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ELECTROMAGNETICS
STEVEN W. ELLINGSON
VOLUME 2 | Electromagnetics_Vol2.pdf |
ELECTROMAGNETICS
VOLUME 2 | Electromagnetics_Vol2.pdf |
Publication of this book was made possible in part by the Virginia Tech
University Libraries’ Open Education Initiative Faculty Grant program:
http://guides.lib.vt.edu/oer/grants
Books in this series
Electromagnetics, V olume 1, https://doi.org/10.21061/electromagnetics-vol-1
Electromagnetics, V olume 2, https://doi... | Electromagnetics_Vol2.pdf |
ELECTROMAGNETICS
STEVEN W. ELLINGSON
VOLUME 2 | Electromagnetics_Vol2.pdf |
Copyright © 2020 Steven W. Ellingson
iv
This work is published by Virginia Tech Publishing, a division of the University Libraries at Virginia Tech, 560 Drillfield
Drive, Blacksburg, V A 24061, USA (publishing@vt.edu).
Suggested citation: Ellingson, Steven W. (2020) Electromagnetics, V ol. 2. Blacksburg, V A: Virgini... | Electromagnetics_Vol2.pdf |
and includes alternative text which allows for machine-readability. The LaTeX source files also include alternative text for
all images and figures.
Publication Cataloging Information
Ellingson, Steven W., author
Electromagnetics (V olume 2) / Steven W. Ellingson
Pages cm
ISBN 978-1-949373-91-2 (print)
... | Electromagnetics_Vol2.pdf |
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transform, and build upon the material for any purpose, even commercially, as long as you
follow the terms of the license: https://creative... | Electromagnetics_Vol2.pdf |
v
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Contents
Pr
eface ix
1 Preliminary Concepts 1
1.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Coordinate Systems . . . . . . . . . . . . . . . . . . ... | Electromagnetics_Vol2.pdf |
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 . . . . . . . . . . . . . . . . . . . . . . . ... | Electromagnetics_Vol2.pdf |
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 P... | Electromagnetics_Vol2.pdf |
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 . . . . . . . . .... | Electromagnetics_Vol2.pdf |
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 A... | Electromagnetics_Vol2.pdf |
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 . . . . . . . . . . . . . . . . . ... | Electromagnetics_Vol2.pdf |
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 . . . . . . . . . . . . . . . . . . . . . . ... | Electromagnetics_Vol2.pdf |
8 Optical Fiber 138
8.1 Optical Fiber: Method of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2 Acceptance Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.3 Dispersion in Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | Electromagnetics_Vol2.pdf |
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 Ant... | Electromagnetics_Vol2.pdf |
viii CONTENTS
10.7 Directivity and Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.8 Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
10.9 Equivalent Circuit Model for Reception . . . . . . . . . . . . . . . . . . . . . .... | Electromagnetics_Vol2.pdf |
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 . . . . . . . . .... | Electromagnetics_Vol2.pdf |
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: elect... | Electromagnetics_Vol2.pdf |
• 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 conduct... | Electromagnetics_Vol2.pdf |
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
add... | Electromagnetics_Vol2.pdf |
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
eng... | Electromagnetics_Vol2.pdf |
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
... | Electromagnetics_Vol2.pdf |
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
... | Electromagnetics_Vol2.pdf |
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 b... | Electromagnetics_Vol2.pdf |
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... | Electromagnetics_Vol2.pdf |
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 Un... | Electromagnetics_Vol2.pdf |
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 ... | Electromagnetics_Vol2.pdf |
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 wor... | Electromagnetics_Vol2.pdf |
xii PREF ACE
About the Author
[m0153]
Steven W . Ellingson (ellingson@vt.edu) 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... | Electromagnetics_Vol2.pdf |
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 p... | Electromagnetics_Vol2.pdf |
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: S... | Electromagnetics_Vol2.pdf |
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 | Electromagnetics_Vol2.pdf |
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 do... | Electromagnetics_Vol2.pdf |
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
... | Electromagnetics_Vol2.pdf |
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),
temp... | Electromagnetics_Vol2.pdf |
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. | Electromagnetics_Vol2.pdf |
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... | Electromagnetics_Vol2.pdf |
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.
• Curv... | Electromagnetics_Vol2.pdf |
∫
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
un... | Electromagnetics_Vol2.pdf |
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 indicat... | Electromagnetics_Vol2.pdf |
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 partic... | Electromagnetics_Vol2.pdf |
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 ul... | Electromagnetics_Vol2.pdf |
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,... | Electromagnetics_Vol2.pdf |
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 mo... | Electromagnetics_Vol2.pdf |
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 materia... | Electromagnetics_Vol2.pdf |
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 mag... | Electromagnetics_Vol2.pdf |
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... | Electromagnetics_Vol2.pdf |
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
Regi... | Electromagnetics_Vol2.pdf |
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 ... | Electromagnetics_Vol2.pdf |
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 | Electromagnetics_Vol2.pdf |
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 = ... | Electromagnetics_Vol2.pdf |
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 ... | Electromagnetics_Vol2.pdf |
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... | Electromagnetics_Vol2.pdf |
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 al... | Electromagnetics_Vol2.pdf |
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 relationsh... | Electromagnetics_Vol2.pdf |
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−
y... | Electromagnetics_Vol2.pdf |
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... | Electromagnetics_Vol2.pdf |
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
valu... | Electromagnetics_Vol2.pdf |
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 t... | Electromagnetics_Vol2.pdf |
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 (... | Electromagnetics_Vol2.pdf |
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 ... | Electromagnetics_Vol2.pdf |
(“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 ... | Electromagnetics_Vol2.pdf |
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 v... | Electromagnetics_Vol2.pdf |
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 e... | Electromagnetics_Vol2.pdf |
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 g... | Electromagnetics_Vol2.pdf |
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... | Electromagnetics_Vol2.pdf |
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 mer... | Electromagnetics_Vol2.pdf |
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 ... | Electromagnetics_Vol2.pdf |
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 ver... | Electromagnetics_Vol2.pdf |
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 d... | Electromagnetics_Vol2.pdf |
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
o... | Electromagnetics_Vol2.pdf |
≈ 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 (amou... | Electromagnetics_Vol2.pdf |
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 descript... | Electromagnetics_Vol2.pdf |
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,... | Electromagnetics_Vol2.pdf |
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
torqu... | Electromagnetics_Vol2.pdf |
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 fi... | Electromagnetics_Vol2.pdf |
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 ... | Electromagnetics_Vol2.pdf |
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 exampl... | Electromagnetics_Vol2.pdf |
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 lo... | Electromagnetics_Vol2.pdf |
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. | Electromagnetics_Vol2.pdf |
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.... | Electromagnetics_Vol2.pdf |
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, an... | Electromagnetics_Vol2.pdf |
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 o... | Electromagnetics_Vol2.pdf |
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 magn... | Electromagnetics_Vol2.pdf |
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 d... | Electromagnetics_Vol2.pdf |
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... | Electromagnetics_Vol2.pdf |
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... | Electromagnetics_Vol2.pdf |
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... | Electromagnetics_Vol2.pdf |
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 (... | Electromagnetics_Vol2.pdf |
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 referr... | Electromagnetics_Vol2.pdf |
• “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... | Electromagnetics_Vol2.pdf |
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... | Electromagnetics_Vol2.pdf |
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 ma... | Electromagnetics_Vol2.pdf |
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 ma... | Electromagnetics_Vol2.pdf |
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∑
... | Electromagnetics_Vol2.pdf |
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