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ELECTROMAGNETICS
STEVEN W. ELLINGSON
VOLUME 2
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ELECTROMAGNETICS
VOLUME 2
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Publication of this book was made possible in part by the Virginia Tech
University Libraries’ Open Education Initiative Faculty Grant program:
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Books in this series
Electromagnetics, V olume 1, https://doi.org/10.21061/electromagnetics-vol-1
Electromagnetics, V olume 2, https://doi.org/10.21061/electromagnetics-vol-2
/T_he Open Electromagnetics Project, https://www.faculty.ece.vt.edu/swe/oem
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ELECTROMAGNETICS
STEVEN W. ELLINGSON
VOLUME 2
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Copyright © 2020 Steven W. Ellingson
iv
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and includes alternative text which allows for machine-readability. The LaTeX source files also include alternative text for
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Publication Cataloging Information
Ellingson, Steven W., author
Electromagnetics (V olume 2) / Steven W. Ellingson
Pages cm
ISBN 978-1-949373-91-2 (print)
ISBN 978-1-949373-92-9 (ebook)
DOI: https://doi.org/10.21061/electromagnetics-vol-2
1. Electromagnetism. 2. Electromagnetic theory.
I. Title
QC760.E445 2020
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Contents
Pr
eface ix
1 Preliminary Concepts 1
1.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Electromagnetic Field Theory: A Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Magnetostatics Redux 11
2.1 Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Magnetic Force on a Current-Carrying Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 T orque Induced by a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 The Biot-Savart Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
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2.4 The Biot-Savart Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Force, Energy , and Potential Difference in a Magnetic Field . . . . . . . . . . . . . . . . . . . 20
3 W ave Propagation in General Media 25
3.1 Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Poynting V ector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 W ave Equations for Lossy Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Complex Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Loss T angent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Plane W aves in Lossy Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.7 W ave Power in a Lossy Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
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3.7 W ave Power in a Lossy Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.8 Decibel Scale for Power Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.9 Attenuation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.10 Poor Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.11 Good Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.12 Skin Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Current Flow in Imperfect Conductors 48
4.1 AC Current Flow in a Good Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Impedance of a Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Surface Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
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4.3 Surface Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 W ave Reflection and T ransmission 56
5.1 Plane W aves at Normal Incidence on a Planar Boundary . . . . . . . . . . . . . . . . . . . . . 56
5.2 Plane W aves at Normal Incidence on a Material Slab . . . . . . . . . . . . . . . . . . . . . . 60
5.3 T otal Transmission Through a Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4 Propagation of a Uniform Plane W ave in an Arbitrary Direction . . . . . . . . . . . . . . . . . 67
vi
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CONTENTS vii
5.5 Decomposition of a W ave into TE and TM Components . . . . . . . . . . . . . . . . . . . . . 70
5.6 Plane W aves at Oblique Incidence on a Planar Boundary: TE Case . . . . . . . . . . . . . . . 72
5.7 Plane W aves at Oblique Incidence on a Planar Boundary: TM Case . . . . . . . . . . . . . . . 76
5.8 Angles of Reflection and Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.9 TE Reflection in Non-magnetic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.10 TM Reflection in Non-magnetic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.11 T otal Internal Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.12 Evanescent W aves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6 W aveguides 95
6.1 Phase and Group V elocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
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6 W aveguides 95
6.1 Phase and Group V elocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 Parallel Plate W aveguide: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3 Parallel Plate W aveguide: TE Case, Electric Field . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4 Parallel Plate W aveguide: TE Case, Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . 102
6.5 Parallel Plate W aveguide: TM Case, Electric Field . . . . . . . . . . . . . . . . . . . . . . . . 104
6.6 Parallel Plate W aveguide: The TM 0 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.7 General Relationships for Unidirectional W aves . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.8 Rectangular W aveguide: TM Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.9 Rectangular W aveguide: TE Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
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6.9 Rectangular W aveguide: TE Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.10 Rectangular W aveguide: Propagation Characteristics . . . . . . . . . . . . . . . . . . . . . . 117
7 T ransmission Lines Redux 121
7.1 Parallel Wire Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2 Microstrip Line Redux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3 Attenuation in Coaxial Cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.4 Power Handling Capability of Coaxial Cable . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.5 Why 50 Ohms? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8 Optical Fiber 138
8.1 Optical Fiber: Method of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
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8 Optical Fiber 138
8.1 Optical Fiber: Method of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2 Acceptance Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.3 Dispersion in Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9 Radiation 145
9.1 Radiation from a Current Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2 Magnetic V ector Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.3 Solution of the W ave Equation for Magnetic V ector Potential . . . . . . . . . . . . . . . . . . 150
9.4 Radiation from a Hertzian Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.5 Radiation from an Electrically-Short Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
9.6 Far-Field Radiation from a Thin Straight Filament of Current . . . . . . . . . . . . . . . . . . 159
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9.6 Far-Field Radiation from a Thin Straight Filament of Current . . . . . . . . . . . . . . . . . . 159
9.7 Far-Field Radiation from a Half-W ave Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.8 Radiation from Surface and V olume Distributions of Current . . . . . . . . . . . . . . . . . . 162
10 Antennas 166
10.1 How Antennas Radiate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.2 Power Radiated by an Electrically-Short Dipole . . . . . . . . . . . . . . . . . . . . . . . . . 168
10.3 Power Dissipated by an Electrically-Short Dipole . . . . . . . . . . . . . . . . . . . . . . . . 169
10.4 Reactance of the Electrically-Short Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
10.5 Equivalent Circuit Model for Transmission; Radiation Efficiency . . . . . . . . . . . . . . . . 173
10.6 Impedance of the Electrically-Short Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
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viii CONTENTS
10.7 Directivity and Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.8 Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
10.9 Equivalent Circuit Model for Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
10.10 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.11 Potential Induced in a Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
10.12 Equivalent Circuit Model for Reception, Redux . . . . . . . . . . . . . . . . . . . . . . . . . 194
10.13 Effective Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
10.14 Friis Transmission Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
A Constitutive Parameters of Some Common Materials 205
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A Constitutive Parameters of Some Common Materials 205
A.1 Permittivity of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
A.2 Permeability of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
A.3 Conductivity of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
B Mathematical Formulas 209
B.1 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.2 V ector Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.3 V ector Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
C Physical Constants 212
Index 213
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Preface
About
This Book
[m0213]
Goals for this book. This book is intended to serve
as a primary textbook for the second semester of a
two-semester course in undergraduate engineering
electromagnetics. The presumed textbook for the first
semester is Electromagnetics V ol. 1, 1 which addresses
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. The
book you are now reading – Electromagnetics V ol. 2 –
addresses the following topics:
• Chapter 1 (“Preliminary Concepts”) provides a
brief summary of conventions for units, notation,
and coordinate systems, and a synopsis of
electromagnetic field theory from V ol. 1.
• Chapter 2 (“Magnetostatics Redux”) extends the
coverage of magnetostatics in V ol. 1 to include
magnetic forces, rudimentary motors, and the
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• Chapter 2 (“Magnetostatics Redux”) extends the
coverage of magnetostatics in V ol. 1 to include
magnetic forces, rudimentary motors, and the
Biot-Savart law .
• Chapter 3 (“W ave Propagation in General
Media”) addresses Poynting’s theorem, theory of
wave propagation in lossy media, and properties
of imperfect conductors.
• Chapter 4 (“Current Flow in Imperfect
Conductors”) addresses the frequency-dependent
distribution of current in wire conductors and
subsequently the AC impedance of wires.
1 S.W . Ellingson, Electromagnetics V ol. 1, VT Publishing, 2018.
CC BY -SA 4.0. ISBN 9780997920185.
https://doi.org/10.21061/electromagnetics- vol- 1
• Chapter 5 (“W ave Reflection and Transmission”)
addresses scattering of plane waves from planar
interfaces.
• Chapter 6 (“W aveguides”) provides an
introduction to waveguide theory via the parallel
plate and rectangular waveguides.
• Chapter 7 (“Transmission Lines Redux”) extends
the coverage of transmission lines in V ol. 1 to
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plate and rectangular waveguides.
• Chapter 7 (“Transmission Lines Redux”) extends
the coverage of transmission lines in V ol. 1 to
include parallel wire lines, the theory of
microstrip lines, attenuation, and power-handling
capabilities. The inevitable but hard-to-answer
question “What’s so special about 50 Ω?” is
addressed at the end of this chapter.
• Chapter 8 (“Optical Fiber”) provides an
introduction to multimode fiber optics, including
the concepts of acceptance angle and modal
dispersion.
• Chapter 9 (“Radiation”) provides a derivation of
the electromagnetic fields radiated by a current
distribution, emphasizing the analysis of line
distributions using the Hertzian dipole as a
differential element.
• Chapter 10 (“ Antennas”) provides an
introduction to antennas, emphasizing equivalent
circuit models for transmission and reception
and characterization in terms of directivity and
pattern. This chapter concludes with the Friis
transmission equation.
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and characterization in terms of directivity and
pattern. This chapter concludes with the Friis
transmission equation.
Appendices covering material properties,
mathematical formulas, and physical constants are
repeated from V ol. 1, with a few additional items.
T arget audience. This book is intended for electrical
engineering students in the third year of a bachelor of
science degree program. It is assumed that students
have successfully completed one semester of
Electromagnetics V ol. 2. c⃝ 2020 S.W . Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics- vol- 2
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x PREF ACE
engineering electromagnetics, nominally using V ol. 1.
However, the particular topics and sequence of topics
in V ol. 1 are not an essential prerequisite, and in any
event this book may be useful as a supplementary
reference when a different textbook is used. 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 are
proficient in basic engineering mathematics, including
complex numbers, trigonometry , vectors, partial
differential equations, and multivariate calculus.
Notation, examples, and highlights. Section 1.2
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
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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
assembled (“remixed”) to create new and different
versions of the book. The text “ [m0213]” 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
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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 cite Wikipedia 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
educators 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
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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 the additional references available as links
from Wikipedia pages. W e 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 V irginia T ech Publishing, University
Libraries, V irginia T ech:
Acquisitions/Developmental Editor & Project
Manager: Anita W alz
Advisors: Peter Potter, Corinne Guimont
Cover, Print Production: Robert Browder
Other V irginia T ech contributors:
Accessibility: Christa Miller, Corinne Guimont,
Sarah Mease
Assessment: Anita W alz
V irginia T ech Students:
Alt text writer: Michel Comer
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Accessibility: Christa Miller, Corinne Guimont,
Sarah Mease
Assessment: Anita W alz
V irginia T ech Students:
Alt text writer: Michel Comer
Figure designers: Kruthika Kikkeri, Sam Lally ,
Chenhao W ang
Copyediting:
Longleaf Press
External reviewers:
Randy Haupt, Colorado School of Mines
Karl W arnick, Brigham Y oung University
Anonymous faculty member, research university
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xi
Also,
thanks are due to the students of the Fall 2019
section of ECE3106 at V irginia T ech who used the
beta version of this book and provided useful
feedback.
Finally , we acknowledge all those who have
contributed their art to Wikimedia Commons
(https://commons.wikimedia.org/) under open
licenses, allowing their work to appear as figures in
this book. These contributors are acknowledged in
figures and in the “Image Credits” section at the end
of each chapter. Thanks to each of you for your
selfless effort.
About the Open Electromagnetics
Project
[m0148]
The Open Electromagnetics Project
(https://www
.faculty .ece.vt.edu/swe/oem/) was
established at V irginia T ech in 2017 with the goal of
creating no-cost openly-licensed textbooks for
courses in undergraduate engineering
electromagnetics. While a number of very fine
traditional textbooks are available on this topic, we
feel that it has become unreasonable to insist that
students pay hundreds of dollars per book when
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feel that it has become unreasonable to insist that
students pay hundreds of dollars per book when
effective alternatives can be provided using modern
media at little or no cost to the student. This project is
equally motivated by the desire for the freedom to
adopt, modify , and improve educational resources.
This work is distributed under a Creative Commons
BY SA license which allows – and we hope
encourages – others to adopt, modify , improve, and
expand the scope of our work.
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xii PREF ACE
About the Author
[m0153]
Steven W . Ellingson ([email protected]) is an
Associate
Professor at V irginia T ech in Blacksburg,
V irginia, in the United States. He received PhD and
MS degrees in Electrical Engineering from the Ohio
State University and a BS in Electrical and Computer
Engineering from Clarkson University . He was
employed by the U.S. Army , Booz-Allen & Hamilton,
Raytheon, and the Ohio State University
ElectroScience Laboratory before joining the faculty
of V irginia T ech, where he teaches courses in
electromagnetics, radio frequency electronics,
wireless communications, and signal processing. His
research includes topics in wireless communications,
radio science, and radio frequency instrumentation.
Professor Ellingson serves as a consultant to industry
and government and is the author of Radio Systems
Engineering (Cambridge University Press, 2016) and
Electromagnetics V ol. 1 (VT Publishing, 2018).
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Chapter 1
Pr
eliminary Concepts
1.1 Units
[m0072]
The term “unit” refers to the measure used to express
a ph
ysical quantity . For example, the mean radius of
the Earth is about 6,371,000 meters; in this case, the
unit is the meter.
A number like “6,371,000” becomes a bit
cumbersome to write, so it is common to use a prefix
to modify the unit. For example, the radius of the
Earth is more commonly said to be 6371 kilometers,
where one kilometer is understood to mean
1000 meters. It is common practice to use prefixes,
such as “kilo-, ” that yield values in the range of 0.001
to 10,000. A list of standard prefixes appears in
T able 1.1.
Prefix Abbreviation Multiply by:
exa E 1018
peta
P 1015
tera T 1012
giga G 109
mega M 106
kilo k 103
milli m 10−3
micro µ 10−6
nano n 10−9
pico p 10−12
femto f 10−15
atto a 10−18
T able 1.1: Prefixes used to modify units.
Unit Abbreviation Quantifies:
ampere A electric current
coulomb
C electric charge
farad F capacitance
henry H inductance
hertz Hz frequency
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Unit Abbreviation Quantifies:
ampere A electric current
coulomb
C electric charge
farad F capacitance
henry H inductance
hertz Hz frequency
joule J energy
meter m distance
newton N force
ohm Ω resistance
second s time
tesla T magnetic flux density
volt V electric potential
watt W power
weber Wb magnetic flux
T able 1.2: Some units that are commonly used in elec-
tromagnetics.
Writing out the names of units can also become
tedious. For this reason, it is common to use standard
abbreviations; e.g., “6731 km” as opposed to
“6371 kilometers, ” where “k” is the standard
abbreviation for the prefix “kilo” and “m” is the
standard abbreviation for “meter. ” A list of
commonly-used base units and their abbreviations are
shown in T able 1.2.
T o avoid ambiguity , it is important to always indicate
the units of a quantity; e.g., writing “6371 km” as
opposed to “6371. ” Failure to do so is a common
source of error and misunderstandings. An example is
the expression:
l= 3t
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opposed to “6371. ” Failure to do so is a common
source of error and misunderstandings. An example is
the expression:
l= 3t
where lis length and tis time. It could be that lis in
Electromagnetics V ol. 2. c⃝ 2020 S.W . Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics- vol- 2
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2 CHAPTER 1. PRELIMINAR Y CONCEPTS
meters and tis in seconds, in which case “3” really
means “3 m/s. ” However, if it is intended that lis in
kilometers and tis in hours, then “3” really means
“3 km/h, ” and the equation is literally different. T o
patch this up, one might write “l = 3t m/s”; however,
note that this does not resolve the ambiguity we just
identified – i.e., we still don’t know the units of the
constant “3. ” Alternatively , one might write “l = 3t
where lis in meters and tis in seconds, ” which is
unambiguous but becomes quite awkward for more
complicated expressions. A better solution is to write
instead:
l= (3 m/s) t
or even better:
l= at where a= 3 m/s
since this separates the issue of units from the perhaps
more-important fact that lis proportional to tand the
constant of proportionality (a) is known.
The meter is the fundamental unit of length in the
International System of Units, known by its French
acronym “SI” and sometimes informally referred to
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International System of Units, known by its French
acronym “SI” and sometimes informally referred to
as the “metric system. ”
In this work, we will use SI units exclusively .
Although
SI is probably the most popular for
engineering use overall, other systems remain in
common use. For example, the English system, where
the radius of the Earth might alternatively be said to
be about 3959 miles, continues to be used in various
applications and to a lesser or greater extent in
various regions of the world. An alternative system in
common use in physics and material science
applications is the CGS (“centimeter-gram-second”)
system. The CGS system is similar to SI, but with
some significant differences. For example, the base
unit of energy in the CGS system is not the “joule”
but rather the “erg, ” and the values of some physical
constants become unitless. Therefore – once again –
it is very important to include units whenever values
are stated.
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constants become unitless. Therefore – once again –
it is very important to include units whenever values
are stated.
SI defines seven fundamental units from which all
other units can be derived. These fundamental units
are distance in meters (m), time in seconds (s),
current in amperes (A), mass in kilograms (kg),
temperature in kelvin (K), particle count in moles
(mol), and luminosity in candela (cd). SI units for
electromagnetic quantities such as coulombs (C) for
charge and volts (V) for electric potential are derived
from these fundamental units.
A frequently-overlooked feature of units is their
ability to assist in error-checking mathematical
expressions. For example, the electric field intensity
may be specified in volts per meter (V/m), so an
expression for the electric field intensity that yields
units of V/m is said to be “dimensionally correct” (but
not necessarily correct), whereas an expression that
cannot be reduced to units of V/m cannot be correct.
Additional Reading:
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not necessarily correct), whereas an expression that
cannot be reduced to units of V/m cannot be correct.
Additional Reading:
• “International System of Units” on Wikipedia.
• “Centimetre-gram-second system of units” on
Wikipedia.
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1.2. NOT A TION 3
1.2 Notation
[m0005]
The list below describes notation used in this book.
• V ector
s: Boldface is used to indicate a vector;
e.g., the electric field intensity vector will
typically appear as E. Quantities not in boldface
are scalars. When writing by hand, it is common
to write “ E” or “ − →E” in lieu of “E. ”
• Unit vectors: A circumflex is used to indicate a
unit vector; i.e., a vector having magnitude equal
to one. For example, the unit vector pointing in
the +xdirection will be indicated as ˆx. In
discussion, the quantity “ ˆx” is typically spoken
“x hat. ”
• Time: The symbol tis used to indicate time.
• P osition: The symbols (x,y,z ), (ρ,φ,z ), and
(r,θ,φ ) indicate positions using the Cartesian,
cylindrical, and spherical coordinate systems,
respectively . It is sometimes convenient to
express position in a manner which is
independent of a coordinate system; in this case,
we typically use the symbol r. For example,
r = ˆxx+ ˆyy+ ˆzzin the Cartesian coordinate
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independent of a coordinate system; in this case,
we typically use the symbol r. For example,
r = ˆxx+ ˆyy+ ˆzzin the Cartesian coordinate
system.
• Phasors: A tilde is used to indicate a phasor
quantity; e.g., a voltage phasor might be
indicated as ˜V, and the phasor representation of
E will be indicated as ˜E.
• Curves, surfaces, and volumes: These
geometrical entities will usually be indicated in
script; e.g., an open surface might be indicated
as S and the curve bounding this surface might
be indicated as C. Similarly , the volume enclosed
by a closed surface S may be indicated as V.
• Integrations over curves, surfaces, and volumes
will usually be indicated using a single integral
sign with the appropriate subscript. For example:
∫
C
· · · dl is an integral over the curve C
∫
S
· · · ds is an integral over the surface S
∫
V
· · · dv is an integral over the volume V.
• Integrations over closed
curves and surfaces will
be
indicated using a circle superimposed on the
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∫
V
· · · dv is an integral over the volume V.
• Integrations over closed
curves and surfaces will
be
indicated using a circle superimposed on the
integral sign. For example:
∮
C
· · · dl is an integral over the closed curve C
∮
S
··· ds is an integral over the closed surface S
A “closed curve” is one which forms an
unbroken loop; e.g., a circle. A “closed surface”
is one which encloses a volume with no
openings; e.g., a sphere.
• The symbol “ ∼=” means “approximately equal
to. ” This symbol is used when equality exists,
but is not being expressed with exact numerical
precision. For example, the ratio of the
circumference of a circle to its diameter is π,
where π∼= 3.14.
• The symbol “≈ ” also indicates “approximately
equal to, ” but in this case the two quantities are
unequal even if expressed with exact numerical
precision. For example, ex = 1 + x+ x2/2 + ...
as an infinite series, but ex ≈ 1 + xfor x≪ 1.
Using this approximation, e0.1 ≈ 1.1, which is
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precision. For example, ex = 1 + x+ x2/2 + ...
as an infinite series, but ex ≈ 1 + xfor x≪ 1.
Using this approximation, e0.1 ≈ 1.1, which is
in good agreement with the actual value
e0.1 ∼= 1.1052.
• The symbol “∼ ” indicates “on the order of, ”
which is a relatively weak statement of equality
indicating that the indicated quantity is within a
factor of 10 or so of the indicated value. For
example, µ∼ 105 for a class of iron alloys, with
exact values being larger or smaller by a factor
of 5 or so.
• The symbol “≜ ” means “is defined as” or “is
equal as the result of a definition. ”
• Complex numbers: j ≜ √
−1.
• See
Appendix C for notation used to identify
commonly-used physical constants.
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4 CHAPTER 1. PRELIMINAR Y CONCEPTS
1.3 Coordinate Systems
[m0180]
The coordinate systems most commonly used in
engineering
analysis are the Cartesian, cylindrical,
and spherical systems. These systems are illustrated
in Figures 1.1, 1.2, and 1.3, respectively . Note that the
use of variables is not universal; in particular, it is
common to encounter the use of rin lieu of ρfor the
radial coordinate in the cylindrical system, and the
use of Rin lieu of rfor the radial coordinate in the
spherical system.
Additional Reading:
• “Cylindrical coordinate system” on Wikipedia.
• “Spherical coordinate system” on Wikipedia.
y
x
z
y
x
z
c⃝ K. Kikkeri CC BY SA 4.0
Figure 1.1: Cartesian coordinate system.
y
x
ϕ
z
ρ
ϕ
z
ρ
z
c⃝ K. Kikkeri CC BY SA 4.0
Figure 1.2: Cylindrical coordinate system.
y
x
ϕz
r
θθ
ϕ
r
c⃝ K. Kikkeri CC BY SA 4.0
Figure 1.3: Spherical coordinate system.
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1.4. ELECTROMAGNETIC FIELD THEOR Y : A REVIEW 5
1.4 Electromagnetic Field
Theory: A Review
[m0179]
This book is the second in a series of textbooks on
electromagnetics.
This section presents a summary of
electromagnetic field theory concepts presented in the
previous volume.
Electric charge and current. Charge is the ultimate
source of the electric field and has SI base units of
coulomb (C). An important source of charge is the
electron, whose charge is defined to be negative.
However, the term “charge” generally refers to a large
number of charge carriers of various types, and whose
relative net charge may be either positive or negative.
Distributions of charge may alternatively be
expressed in terms of line charge density ρl (C/m),
surface charge density ρs (C/m2), or volume charge
density ρv (C/m3). Electric current describes the net
motion of charge. Current is expressed in SI base
units of amperes (A) and may alternatively be
quantified in terms of surface current density Js
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motion of charge. Current is expressed in SI base
units of amperes (A) and may alternatively be
quantified in terms of surface current density Js
(A/m) or volume current density J (A/m2).
Electrostatics. Electrostatics is the theory of the
electric field subject to the constraint that charge does
not accelerate. That is, charges may be motionless
(“static”) or move without acceleration (“steady
current”).
The electric field may be interpreted in terms of
energy or flux. The energy interpretation of the
electric field is referred to as electric field intensity E
(SI base units of N/C or V/m), and is related to the
energy associated with charge and forces between
charges. One finds that the electric potential (SI base
units of V) over a path C is given by
V = −
∫
C
E · dl (1.1)
The principle of independence of path means that
only the endpoints of C in Equation 1.1, and no other
details of C, matter. This leads to the finding that the
electrostatic field is conservative; i.e.,
∮
C
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details of C, matter. This leads to the finding that the
electrostatic field is conservative; i.e.,
∮
C
E · dl = 0 (1.2)
This is referred to as Kirchoff ’s voltage law for
electrostatics. The inverse of Equation 1.1 is
E = −∇V (1.3)
That is, the electric field intensity points in the
direction in which the potential is most rapidly
decreasing, and the magnitude is equal to the rate of
change in that direction.
The flux interpretation of the electric field is referred
to as electric flux density D (SI base units of C/m 2),
and quantifies the effect of charge as a flow emanating
from the charge. Gauss’ law for electric fields states
that the electric flux through a closed surface is equal
to the enclosed charge Qencl; i.e.,
∮
S
D · ds = Qencl (1.4)
Within a material region, we find
D = ǫE (1.5)
where ǫis the permittivity (SI base units of F/m) of
the material. In free space, ǫis equal to
ǫ0 ≜ 8.854 × 10−12 F/m (1.6)
It is often convenient to quantify the permittivity of
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the material. In free space, ǫis equal to
ǫ0 ≜ 8.854 × 10−12 F/m (1.6)
It is often convenient to quantify the permittivity of
material in terms of the unitless relative permittivity
ǫr ≜ ǫ/ǫ0.
Both E and D are useful as they lead to distinct and
independent boundary conditions at the boundary
between dissimilar material regions. Let us refer to
these regions as Regions 1 and 2, having fields
(E1,D1) and (E2,D2), respectively . Given a vector
ˆn perpendicular to the boundary and pointing into
Region 1, we find
ˆn × [E1 − E2] = 0 (1.7)
i.e., the tangential component of the electric field is
continuous across a boundary , and
ˆn · [D1 − D2] = ρs (1.8)
i.e., any discontinuity in the normal component of the
electric field must be supported by a surface charge
distribution on the boundary .
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6 CHAPTER 1. PRELIMINAR Y CONCEPTS
Magnetostatics. Magnetostatics is the theory of the
magnetic field in response to steady current or the
intrinsic magnetization of materials. Intrinsic
magnetization is a property of some materials,
including permanent magnets and magnetizable
materials.
Like the electric field, the magnetic field may be
quantified in terms of energy or flux. The flux
interpretation of the magnetic field is referred to as
magnetic flux density B (SI base units of Wb/m 2),
and quantifies the field as a flow associated with, but
not emanating from, the source of the field. The
magnetic flux Φ (SI base units of Wb) is this flow
measured through a specified surface. Gauss’ law for
magnetic fields states that
∮
S
B · ds = 0 (1.9)
i.e., the magnetic flux through a closed surface is
zero. Comparison to Equation 1.4 leads to the
conclusion that the source of the magnetic field
cannot be localized; i.e., there is no “magnetic
charge” analogous to electric charge. Equation 1.9
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conclusion that the source of the magnetic field
cannot be localized; i.e., there is no “magnetic
charge” analogous to electric charge. Equation 1.9
also leads to the conclusion that magnetic field lines
form closed loops.
The energy interpretation of the magnetic field is
referred to as magnetic field intensity H (SI base units
of A/m), and is related to the energy associated with
sources of the magnetic field. Ampere’s law for
magnetostatics states that
∮
C
H · dl = Iencl (1.10)
where Iencl is the current flowing past any open
surface bounded by C.
Within a homogeneous material region, we find
B = µH (1.11)
where µis the permeability (SI base units of H/m) of
the material. In free space, µis equal to
µ0 ≜ 4π× 10−7 H/m. (1.12)
It is often convenient to quantify the permeability of
material in terms of the unitless relative permeability
µr ≜ µ/µ0.
Both B and H are useful as they lead to distinct and
independent boundary conditions at the boundaries
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µr ≜ µ/µ0.
Both B and H are useful as they lead to distinct and
independent boundary conditions at the boundaries
between dissimilar material regions. Let us refer to
these regions as Regions 1 and 2, having fields
(B1,H1) and (B2,H2), respectively . Given a vector
ˆn perpendicular to the boundary and pointing into
Region 1, we find
ˆn · [B1 − B2] = 0 (1.13)
i.e., the normal component of the magnetic field is
continuous across a boundary , and
ˆn × [H1 − H2] = Js (1.14)
i.e., any discontinuity in the tangential component of
the magnetic field must be supported by current on
the boundary .
Maxwell’s equations. Equations 1.2, 1.4, 1.9, and
1.10 are Maxwell’s equations for static fields in
integral form. As indicated in T able 1.3, these
equations may alternatively be expressed in
differential form. The principal advantage of the
differential forms is that they apply at each point in
space (as opposed to regions defined by C or S), and
subsequently can be combined with the boundary
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space (as opposed to regions defined by C or S), and
subsequently can be combined with the boundary
conditions to solve complex problems using standard
methods from the theory of differential equations.
Conductivity . Some materials consist of an
abundance of electrons which are loosely-bound to
the atoms and molecules comprising the material.
The force exerted on these electrons by an electric
field may be sufficient to overcome the binding force,
resulting in motion of the associated charges and
subsequently current. This effect is quantified by
Ohm’s law for electromagnetics:
J = σE (1.15)
where J in this case is the conduction current
determined by the conductivity σ(SI base units of
S/m). Conductivity is a property of a material that
ranges from negligible (i.e., for “insulators”) to very
large for good conductors, which includes most
metals.
A perfect conductor is a material within which E is
essentially zero regardless of J. For such material,
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metals.
A perfect conductor is a material within which E is
essentially zero regardless of J. For such material,
σ→ ∞. Perfect conductors are said to be
equipotential regions; that is, the potential difference
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1.4. ELECTROMAGNETIC FIELD THEOR Y : A REVIEW 7
Electrostatics / Time-V arying
Magnetostatics (Dynamic)
Electric & magnetic independent possibly coupled
fields are...
Maxwell’s eqns.
∮
S D · ds = Qenc l
∮
S D · ds = Qencl
(integral)
∮
C E · dl = 0
∮
C E · dl = − ∂
∂t
∫
S B · ds∮
S B · ds = 0
∮
S B · d
s = 0∮
C H · dl = Iencl
∮
C H · dl = Iencl+
∫
S
∂
∂tD · ds
Maxwell’s eqns. ∇ · D = ρv ∇ · D = ρv
(differential) ∇ × E = 0 ∇ × E = − ∂
∂tB
∇ · B = 0 ∇ · B = 0
∇ × H = J ∇ × H = J+ ∂
∂tD
T able 1.3: Comparison of Maxwell’s equations for static and time-v arying electromagnetic fields. Differences
in the time-varying case relative to the static case are highlighted in blue.
between
any two points within a perfect conductor is
zero, as can be readily verified using Equation 1.1.
Time-varying fields. F araday’s law states that a
time-varying magnetic flux induces an electric
potential in a closed loop as follows:
V = − ∂
∂tΦ (1.16)
Setting
this equal to the left side of Equation 1.2 leads
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potential in a closed loop as follows:
V = − ∂
∂tΦ (1.16)
Setting
this equal to the left side of Equation 1.2 leads
to the Maxwell-F araday equation in integral form:
∮
C
E · dl = − ∂
∂t
∫
S
B · ds (1.17)
where C is
the closed path defined by the edge of the
open surface S. Thus, we see that a time-varying
magnetic flux is able to generate an electric field. W e
also observe that electric and magnetic fields become
coupled when the magnetic flux is time-varying.
An analogous finding leads to the general form of
Ampere’s law:
∮
C
H · dl = Iencl +
∫
S
∂
∂tD · ds (1.18)
where
the new term is referred to as displacement
current. Through the displacement current, a
time-varying electric flux may be a source of the
magnetic field. In other words, we see that the electric
and magnetic fields are coupled when the electric flux
is time-varying.
Gauss’ law for electric and magnetic fields, boundary
conditions, and constitutive relationships
(Equations 1.5, 1.11, and 1.15) are the same in the
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Gauss’ law for electric and magnetic fields, boundary
conditions, and constitutive relationships
(Equations 1.5, 1.11, and 1.15) are the same in the
time-varying case.
As indicated in T able 1.3, the time-varying version of
Maxwell’s equations may also be expressed in
differential form. The differential forms make clear
that variations in the electric field with respect to
position are associated with variations in the magnetic
field with respect to time (the Maxwell-Faraday
equation), and vice-versa (Ampere’s law).
Time-harmonic waves in source-free and lossless
media. The coupling between electric and magnetic
fields in the time-varying case leads to wave
phenomena. This is most easily analyzed for fields
which vary sinusoidally , and may thereby be
expressed as phasors. 1 Phasors, indicated in this
book by the tilde (“ ˜”), are complex-valued
quantities representing the magnitude and phase of
the associated sinusoidal waveform. Maxwell’s
equations in differential phasor form are:
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quantities representing the magnitude and phase of
the associated sinusoidal waveform. Maxwell’s
equations in differential phasor form are:
∇ · ˜D = ˜ρv (1.19)
∇ × ˜E = −jω˜B (1.20)
∇ · ˜B = 0 (1.21)
∇ × ˜H = ˜J + jω˜D (1.22)
where ω≜ 2πf, and where f is frequency (SI base
1 Sinus oidally-varying fields are sometimes also said to be time-
harmonic.
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8 CHAPTER 1. PRELIMINAR Y CONCEPTS
units of Hz). In regions which are free of sources (i.e.,
charges and currents) and consisting of loss-free
media (i.e., σ= 0), these equations reduce to the
following:
∇ · ˜E = 0 (1.23)
∇ × ˜E = −jωµ˜H (1.24)
∇ · ˜H = 0 (1.25)
∇ × ˜H = + jωǫ˜E (1.26)
where we have used the relationships D = ǫE and
B = µH to eliminate the flux densities D and B,
which are now redundant. Solving Equations
1.23–1.26 for E and H, we obtain the vector wave
equations:
∇2 ˜E + β2 ˜E = 0 (1.27)
∇2 ˜H + β2 ˜H = 0 (1.28)
where
β ≜ ω√
µǫ (1.29)
W
aves in source-free and lossless media are solutions
to the vector wave equations.
Uniform plane waves in source-free and lossless
media. An important subset of solutions to the vector
wave equations are uniform plane waves. Uniform
plane waves result when solutions are constrained to
exhibit constant magnitude and phase in a plane. For
example, if this plane is specified to be perpendicular
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exhibit constant magnitude and phase in a plane. For
example, if this plane is specified to be perpendicular
to z(i.e., ∂/∂x = ∂/∂y = 0) then solutions for ˜E
have the form:
˜E = ˆx ˜Ex + ˆy ˜Ey (1.30)
where
˜Ex = E+
x0e−jβz + E−
x0e+jβz (1.31)
˜Ey = E+
y0e−jβz + E−
y0e+jβz (1.32)
and where E+
x0, E−
x0, E+
y0, and E−
y0 are constant
complex-valued coefficients which depend on sources
and boundary conditions. The first term and second
terms of Equations 1.31 and 1.32 correspond to waves
traveling in the +ˆz and −ˆz directions, respectively .
Because ˜H is a solution to the same vector wave
equation, the solution for H is identical except with
different coefficients.
The scalar components of the plane waves described
in Equations 1.31 and 1.32 exhibit the same
characteristics as other types of waves, including
sound waves and voltage and current waves in
transmission lines. In particular, the phase velocity of
waves propagating in the +ˆz and −ˆz direction is
vp = ω
β = 1√µǫ (1.33)
and
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transmission lines. In particular, the phase velocity of
waves propagating in the +ˆz and −ˆz direction is
vp = ω
β = 1√µǫ (1.33)
and
the wavelength is
λ= 2π
β (1.34)
By
requiring solutions for ˜E and ˜H to satisfy the
Maxwell curl equations (i.e., the Maxwell-Faraday
equation and Ampere’s law), we find that ˜E, ˜H, and
the direction of propagation ˆk are mutually
perpendicular. In particular, we obtain the plane wave
relationships:
˜E = −ηˆk × ˜H (1.35)
˜H = 1
η
ˆk × ˜E (1.36)
where
η≜
√ µ
ǫ (1.37)
is
the wave impedance, also known as the intrinsic
impedance of the medium, and ˆk is in the same
direction as ˜E × ˜H.
The power density associated with a plane wave is
S =
⏐⏐
⏐˜E
⏐
⏐
⏐
2
2η (1.38)
where Shas
SI base units of W/m 2, and here it is
assumed that ˜E is in peak (as opposed to rms) units.
Commonly-assumed properties of materials.
Finally , a reminder about commonly-assumed
properties of the material constitutive parameters ǫ, µ,
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Commonly-assumed properties of materials.
Finally , a reminder about commonly-assumed
properties of the material constitutive parameters ǫ, µ,
and σ. W e often assume these parameters exhibit the
following properties:
• Homogeneity. A material that is homogeneous is
uniform over the space it occupies; that is, the
values of its constitutive parameters are constant
at all locations within the material.
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1.4. ELECTROMAGNETIC FIELD THEOR Y : A REVIEW 9
• Isotropy. A material that is isotropic behaves in
precisely the same way regardless of how it is
oriented with respect to sources, fields, and other
materials.
• Linearity. A material is said to be linear if its
properties do not depend on the sources and
fields applied to the material. Linear media
exhibit superposition; that is, the response to
multiple sources is equal to the sum of the
responses to the sources individually .
• Time-invariance. A material is said to be
time-invariant if its properties do not vary as a
function of time.
Additional Reading:
• “Maxwell’s Equations” on Wikipedia.
• “W ave Equation” on Wikipedia.
• “Electromagnetic W ave Equation” on Wikipedia.
• “Electromagnetic radiation” on Wikipedia.
[m0181]
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10 CHAPTER 1. PRELIMINAR Y CONCEPTS
Image Credits
Fig. 1.1: c⃝ K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0006 fCartesianBasis.svg,
CC
BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 1.2: c⃝ K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0096 fCylindricalCoordinates.svg,
CC
BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 1.3: c⃝ K. Kikkeri, https://commons.wikimedia.org/wiki/File:Spherical Coordinate System.svg,
CC
BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
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Chapter 2
Magnetostatics
Redux
2.1 Lorentz Force
[m0015]
The Lorentz force is the force experienced by
charge in the presence of electric and magnetic
fields.
Consider a particle having charge q. The force Fe
experienced by the particle in the presence of electric
field intensity E is
Fe = qE
The force Fm experienced by the particle in the
presence of magnetic flux density B is
Fm = qv × B
where v is the velocity of the particle. The Lorentz
force experienced by the particle is simply the sum of
these forces; i.e.,
F = Fe + Fm
= q(E + v × B) (2.1)
The term “Lorentz force” is simply a concise way to
refer to the combined contributions of the electric and
magnetic fields.
A common application of the Lorentz force concept is
in analysis of the motions of charged particles in
electromagnetic fields. The Lorentz force causes
charged particles to exhibit distinct rotational
(“cyclotron”) and translational (“drift”) motions. This
is illustrated in Figures 2.1 and 2.2.
Additional Reading:
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(“cyclotron”) and translational (“drift”) motions. This
is illustrated in Figures 2.1 and 2.2.
Additional Reading:
• “Lorentz force” on Wikipedia.
c⃝ St annered CC BY 2.5.
Figure 2.1: Motion of a particle bearing (left ) posi-
tive charge and (right ) negative charge. T op: Magnetic
field directed toward the viewer; no electric field. Bot-
tom: Magnetic field directed toward the viewer; elec-
tric field oriented as shown.
Electromagnetics V ol. 2. c⃝ 2020 S.W . Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics- vol- 2
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12 CHAPTER 2. MAGNETOST A TICS REDUX
c⃝ M. Biaek CC BY -SA 4.0.
Figure 2.2: Electrons moving in a circle in a magnetic
field (cyclotron motion). The electrons are produced
by an electron gun at bottom, consisting of a hot cath-
ode, a metal plate heated by a filament so it emits elec-
trons, and a metal anode at a high voltage with a hole
which accelerates the electrons into a beam. The elec-
trons are normally invisible, but enough air has been
left in the tube so that the air molecules glow pink
when struck by the fast-moving electrons.
2.2 Magnetic Force on a
Current-Carrying Wire
[m0017]
Consider an infinitesimally-thin and
perfectly-conducting
wire bearing a current I (SI base
units of A) in free space. Let B (r) be the impressed
magnetic flux density at each point r in the region of
space occupied by the wire. By impressed, we mean
that the field exists in the absence of the
current-carrying wire, as opposed to the field that is
induced by this current. Since current consists of
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that the field exists in the absence of the
current-carrying wire, as opposed to the field that is
induced by this current. Since current consists of
charged particles in motion, we expect that B(r) will
exert a force on the current. Since the current is
constrained to flow on the wire, we expect this force
will also be experienced by the wire. Let us now
consider this force.
T o begin, recall that the force exerted on a particle
bearing charge qhaving velocity v is
Fm(r) = qv (r) × B (r) (2.2)
Thus, the force exerted on a differential amount of
charge dqis
dFm(r) = dqv (r) × B (r) (2.3)
Let dl (r) represent a differential-length segment of
the wire at r, pointing in the direction of current flow .
Then
dqv (r) = Idl (r) (2.4)
(If this is not clear, it might help to consider the units:
On the left, C·m/s = (C/s)·m = A·m, as on the right.)
Subsequently ,
dFm(r) = Idl (r) × B (r) (2.5)
There are three important cases of practical interest.
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Subsequently ,
dFm(r) = Idl (r) × B (r) (2.5)
There are three important cases of practical interest.
First, consider a straight segment l forming part of a
closed loop of current in a spatially-uniform
impressed magnetic flux density B (r) = B0. In this
case, the force exerted by the magnetic field on such a
segment is given by Equation 2.5 with dl replaced by
l; i.e.:
Fm = Il × B0
(2.6)
Summarizing,
The force experienced by a straight segment of
current-carrying
wire in a spatially-uniform mag-
netic field is given by Equation 2.6.
The second case of practical interest is a rigid closed
loop of current in a spatially-uniform magnetic flux
density B0. If the loop consists of straight sides –
e.g., a rectangular loop – then the force applied to the
loop is the sum of the forces applied to each side
separately , as determined by Equation 2.6. However,
we wish to consider loops of arbitrary shape. T o
accommodate arbitrarily-shaped loops, let C be the
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we wish to consider loops of arbitrary shape. T o
accommodate arbitrarily-shaped loops, let C be the
path through space occupied by the loop. Then the
force experienced by the loop is
F =
∫
C
dFm(r)
=
∫
C
Idl (r) × B0 (2.7)
Since I and B0 are constants, they may be extracted
from the integral:
F = I
[ ∫
C
dl (r)
]
× B0 (2.8)
Note the quantity in square brackets is zero.
Therefore:
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2.2. MAGNETIC FORCE ON A CURRENT -CARR YING WIRE 13
The net force on a current-carrying loop of wire
in
a uniform magnetic field is zero.
Note that this does not preclude the possibility that
the rigid loop rotates; for example, the force on
opposite sides of the loop may be equal and opposite.
What we have found is merely that the force will not
lead to a translational net force on the loop; e.g.,
force that would propel the loop away from its current
position in space. The possibility of rotation without
translation leads to the most rudimentary concept for
an electric motor. Practical electric motors use
variations on essentially this same idea; see
“ Additional Reading” for more information.
The third case of practical interest is the force
experienced by two parallel infinitesimally-thin wires
in free space, as shown in Figure 2.3. Here the wires
are infinite in length (we’ll return to that in a
moment), lie in the x= 0 plane, are separated by
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are infinite in length (we’ll return to that in a
moment), lie in the x= 0 plane, are separated by
distance d, and carry currents I1 and I2, respectively .
The current in wire 1 gives rise to a magnetic flux
density B1. The force exerted on wire 2 by B1 is:
F2 =
∫
C
[I2dl (r) × B1 (r)] (2.9)
where C is the path followed by I2, and dl (r) = ˆzdz.
A simple way to determine B1 in this situation is as
follows. First, if wire 1 had been aligned along the
x= y= 0 line, then the magnetic flux density
everywhere would be
ˆφµ0I1
2πρ
In
the present problem, wire 1 is displaced by d/2 in
the −ˆy direction. Although this would seem to make
the new expression more complicated, note that the
only positions where values of B1 (r) are required are
those corresponding to C; i.e., points on wire 2. For
these points,
B1 (r) = −ˆxµ0I1
2πd along C (2.10)
That
is, the relevant distance is d(not ρ), and the
direction of B1 (r) for points along C is −ˆx (not ˆφ).
Returning to Equation 2.9, we obtain:
F2 =
∫
C
[
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direction of B1 (r) for points along C is −ˆx (not ˆφ).
Returning to Equation 2.9, we obtain:
F2 =
∫
C
[
I2 ˆzdz×
(
−ˆxµ0I1
2πd
)]
= −ˆy µ0I1I2
2πd
∫
C
dz (2.11)
The
remaining integral is simply the length of wire 2
that we wish to consider. Infinitely-long wires will
therefore result in infinite force. This is not a very
interesting or useful result. However, the force per
unit length of wire is finite, and is obtained simply by
dropping the integral in the previous equation. W e
obtain: F2
∆l = −ˆy µ0I1I2
2πd (2.12)
where ∆
lis the length of the section of wire 2 being
considered. Note that when the currents I1 and I2
flow in the same direction (i.e., have the same sign),
the magnetic force exerted by the current on wire 1
pulls wire 2 toward wire 1.
The same process can be used to determine the
magnetic force F1 exerted by the current in wire 1 on
wire 2. The result is
F1
∆l = + ˆy µ0I1I2
2πd (2.13)
c⃝ Y. Zhao CC BY -SA 4.0
Figure 2.3: Parallel current-carrying wires.
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14 CHAPTER 2. MAGNETOST A TICS REDUX
When the currents I1 and I2 flow in the same
direction (i.e., when the product I1I2 is positive), then
the magnetic force exerted by the current on wire 2
pulls wire 1 toward wire 2.
W e are now able to summarize the results as follows:
If currents in parallel wires flow in the same di-
rection,
then the wires attract; whereas if the cur-
rents flow in opposite directions, then the wires
repel.
Also:
The magnitude of the associated force is
µ0I1I2/2πdfor
wires separated by distance din
non-magnetic media.
If the wires are fixed in position and not able to move,
these forces represent stored (potential) energy . It is
worth noting that this is precisely the energy which is
stored by an inductor – for example, the two wire
segments here might be interpreted as segments in
adjacent windings of a coil-shaped inductor.
Example 2.1. DC power cable.
A power cable connects a 12 V battery to a load
exhibiting an impedance of 10 Ω. The
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Example 2.1. DC power cable.
A power cable connects a 12 V battery to a load
exhibiting an impedance of 10 Ω. The
conductors are separated by 3 mm by a plastic
insulating jacket. Estimate the force between the
conductors.
Solution. The current flowing in each conductor
is 12 V divided by 10 Ω, which is 1.2 A. In terms
of the theory developed in this section, a current
I1 = +1.2 A flows from the positive terminal of
the battery to the load on one conductor, and a
current I2 = −1.2 A returns to the battery on the
other conductor. The change in sign indicates
that the currents at any given distance from the
battery are flowing in opposite directions. Also
from the problem statement, d= 3 mm and the
insulator is presumably non-magnetic.
Assuming the conductors are approximately
straight, the force between conductors is
≈ µ0I1I2
2πd
∼= −96 .0 µN
with the negative sign indicating that the wires
repel.
Note
in the above example that this force is quite
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≈ µ0I1I2
2πd
∼= −96 .0 µN
with the negative sign indicating that the wires
repel.
Note
in the above example that this force is quite
small, which explains why it is not always observed.
However, this force becomes significant when the
current is large or when many sets of conductors are
mechanically bound together (amounting to a larger
net current), as in a motor.
Additional Reading:
• “Electric motor” on Wikipedia.
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2.3. TORQUE INDUCED BY A MAGNETIC FIELD 15
2.3 T orque Induced by a
Magnetic Field
[m0024]
A magnetic field exerts a force on current. This force
is
exerted in a direction perpendicular to the direction
of current flow . For this reason, current-carrying
structures in a magnetic field tend to rotate. A
convenient description of force associated with
rotational motion is torque. In this section, we define
torque and apply this concept to a closed loop of
current. These concepts apply to a wide range of
practical devices, including electric motors.
Figure 2.4 illustrates the concept of torque. T orque
depends on the following:
• A local origin r0,
• A point r which is connected to r0 by a
perfectly-rigid mechanical structure, and
• The force F applied at r.
In terms of these parameters, the torque T is:
T ≜ d × F (2.14)
where the lever arm d ≜ r − r0 gives the location of
r relative to r0. Note that T is a position-free vector
c⃝ C. W ang CC BY -SA 4.0
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where the lever arm d ≜ r − r0 gives the location of
r relative to r0. Note that T is a position-free vector
c⃝ C. W ang CC BY -SA 4.0
Figure 2.4: T orque associated with a single lever arm.
which points in a direction perpendicular to both d
and F.
Note that T does not point in the direction of rotation.
Nevertheless, T indicates the direction of rotation
through a “right hand rule”: If you point the thumb of
your right hand in the direction of T, then the curled
fingers of your right hand will point in the direction of
torque-induced rotation.
Whether rotation actually occurs depends on the
geometry of the structure. For example, if T aligns
with the axis of a perfectly-rigid mechanical shaft,
then all of the work done by F will be applied to
rotation of the shaft on this axis. Otherwise, torque
will tend to rotate the shaft in other directions as well.
If the shaft is not free to rotate in these other
directions, then the effective torque – that is, the
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If the shaft is not free to rotate in these other
directions, then the effective torque – that is, the
torque that contributes to rotation of the shaft – is
reduced.
The magnitude of T has SI base units of N·m and
quantifies the energy associated with the rotational
force. As you might expect, the magnitude of the
torque increases with increasing lever arm magnitude
|d|. In other words, the torque resulting from a
constant applied force increases with the length of the
lever arm.
T orque, like the translational force F, satisfies
superposition. That is, the torque resulting from
forces applied to multiple rigidly-connected lever
arms is the sum of the torques applied to the lever
arms individually .
Now consider the current loop shown in Figure 2.5.
The loop is perfectly rigid and is rigidly attached to a
non-conducting shaft. The assembly consisting of the
loop and the shaft may rotate without friction around
the axis of the shaft. The loop consists of four straight
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loop and the shaft may rotate without friction around
the axis of the shaft. The loop consists of four straight
segments that are perfectly-conducting and
infinitesimally-thin. A spatially-uniform and static
impressed magnetic flux density B0 = ˆxB0 exists
throughout the domain of the problem. (Recall that an
impressed field is one that exists in the absence of any
other structure in the problem.) What motion, if any ,
is expected?
Recall that the net translational force on a current
loop in a spatially-uniform and static magnetic field is
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16 CHAPTER 2. MAGNETOST A TICS REDUX
zero (Section 2.2). However, this does not preclude
the possibility of different translational forces acting
on each of the loop segments resulting in a rotation of
the shaft. Let us first calculate these forces. The force
FA on segment A is
FA = IlA × B0 (2.15)
where lA is a vector whose magnitude is equal to the
length of the segment and which points in the
direction of the current. Thus,
FA = I(ˆzL) × (ˆxB0)
= ˆyILB0 (2.16)
Similarly , the force FC on segment C is
FC = I(−ˆzL) × (ˆxB0)
= −ˆyILB0 (2.17)
The forces FB and FD on segments B and D,
respectively , are:
FB = I(−ˆxL) × (ˆxB0) = 0 (2.18)
and
FD = I(+ˆxL) × (ˆxB0) = 0 (2.19)
Thus, the force exerted on the current loop by the
impressed magnetic field will lead to rotation in the
+ ˆφdirection.
z
x
A
B
C
D
L
W
I
B0=
x̂B0
current
source
wir
e loop
non-conducting shaft
c⃝ C. W ang CC BY -SA 4.0
Figure 2.5: A rudimentary electric motor consisting
of a single current loop.
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source
wir
e loop
non-conducting shaft
c⃝ C. W ang CC BY -SA 4.0
Figure 2.5: A rudimentary electric motor consisting
of a single current loop.
W e calculate the associated torque T as
T = TA + TB + TC + TD (2.20)
where TA, TB, TC, and TD are the torques
associated with segments A, B, C, and D, respectively .
For example, the torque associated with segment A is
TA = W
2 ˆx × FA
= ˆzLW
2 IB0 (2.21)
Similarly
,
TB = 0 since FB = 0 (2.22)
TC = ˆzLW
2 IB0 (2.23)
TD =
0 since FD = 0 (2.24)
Summing these contributions, we find
T = ˆzLWIB0 (2.25)
Note that T points in the +ˆz direction, indicating
rotational force exerted in the + ˆφdirection, as
expected. Also note that the torque is proportional to
the area LW of the loop, is proportional to the current
I, and is proportional to the magnetic field magnitude
B0.
The analysis that we just completed was static; that is,
it applies only at the instant depicted in Figure 2.5. If
the shaft is allowed to turn without friction, then the
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it applies only at the instant depicted in Figure 2.5. If
the shaft is allowed to turn without friction, then the
loop will rotate in the + ˆφdirection. So, what will
happen to the forces and torque? First, note that FA
and FC are always in the +ˆy and −ˆy directions,
respectively , regardless of the rotation of the loop.
Once the loop rotates away from the position shown
in Figure 2.5, the forces FB and FD become
non-zero; however, they are always equal and
opposite, and so do not affect the rotation. Thus, the
loop will rotate one-quarter turn and then come to
rest, perhaps with some damped oscillation around
the rest position depending on the momentum of the
loop. At the rest position, the lever arms for
segments A and C are pointing in the same directions
as FA and FC, respectively . Therefore, the cross
product of the lever arm and translational force for
each segment is zero and subsequently
TA = TC = 0. Once stopped in this position, both
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product of the lever arm and translational force for
each segment is zero and subsequently
TA = TC = 0. Once stopped in this position, both
the net translational force and the net torque are zero.
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2.3. TORQUE INDUCED BY A MAGNETIC FIELD 17
c⃝ Ab normaal CC BY -SA 3.0
Figure 2.6: This DC electric motor uses brushes (here,
the motionless leads labeled “+ ” and “− ”) combined
with the motion of the shaft to periodically alternate
the direction of current between two coils, thereby cre-
ating nearly constant torque.
If such a device is to be used as a motor, it is
necessary to find a way to sustain the rotation. There
are several ways in which this might be accomplished.
First, one might make I variable in time. For
example, the direction of I could be reversed as the
loop passes the quarter-turn position. This reverses
FA and FC, propelling the loop toward the half-turn
position. The direction of I can be changed again as
the loop passes half-turn position, propelling the loop
toward the three-quarter-turn position. Continuing
this periodic reversal of the current sustains the
rotation. Alternatively , one may periodically reverse
the direction of the impressed magnetic field to the
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rotation. Alternatively , one may periodically reverse
the direction of the impressed magnetic field to the
same effect. These methods can be combined or
augmented using multiple current loops or multiple
sets of time-varying impressed magnetic fields. Using
an appropriate combination of current loops,
magnetic fields, and waveforms for each, it is possible
to achieve sustained torque throughout the rotation.
An example is shown in Figure 2.6.
Additional Reading:
• “T orque” on Wikipedia.
• “Electric motor” on Wikipedia.
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18 CHAPTER 2. MAGNETOST A TICS REDUX
2.4 The Biot-Savart Law
[m0066]
The Biot-Savart law (BSL) provides a method to
calculate
the magnetic field due to any distribution of
steady (DC) current. In magnetostatics, the general
solution to this problem employs Ampere’s law; i.e.,
∫
C
H · dl = Iencl (2.26)
in integral form or
∇ × H = J (2.27)
in differential form. The integral form is relatively
simple when the problem exhibits a high degree of
symmetry , facilitating a simple description in a
particular coordinate system. An example is the
magnetic field due to a straight and infinitely-long
current filament, which is easily determined by
solving the integral equation in cylindrical
coordinates. However, many problems of practical
interest do not exhibit the necessary symmetry . A
commonly-encountered example is the magnetic field
due to a single loop of current, which will be
addressed in Example 2.2. For such problems, the
differential form of Ampere’s law is needed.
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due to a single loop of current, which will be
addressed in Example 2.2. For such problems, the
differential form of Ampere’s law is needed.
BSL is the solution to the differential form of
Ampere’s law for a differential-length current
element, illustrated in Figure 2.7. The current element
is I dl, where I is the magnitude of the current (SI
base units of A) and dl is a differential-length vector
indicating the direction of the current at the “source
point” r′. The resulting contribution to the magnetic
field intensity at the “field point” r is
dH(r) = I dl 1
4πR2 × ˆR (2.28)
where
R = ˆRR≜ r − r′ (2.29)
In
other words, R is the vector pointing from the
source point to the field point, and dH at the field
point is given by Equation 2.28. The magnetic field
due to a current-carrying wire of any shape may be
obtained by integrating over the length of the wire:
H(r) =
∫
C
dH(r) = I
4π
∫
C
dl × ˆR
R2 (2.30)
In
addition to obviating the need to solve a differential
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H(r) =
∫
C
dH(r) = I
4π
∫
C
dl × ˆR
R2 (2.30)
In
addition to obviating the need to solve a differential
equation, BSL provides some useful insight into the
behavior of magnetic fields. In particular,
Equation 2.28 indicates that magnetic fields follow
the inverse square law – that is, the magnitude of the
magnetic field due to a differential current element
decreases in proportion to the inverse square of
distance (R−2). Also, Equation 2.28 indicates that the
direction of the magnetic field due to a differential
current element is perpendicular to both the direction
of current flow ˆl and the vector ˆR pointing from the
source point to field point. This observation is quite
useful in anticipating the direction of magnetic field
vectors in complex problems.
It may be helpful to note that BSL is analogous to
Coulomb’s law for electric fields, which is a solution
to the differential form of Gauss’ law , ∇ · D = ρv.
However, BSL applies only under magnetostatic
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to the differential form of Gauss’ law , ∇ · D = ρv.
However, BSL applies only under magnetostatic
conditions. If the variation in currents or magnetic
fields over time is significant, then the problem
becomes significantly more complicated. See
“Jefimenko’s Equations” in “ Additional Reading” for
more information.
Example 2.2. Magnetic field along the axis of a
circular loop of current.
Consider a ring of radius ain the z= 0 plane,
centered on the origin, as shown in Figure 2.8.
As indicated in the figure, the current I flows in
R R
d
H(r)
dl
@r'
I
c⃝ C. W ang CC BY -SA 4.0
Figure 2.7: Use of the Biot-Savart law to calculate the
magnetic field due to a line current.
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2.4. THE BIOT -SA V AR T LA W 19
the ˆφdirection. Find the magnetic field intensity
along
the zaxis.
Solution. The source current position is given in
cylindrical coordinates as
r′ = ˆρa (2.31)
The position of a field point along the zaxis is
r = ˆzz (2.32)
Thus,
ˆRR≜ r − r′ = −ˆρa+ ˆzz (2.33)
and
R≜ |r − r′| =
√
a2 + z2 (2.34)
Equation
2.28 becomes:
dH(ˆzz) = I ˆφadφ
4π[a2 + z2] × ˆzz− ˆρa√
a2 + z2
= Ia
4π
ˆza− ˆρz
[a2 + z2]3/ 2 dφ (2.35)
No
w integrating over the current:
H(ˆzz) =
∫ 2π
0
Ia
4π
ˆza− ˆρz
[a2 + z2]3/ 2 dφ (2.36)
= I
a
4π[a2 + z2]3/ 2
∫ 2π
0
(ˆza− ˆρz) dφ
(2.37)
= I
a
4π[a2 + z2]3/ 2
(
ˆza
∫ 2π
0
dφ− z
∫ 2
π
0
ˆρdφ
)
(2.38)
The second integral is equal to zero. T o see this,
note that the integral is simply summing values
of ˆρfor all possible values of φ. Since
ˆρ(φ+ π) = −ˆρ(φ), the integrand for any given
value of φis equal and opposite the integrand π
radians later. (This is one example of a symmetry
argument.)
c⃝ K. Kikkeri CC BY SA 4.0 (modified)
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value of φis equal and opposite the integrand π
radians later. (This is one example of a symmetry
argument.)
c⃝ K. Kikkeri CC BY SA 4.0 (modified)
Figure 2.8: Calculation of the magnetic field along
the zaxis due to a circular loop of current centered in
the z= 0 plane.
The first integral in the previous equation is
equal
to 2π. Thus, we obtain
H(ˆzz) = ˆz Ia2
2 [a2 + z2]3/ 2 (2.39)
Note
that the result is consistent with the
associated “right hand rule” of magnetostatics:
That is, the direction of the magnetic field is in
the direction of the curled fingers of the right
hand when the thumb of the right hand is aligned
with the location and direction of current. It is a
good exercise to confirm that this result is also
dimensionally correct.
Equation 2.28 extends straightforwardly to other
distributions of current. For example, the magnetic
field due to surface current Js (SI base units of A/m)
can be calculated using Equation 2.28 with I dl
replaced by
Js ds
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field due to surface current Js (SI base units of A/m)
can be calculated using Equation 2.28 with I dl
replaced by
Js ds
where dsis the differential element of surface area.
This can be confirmed by dimensional analysis: I dl
has SI base units of A·m, as does JS ds. Similarly ,
the magnetic field due to volume current J (SI base
units of A/m 2) can be calculated using Equation 2.28
with I dlreplaced by
J dv
where dvis the differential element of volume. For a
single particle with charge q(SI base units of C) and
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20 CHAPTER 2. MAGNETOST A TICS REDUX
velocity v (SI base units of m/s), the relevant quantity
is
qv
since C·m/s = (C/s)·m = A·m. In all of these cases,
Equation 2.28 applies with the appropriate
replacement for I dl.
Note that the quantities qv, I dl, JS ds, and J dv, all
having the same units of A·m, seem to be referring to
the same physical quantity . This physical quantity is
known as current moment. Thus, the “input” to BSL
can be interpreted as current moment, regardless of
whether the current of interest is distributed as a line
current, a surface current, a volumetric current, or
simply as moving charged particles. See “ Additional
Reading” at the end of this section for additional
information on the concept of “moment” in classical
physics.
Additional Reading:
• “Biot-Savart Law” on Wikipedia.
• “Jefimenko’s Equations” on Wikipedia.
• “Moment (physics)” on Wikipedia.
2.5 Force, Energy, and Potential
Difference in a Magnetic
Field
[m0059]
The force Fm experienced
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• “Moment (physics)” on Wikipedia.
2.5 Force, Energy, and Potential
Difference in a Magnetic
Field
[m0059]
The force Fm experienced
by a particle at location r
bearing charge qdue to a magnetic field is
Fm = qv × B(r) (2.40)
where v is the velocity (magnitude and direction) of
the particle, and B(r) is the magnetic flux density at
r. Now we must be careful: In this description, the
motion of the particle is not due to Fm. In fact the
cross product in Equation 2.40 clearly indicates that
Fm and v must be in perpendicular directions.
Instead, the reverse is true: i.e., it is the motion of the
particle that is giving rise to the force. The motion
described by v may be due to the presence of an
electric field, or it may simply be that that charge is
contained within a structure that is itself in motion.
Nevertheless, the force Fm has an associated
potential energy . Furthermore, this potential energy
may change as the particle moves. This change in
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Nevertheless, the force Fm has an associated
potential energy . Furthermore, this potential energy
may change as the particle moves. This change in
potential energy may give rise to an electrical
potential difference (i.e., a “voltage”), as we shall
now demonstrate.
The change in potential energy can be quantified
using the concept of work, W. The incremental work
∆W done by moving the particle a short distance ∆l ,
over which we assume the change in Fm is
negligible, is
∆W ≈ Fm · ˆl∆l (2.41)
where in this case ˆl is the unit vector in the direction
of the motion; i.e., the direction of v. Note that the
purpose of the dot product in Equation 2.41 is to
ensure that only the component of Fm parallel to the
direction of motion is included in the energy tally .
Any component of v which is due to Fm (i.e.,
ultimately due to B) must be perpendicular to Fm, so
∆W for such a contribution must be, from
Equation 2.41, equal to zero. In other words: In the
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ultimately due to B) must be perpendicular to Fm, so
∆W for such a contribution must be, from
Equation 2.41, equal to zero. In other words: In the
absence of a mechanical force or an electric field, the
potential energy of a charged particle remains
constant regardless of how it is moved by Fm. This
surprising result may be summarized as follows:
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2.5. FORCE, ENERGY , AND POTENTIAL DIFFERENCE IN A MAGNETIC FIELD 21
The magnetic field does no work.
Instead,
the change of potential energy associated
with the magnetic field must be completely due to a
change in position resulting from other forces, such as
a mechanical force or the Coulomb force. The
presence of a magnetic field merely increases or
decreases this potential difference once the particle
has moved, and it is this change in the potential
difference that we wish to determine.
W e can make the relationship between potential
difference and the magnetic field explicit by
substituting the right side of Equation 2.40 into
Equation 2.41, yielding
∆W ≈ q[v × B(r)] · ˆl∆l (2.42)
Equation 2.42 gives the work only for a short distance
around r. Now let us try to generalize this result. If
we wish to know the work done over a larger
distance, then we must account for the possibility that
v × B varies along the path taken. T o do this, we may
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distance, then we must account for the possibility that
v × B varies along the path taken. T o do this, we may
sum contributions from points along the path traced
out by the particle, i.e.,
W ≈
N∑
n=1
∆W (rn) (2.43)
where rn are positions defining the path. Substituting
the right side of Equation 2.42, we have
W ≈ q
N∑
n=1
[v × B(rn)] · ˆl(rn)∆l (2.44)
T aking the limit as ∆l → 0, we obtain
W = q
∫
C
[v × B(r)] · ˆl(r)dl (2.45)
where C is the path (previously , the sequence of rn’s)
followed by the particle. Now omitting the explicit
dependence on r in the integrand for clarity:
W = q
∫
C
[v × B] · dl (2.46)
where dl = ˆldlas usual. Now , we are able to
determine the change in potential energy for a
charged particle moving along any path in space,
given the magnetic field.
At this point, it is convenient to introduce the electric
potential difference V21 between the start point (1)
and end point (2) of C. V21 is defined as the work
done by traversing C, per unit of charge; i.e.,
V21 ≜ W
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