66
A students guide to Maxwells Equations
Force
F
Start
End
dl1
dl 8
F
u1
u8
F
Path of object
dl8
Path divided into
N segments
u8
F
8
1
2
Component of F
in direction of dl8
N
3
Figure 3.4 Component of force along object path.
To nd the work in this case,
4
The AmpereMaxwell law
For thousands of years, the only known sources of magnetic elds were
certain iron ores and other materials that had been accidentally or
deliberately magnetized. Then in 1820, French physicist Andre-Marie
Ampere heard that in Denma
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A students guide to Maxwells Equations
3.4 A square loop of side a moves with speed v into a region in which a
magnetic eld of magnitude B0 exists perpendicular to the plane of the
loop, as shown in the gure. Make a plot of the emf induced in the loop
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A students guide to Maxwells Equations
~
~ E
r ~ @ B Applying Faradays law (differential form)
@t
The differential form of Faradays law is very useful in deriving the
electromagnetic wave equation, which you can read about in Chapter 5.
You may also en
85
The AmpereMaxwell law
H
C
~ d~ The magnetic eld circulation
B l
Spend a few minutes moving a magnetic compass around a long, straight
wire carrying a steady current, and heres what youre likely to nd: the
current in the wire produces a magnetic eld tha
The AmpereMaxwell law
l0
87
The permeability of free space
The constant of proportionality between the magnetic circulation on the
left side of the AmpereMaxwell law and the enclosed current and rate of
ux change on the right side is l0, the permeability
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A students guide to Maxwells Equations
toward zero would then give you the circulation of the magnetic eld:
I
B l:
4:1
Magnetic field circulation ~ d~
C
The AmpereMaxwell law tells you that this quantity is proportional to
the enclosed current and rate
89
The AmpereMaxwell law
Ienc
The enclosed electric current
Although the concept of enclosed current sounds simple, the question
of exactly which current to include on the right side of the Ampere
Maxwell law requires careful consideration.
It should be c
90
A students guide to Maxwells Equations
(b)
(a)
I1
C1
I2
(c)
C2
C3
I3
Figure 4.5 Alternative surfaces with boundaries C1, C2, and C3.
direction of integration, your thumb points in the direction of positive
current. Thus, the enclosed current in Figure
The AmpereMaxwell law
R
d
~
dt S E
^ da
n
91
The rate of change of ux
This term is the electric ux analog of the changing magnetic ux term in
Faradays law, which you can read about in Chapter 3. In that case, a
changing magnetic ux through any surface wa
The AmpereMaxwell law
(a)
93
Amperian loop
Current I penetrates
this surface
I
Capacitor
Switch
Battery
(b)
Amperian loop
I
Switch
Capacitor
Battery
Figure 4.7 Alternative surfaces for determining enclosed current.
time. This means that the electric ux th
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A students guide to Maxwells Equations
Applied magnetic
field produced by
current I
I
Magnetic dipole
moments align with
applied field
I
Figure 4.2 Effect of magnetic core on eld inside solenoid.
materials weakly reinforces the applied eld, so these ma
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A students guide to Maxwells Equations
If you understand the crossproduct between d~ and ^ in the
l
r
BiotSavart law, you probably suspect that this is not the case. To verify
that, imagine that ~ has a component pointing directly toward the wire. If
B
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A students guide to Maxwells Equations
Time-varying
electric field
Special Gaussian
surface
+
+
+
+
+Q
Q
Figure 4.8 Changing electric ux between capacitor plates.
The change in electric ux over time is therefore,
Z
d
~ ^ da d Q 1 dQ :
E n
dt S
dt e0
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A students guide to Maxwells Equations
Amperian loop
I
Switch
Capacitor
Battery
Figure 4.6 Charging capacitor.
a magnetic eld around the wires, and the circulation of that eld is given
by Amperes law
I
~ d~ l0 Ienc :
B l
C
A serious problem arises in d
The AmpereMaxwell law
H
C
95
R
~ d~ l0 Ienc e0 d ~ ^ da Applying the
B l
dt S E n
AmpereMaxwell law
(integral form)
Like the electric eld in Gausss law, the magnetic eld in the
AmpereMaxwell law is buried within an integral and coupled to another
vector q
97
The AmpereMaxwell law
Amperian loop go through that location. In other words, the loop radius
should be equal to the distance from the wire at which you intend to nd
the value of the magnetic eld. The following example shows how this
works.
Example 4.1
Faradays law
81
Example 3.5: Given an expression for the induced electric eld, nd the
time rate of change of the magnetic eld.
Problem: Find the rate of change with time of the magnetic eld at a
location at which the induced electric eld is given by
"
2
84
A students guide to Maxwells Equations
includes two sources for the magnetic eld; a steady conduction current
and a changing electric ux through any surface S bounded by path C.
In this chapter, youll nd a discussion of the circulation of the magnetic
65
Faradays law
H
C
~ d~ The path integral of a vector eld
A l
The line integral of a vector eld around a closed path is called the
circulation of the eld. A good way to understand the meaning of this
operation is to consider the work done by a force as i
Faradays law
67
Although the force in this example is uniform, the same analysis
pertains to a vector eld of force that varies in magnitude and direction
along the path. The integral on the right side of Equation 3.7 may be
dened for any vector eld ~ and
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A students guide to Maxwells Equations
~ The induced electric eld
E
The electric eld in Faradays law is similar to the electrostatic eld in its
effect on electric charges, but quite different in its structure. Both types of
electric eld accelerate elec
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A students guide to Maxwells Equations
H
C
~ d~ The electric eld circulation
E l
Since the eld lines of induced electric elds form closed loops, these
elds are capable of driving charged particles around continuous circuits.
Charge moving through a cir
Faradays law
R
d
~
dt S B
^ da
n
69
The rate of change of ux
The right side of the common form of Faradays law may look intimidating at rst glance, but a careful inspection of the terms reveals that the
largest portion of this expression is simply the ma
ECE 3613: EM Fields I
Lecture 18: Magnetic Materials
Professor Roach
11/10/2016
Dr. Tyrone Roach, University of Oklahoma, ECE 3613 Section 01, Fall 2016
1
Additional Sources of Magnetic Fields
Previously, weve considered the source of magnetic
fields to
71
Faradays law
Lenzs law
Theres a great deal of physics wrapped up in the minus sign on the right
side of Faradays law, so it is tting that it has a name: Lenzs law. The
name comes from Heinrich Lenz, a German physicist who had an
important insight conce
Faradays law
73
Example 3.2: Given an expression for the change in orientation of a
conducting loop in a xed magnetic eld, nd the emf induced in the loop.
Problem: A circular loop of radius r0 rotates with angular speed x in a
xed magnetic eld as shown in
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A students guide to Maxwells Equations
(a)
(b)
(c)
Rotating loop
B
Loop of decreasing radius
Magnet motion
N
Induced current
B
Induced current
B
Induced current
Figure 3.5 Magnetic ux and induced current.
through a circuit does not produce an electric
75
Faradays law
3.2 The differential form of Faradays law
The differential form of Faradays law is generally written as
@~
B
~ E
r~
@t
Faradays law:
The left side of this equation is a mathematical description of the curl of
the electric eld the tendency
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A students guide to Maxwells Equations
Example 3.3: Given an expression for the change in size of a conducting
loop in a xed magnetic eld, nd the emf induced in the loop.
Problem: A circular loop lying perpendicular to a xed magnetic eld
decreases in s