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
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A students guide to Maxwells Equations
H
R
d
~ ~
C E d l dt
~ ^ da
B n
S
Applying Faradays
law (integral form)
The following examples show you how to use Faradays law to solve
problems involving changing magnetic ux and induced emf.
Example 3.1: Given
79
Faradays law
~ E
r ~ The curl of the electric eld
Since charge-based electric elds diverge away from points of positive
charge and converge toward points of negative charge, such elds cannot
circulate back on themselves. You can understand that by look
Faradays law
77
which the ow vectors on one side of the point are signicantly different
(in magnitude, direction, or both) from the ow vectors on the opposite
side of the point.
To aid this thought experiment, imagine holding a tiny paddlewheel at
each po
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A students guide to Maxwells Equations
~
r
Del cross the curl
The curl of a vector eld is a measure of the elds tendency to circulate
about a point much like the divergence is a measure of the tendency of
the eld to ow away from a point. Once again we
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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
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
(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
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
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
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A students guide to Maxwells Equations
(a)
(b)
z
z
y
y
x
x
Figure 3.8 Effect of @[email protected] and @[email protected] on value of curl.
Figure 3.8(a) and the x-component of the curl in the equation: this term
involves the change in Az with y and the change in Ay with z. P
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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
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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
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
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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
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
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
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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
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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
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
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
Homework #02
ECE 3613, Electromagnetic Fields I, Fall 2016
Assigned: Tuesday, 09/06/2016; Due: Tuesday, 09/13/2016, 9AM
Each problem is worth a maximum of 10 points (unless otherwise indicated)
Reminder: Two column format for HW assignments now in effect
Homework #02: Problem Solutions
ECE 3613, Electromagnetic Fields I, Fall 2016
A
The minus sign just indicates the flow
of flux is opposite the direction of dS
0.0097727
0.0097727
4
Homework #06 - Revised
ECE 3613, Electromagnetic Fields I, Fall 2016
Assigned: Tuesday, 10/12/2016; Due: Thursday, 10/20/2016, 9AM
Each problem is worth a maximum of 10 points (unless otherwise indicated)
Notes: Reminder, two column format for HW assignme
Homework #07
ECE 3613, Electromagnetic Fields I, Fall 2016
Assigned: Saturday, 10/22/2016;
Due: Thursday, 10/27/2016 (In Class or before 4:30 pm in the ECE Office)
Each problem is worth a maximum of 10 points (unless otherwise indicated)
Notes: Reminder,
Homework #04 (Revised)
ECE 3613, Electromagnetic Fields I, Fall 2016
Assigned: Tuesday, 09/20/2016; Due: Tuesday, 09/27/2016, 9AM
Each problem is worth a maximum of 10 points (unless otherwise indicated)
Notes: Reminder, two column format for HW assignmen
Homework #05
ECE 3613, Electromagnetic Fields I, Fall 2016
Assigned: Tuesday, 10/04/2016; Due: Tuesday, 10/11/2016, 9AM
Each problem is worth a maximum of 10 points (unless otherwise indicated)
Notes: Reminder, two column format for HW assignments now in