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Chapter 28
Sources of Magnetic Field
Copyright © 2009 Pearson Education, Inc.
28-6 Biot-Savart Law
Example 28-12: Current loop.
Determine B
B for points on the axis of a
circular loop of wire of radius R carrying a
current I.
0
r̂
dB 
Id  2
4
r
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28-6 Biot-Savart Law
Example 28-13: B due to a wire segment.
One quarter of a circular loop of wire carries a
current I. The current I enters and leaves on
straight segments of wire, as shown; the straight
wires are along the radial direction from the center
C of the circular portion. Find the magnetic field at
point C.
0
r̂
dB 
Id  2
4
r
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28-7 Magnetic Materials –
Ferromagnetism
Ferromagnetic materials are those that
can become strongly magnetized, such as
iron and nickel.
These materials are made up of tiny
regions called domains; the magnetic field
in each domain is in a single direction.
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28-7 Magnetic Materials –
Ferromagnetism
When the material is
unmagnetized, the
domains are randomly
oriented. They can be
partially or fully aligned
by placing the material
in an external magnetic
field.
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28-7 Magnetic Materials –
Ferromagnetism
A magnet, if undisturbed, will tend to retain its
magnetism. It can be demagnetized by shock or
heat.
The relationship between the external magnetic
field and the internal field in a ferromagnet is
not simple, as the magnetization can vary.
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28-8 Electromagnets and Solenoids –
Applications
Remember that a solenoid is a long coil of
wire. If it is tightly wrapped, the magnetic field
in its interior is almost uniform.
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28-9 Magnetic Fields in Magnetic
Materials; Hysteresis
If a ferromagnetic material is placed in the core
of a solenoid or toroid, the magnetic field is
enhanced by the field created by the
ferromagnet itself. This is usually much greater
than the field created by the current alone.
If we write
B = μI
where μ is the magnetic permeability,
ferromagnets have μ >> μ0, while all other
materials have μ ≈ μ0.
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28-9 Magnetic Fields in Magnetic
Materials; Hysteresis
Not only is the
permeability very large
for ferromagnets, its
value depends on the
external field.
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28-9 Magnetic Fields in Magnetic
Materials; Hysteresis
Furthermore, the induced
field depends on the history
of the material. Starting
with unmagnetized material
and no magnetic field, the
magnetic field can be
increased, decreased,
reversed, and the cycle
repeated. The resulting plot
of the total magnetic field
within the ferromagnet is
called a hysteresis loop.
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28-10 Paramagnetism and Diamagnetism
All materials exhibit some level of magnetic
behavior; most are either paramagnetic (μ
slightly greater than μ0) or diamagnetic (μ
slightly less than μ0). The following is a table
of magnetic susceptibility χm, where
χm = μ/μ0 – 1.
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28-10 Paramagnetism and Diamagnetism
Molecules of paramagnetic materials have a
small intrinsic magnetic dipole moment, and
they tend to align somewhat with an external
magnetic field, increasing it slightly.
Molecules of diamagnetic materials have no
intrinsic magnetic dipole moment; an
external field induces a small dipole moment,
but in such a way that the total field is
slightly decreased.
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Chapter 29
Electromagnetic Induction
and Faraday’s Law
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29-1 Induced EMF
Almost 200 years ago, Faraday looked for
evidence that a magnetic field would induce
an electric current with this apparatus:
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29-1 Induced EMF
He found no evidence when the current was
steady, but did see a current induced when the
switch was turned on or off.
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29-1 Induced EMF
Therefore, a changing magnetic field induces
an emf.
Faraday’s experiment used a magnetic field
that was changing because the current
producing it was changing; the previous
graphic shows a magnetic field that is
changing because the magnet is moving.
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29-2 Faraday’s Law of Induction;
Lenz’s Law
The induced emf in a wire loop is proportional
to the rate of change of magnetic flux through
the loop.
Magnetic flux:
Unit of magnetic flux: weber, Wb:
1 Wb = 1 T·m2.
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29-2 Faraday’s Law of Induction;
Lenz’s Law
This drawing shows the variables in the flux
equation:
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29-2 Faraday’s Law of Induction;
Lenz’s Law
The magnetic flux is analogous to the electric
flux – it is proportional to the total number of
magnetic field lines passing through the loop.
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29-2 Faraday’s Law of Induction;
Lenz’s Law
Conceptual Example 29-1: Determining flux.
A square loop of wire encloses area A1. A uniform
magnetic field B perpendicular to the loop
extends over the area A2. What is the magnetic
flux through the loop A1?
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29-2 Faraday’s Law of Induction;
Lenz’s Law
Faraday’s law of induction: the emf induced in
a circuit is equal to the rate of change of
magnetic flux through the circuit:
d B
 
dt
or
d B
  N
dt
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 N loops
29-2 Faraday’s Law of Induction;
Lenz’s Law
Example 29-2: A loop of wire in a magnetic
field.
A square loop of wire of side l = 5.0 cm is in a
uniform magnetic field B = 0.16 T. What is the
magnetic flux in the loop (a) when B is
perpendicular to the face of the loop and (b)
when B is at an angle of 30° to the area A of the
loop? (c) What is the magnitude of the average
current in the loop if it has a resistance of
0.012 Ω and it is rotated from position (b) to
position (a) in 0.14 s?
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ConcepTest 29.1 Magnetic Flux I
In order to change the
magnetic flux through
the loop, what would
you have to do?
1) drop the magnet
2) move the magnet upward
3) move the magnet sideways
4) only (1) and (2)
5) all of the above
ConcepTest 29.1 Magnetic Flux I
In order to change the
magnetic flux through
the loop, what would
you have to do?
1) drop the magnet
2) move the magnet upward
3) move the magnet sideways
4) only (1) and (2)
5) all of the above
Moving the magnet in any direction would
change the magnetic field through the
loop and thus the magnetic flux.
ConcepTest 29.1 Magnetic Flux II
1) tilt the loop
In order to change the
magnetic flux through
the loop, what would
you have to do?
2) change the loop area
3) use thicker wires
4) only (1) and (2)
5) all of the above
ConcepTest 29.1 Magnetic Flux II
1) tilt the loop
In order to change the
magnetic flux through
the loop, what would
you have to do?
2) change the loop area
3) use thicker wires
4) only (1) and (2)
5) all of the above
Since  = BA cos q , changing the
area or tilting the loop (which varies
the projected area) would change
the magnetic flux through the loop.
29-2 Faraday’s Law of Induction;
Lenz’s Law
The minus sign gives the direction of the
induced emf:
A current produced by an induced emf moves in a
direction so that the magnetic field it produces tends to
restore the changed field.
or:
An induced emf is always in a direction that opposes
the original change in flux that caused it.
Nature hates change!
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29-2 Faraday’s Law of Induction;
Lenz’s Law
How can the flux change?
B
B A cosq


Change any one and the flux changes.
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29-2 Faraday’s Law of Induction;
Lenz’s Law
Magnetic flux will change if the area of the
loop changes.
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29-2 Faraday’s Law of Induction;
Lenz’s Law
Magnetic flux will change if the angle between
the loop and the field changes.
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29-2 Faraday’s Law of Induction;
Lenz’s Law
Conceptual Example 29-3: Induction stove.
In an induction stove, an ac current exists in
a coil that is the “burner” (a burner that
never gets hot). Why will it heat a metal pan
but not a glass container?
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29-2 Faraday’s Law of Induction;
Lenz’s Law
Problem Solving: Lenz’s Law
1. Determine whether the magnetic flux is increasing,
decreasing, or unchanged.
2. The magnetic field due to the induced current
points in the opposite direction to the original field
if the flux is increasing; in the same direction if it is
decreasing; and is zero if the flux is not changing.
3. Use the right-hand rule to determine the direction
of the current.
4. Remember that the external field and the field due
to the induced current are different.
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29-2 Faraday’s Law of Induction;
Lenz’s Law
Conceptual Example 29-4: Practice with
Lenz’s law.
In which direction is the current induced in
the circular loop for each situation?
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ConcepTest 29.2 Moving Bar Magnet
If a north pole moves toward the
1) clockwise
loop from above the page, in what
2) counterclockwise
direction is the induced current?
3) no induced current
ConcepTest 29.2 Moving Bar Magnet
If a north pole moves toward the
1) clockwise
loop from above the page, in what
2) counterclockwise
direction is the induced current?
3) no induced current
The magnetic field of the moving bar
magnet is pointing into the page and
getting larger as the magnet moves
closer to the loop. Thus the induced
magnetic field has to point out of the
page. A counterclockwise induced
current will give just such an induced
magnetic field.
Follow-up: What happens if the magnet is stationary but the loop moves?
29-2 Faraday’s Law of Induction;
Lenz’s Law
Example 29-5: Pulling a coil from
a magnetic field.
A 100-loop square coil of wire, with side
l = 5.00 cm and total resistance 100 Ω, is
positioned perpendicular to a uniform
0.600-T magnetic field. It is quickly
pulled from the field at constant speed
(moving perpendicular to B
B) to a region
where B drops abruptly to zero. At t = 0,
the right edge of the coil is at the edge
of the field. It takes 0.100 s for the whole
coil to reach the field-free region. Find
(a) the rate of change in flux through the
coil, and (b) the emf and current
induced. (c) How much energy is
dissipated in the coil? (d) What was the
average force required (Fext)?
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29-3 EMF Induced in a Moving
Conductor
This image shows another way the magnetic
flux can change:
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29-3 EMF Induced in a Moving
Conductor
The induced current is in a direction that tends
to slow the moving bar – it will take an external
force to keep it moving.
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29-3 EMF Induced in a Moving
Conductor
The induced emf has magnitude
This equation is valid as long as B, l, and
v are mutually perpendicular (if not, it is
true for their perpendicular components).
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ConcepTest 29.3 Moving Wire Loop
A wire loop is being pulled
through a uniform magnetic
field that suddenly ends.
What is the direction of the
induced current?
x x x x x
x x x x x
x x x x x
x x x x x
x x x x x
x x x x x
x x x x x
1) clockwise
2) counterclockwise
3) no induced current
ConcepTest 29.3 Moving Wire Loop
A wire loop is being pulled
through a uniform magnetic
field that suddenly ends.
What is the direction of the
1) clockwise
2) counterclockwise
3) no induced current
induced current?
x x x x x
The B field into the page is disappearing in
x x x x x
the loop, so it must be compensated by an
x x x x x
induced flux also into the page. This can
x x x x x
be accomplished by an induced current in
x x x x x
the clockwise direction in the wire loop.
x x x x x
x x x x x
Follow-up: What happens when the loop is completely out of the field?
29-3 EMF Induced in a Moving
Conductor
Example 29-7: Electromagnetic
blood-flow measurement.
The rate of blood flow in our
body’s vessels can be measured
using the apparatus shown,
since blood contains charged
ions. Suppose that the blood
vessel is 2.0 mm in diameter, the
magnetic field is 0.080 T, and the
measured emf is 0.10 mV. What
is the flow velocity of the blood?
Remind you of the Hall effect?
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29-3 EMF Induced in a Moving
Conductor
Example 29-8: Force on the rod.
To make the rod move to the right at speed v, you
need to apply an external force on the rod to the
right. (a) Explain and determine the magnitude of
the required force. (b) What external power is
needed to move the rod?
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29-4 Electric Generators
A generator is the opposite of a motor – it
transforms mechanical energy into electrical
energy. This is an ac generator:
The axle is rotated by an
external force such as
falling water or steam.
The brushes are in
constant electrical
contact with the slip
rings.
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ConcepTest 29.5 Rotating Wire Loop
If a coil is rotated as shown,
in a magnetic field pointing
to the left, in what direction
is the induced current?
1) clockwise
2) counterclockwise
3) no induced current
ConcepTest 29.5 Rotating Wire Loop
If a coil is rotated as shown,
in a magnetic field pointing
to the left, in what direction
1) clockwise
2) counterclockwise
3) no induced current
is the induced current?
As the coil is rotated into the B field,
the magnetic flux through it increases.
According to Lenz’s law, the induced B
field has to oppose this increase, thus
the new B field points to the right. An
induced counterclockwise current
produces just such a B field.
29-4 Electric Generators
If the loop is rotating with constant angular
velocity ω, the induced emf is sinusoidal:
For a coil of N loops,
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29-4 Electric Generators
Example 29-9: An ac generator.
The armature of a 60-Hz ac
generator rotates in a 0.15-T
magnetic field. If the area of the coil
is 2.0 x 10-2 m2, how many loops
must the coil contain if the peak
output is to be V0 = 170 V?
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29-5 Back EMF and Counter Torque;
Eddy Currents
d B
d
0  V  It R 
 V  I  t  R   A Bsin q 
dt
dt

 
Back EMF
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29-5 Back EMF and Counter Torque;
Eddy Currents
Induced currents can flow
in bulk material as well as
through wires. These are
called eddy currents, and
can dramatically slow a
conductor moving into or
out of a magnetic field.
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