1 Comparison of Electrostatics and Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2 Gauss’ Law for Magnetic Fields: Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.3 Gauss’ Law for Magnetism: Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.4 Ampere’s Circuital Law (Magnetostatics): Integral Form . . . . . . . . . . . . . . . . . . . . . 149
7.5 Magnetic Field of an Infinitely-Long Straight Current-Bearing Wire . . . . . . . . . . . . . . . 150
7.6 Magnetic Field Inside a Straight Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.7 Magnetic Field of a Toroidal Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.8 Magnetic Field of an Infinite Current Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.9 Ampere’s Law (Magnetostatics): Differential Form . . . . . . . . . . . . . . . . . . . . . . . . 159
7.10 Boundary Conditions on the Magnetic Flux Density (B) . . . . . . . . . . . . . . . . . . . . . . 160
7.11 Boundary Conditions on the Magnetic Field Intensity (H) . . . . . . . . . . . . . . . . . . . . . 161
7.12 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.13 Inductance of a Straight Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.14 Inductance of a Coaxial Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.15 Magnetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.16 Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
8 Time-Varying Fields 174
8.1 Comparison of Static and Time-Varying Electromagnetics . . . . . . . . . . . . . . . . . . . . 174
8.2 Electromagnetic Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.3 Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.4 Induction in a Motionless Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.5 Transformers: Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.6 Transformers as Two-Port Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.7 The Electric Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.8 The Maxwell-Faraday Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
8.9 Displacement Current and Ampere’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
9 Plane Waves in Lossless Media 194
9.1 Maxwell’s Equations in Differential Phasor Form . . . . . . . . . . . . . . . . . . . . . . . . . 194
9.2 Wave Equations for Source-Free and Lossless Regions . . . . . . . . . . . . . . . . . . . . . . 196
9.3 Types of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9.4 Uniform Plane Waves: Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.5 Uniform Plane Waves: Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.6 Wave Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.7 Wave Power in a Lossless Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
A Constitutive Parameters of Some Common Materials 213
A.1 Permittivity of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.2 Permeability of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
CONTENTS xi
A.3 Conductivity of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
B Mathematical Formulas 217
B.1 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
B.2 Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
B.3 Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
C Physical Constants 220
Index 221
Preface
About This Book
[m0146]
Goals for this book. This book is intended to serve
as a primary textbook for a one-semester introductory
course in undergraduate engineering
electromagnetics, including the following topics:
electric and magnetic fields; electromagnetic
properties of materials; electromagnetic waves; and
devices that operate according to associated
electromagnetic principles including resistors,
capacitors, inductors, transformers, generators, and
transmission lines.
This book employs the “transmission lines first”
approach, in which transmission lines are introduced
using a lumped-element equivalent circuit model for a
differential length of transmission line, leading to
one-dimensional wave equations for voltage and
current.1 This is sufficient to address transmission
line concepts, including characteristic impedance,
input impedance of terminated transmission lines, and
impedance matching techniques. Attention then turns
to electrostatics, magnetostatics, time-varying fields,
and waves, in that order.
What’s new. This version of the book is the second
public release of this book. The first release, known
as “Volume 1 (Beta),” was released in January 2018.
Improvements from the beta version include the
following:
• Correction of errors identified in the beta version
errata and many minor improvements.
• Addition of an index.
1Are you an instructor who is not a fan of the “transmission
lines first” approach? Then see “What are those little numbers
in square brackets?” later in this section.
• Accessibility features: Figures now include “alt
text” suitable for screen reading software.
• Addition of a separate manual of examples and
solutions (see the web site).
• Addition of source files for the book (see the
web site).
Target audience. This book is intended for electrical
engineering students in the third year of a bachelor of
science degree program. It is assumed that readers are
familiar with the fundamentals of electric circuits and
linear systems, which are normally taught in the
second year of the degree program. It is also assumed
that readers have received training in basic
engineering mathematics, including complex
numbers, trigonometry, vectors, partial differential
equations, and multivariate calculus. Review of the
relevant principles is provided at various points in the
text. In a few cases (e.g., phasors, vectors) this review
consists of a separate stand-alone section.
Notation, examples, and highlights. Section 1.7
summarizes the mathematical notation used in this
book. Examples are set apart from the main text as
follows:
Example 0.1. This is an example.
“Highlight boxes” are used to identify key ideas as
follows:
This is a key idea.
What are those little numbers in square brackets?
This book is a product of the
Open Electromagnetics Project. This project provides
a large number of sections (“modules”) which are
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
xiii
assembled (“remixed”) to create new and different
versions of the book. The text “[m0146]” that you see at
the beginning of this section uniquely identifies the
module within the larger set of modules provided by
the project. This identification is provided because
different remixes of this book may exist, each
consisting of a different subset and arrangement of
these modules. Prospective authors can use this
identification as an aid in creating their own remixes.
Why do some sections of this book seem to repeat
material presented in previous sections? In some
remixes of this book, authors might choose to
eliminate or reorder modules. For this reason, the
modules are written to “stand alone” as much as
possible. As a result, there may be some redundancy
between sections that would not be present in a
traditional (non-remixable) textbook. While this may
seem awkward to some at first, there are clear
benefits: In particular, it never hurts to review relevant
past material before tackling a new concept. And,
since the electronic version of this book is being
offered at no cost, there is not much gained by
eliminating this useful redundancy.
Why citeWikipedia pages as additional reading?
Many modules cite Wikipedia entries as sources of
additional information. Wikipedia represents both the
best and worst that the Internet has to offer. Most
authors of traditional textbooks would agree that
citing Wikipedia pages as primary sources is a bad
idea, since quality is variable and content is subject to
change over time. On the other hand, many Wikipedia
pages are excellent, and serve as useful sources of
relevant information that is not strictly within the
scope of the curriculum. Furthermore, students
benefit from seeing the same material presented
differently, in a broader context, and with additional
references cited by Wikipedia pages. We trust
instructors and students to realize the potential pitfalls
of this type of resource and to be alert for problems.
Acknowledgments. Here’s a list of talented and
helpful people who contributed to this book:
The staff of VT Publishing, University Libraries,
Virginia Tech:
Editor: Anita Walz
Advisors: Peter Potter, Corinne Guimont
Cover: Robert Browder, Anita Walz
Other VT contributors:
Assessment: Tiffany Shoop, Anita Walz
Accessibility: Christa Miller
Virginia Tech students:
Alt text writer: Stephanie Edwards
Figure designer: Michaela Goldammer
Figure designer: Kruthika Kikkeri
Figure designer: Youmin Qin
Copyediting:
Melissa Ashman, Kwantlen Polytechnic University
External reviewers:
Samir El-Ghazaly, University of Arkansas
Stephen Gedney, University of Colorado Denver
Randy Haupt, Colorado School of Mines
Karl Warnick, Brigham Young University
Also, thanks are due to the students of the Spring
2018 and Summer 2018 sections of ECE3105 at
Virginia Tech who used the beta version of this book
and provided useful feedback. Thanks also to Justin
Yonker, instructor of the Summer 2018 section.
