Physics 122B
Electricity and Magnetism
Lecture 21 (Knight:33.1-33.4)
Electromagnetic Induction
Martin Savage
Lecture 21 Announcements
 Lecture HW due tonight at 10 PM.
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Physics 122B - Lecture 21
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Induced Magnetic Dipoles
When an unmagnetized ferromagnetic material
is placed in an externally applied magnetic field,
magnetic domains in the material that are
aligned with the field are energetically
favored.
This causes such aligned domains to grow,
and for domains that are nearly aligned to
rotate their magnetic moments to match the
field direction. The net result is that a
magnetic dipole moment is induced in the
material, with a new south pole close to the
north pole of the external magnet.
If, when the field is removed, some fraction
of the magnetic dipole moment remains, the
material has become a permanent magnet.
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Hysteresis*
Some ferromagnetic
materials can be permanently
magnetized, and “remember”
their history of magnetization.
The “hysteresis curve” shows
the response of a ferromagnetic
material to an external applied
field. As the external field is
applied, the material at first
has increased magnetization,
but then reaches a limit at (a)
and saturates. When the
external field drops to zero at
(b), the material retains about
60% of its maximum
magnetization.
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Unmagnetized
Physics 122B - Lecture 21
Partially
magnetized
Saturated
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Nuclear Magnetism
A single proton (like the one in every
hydrogen nucleus) has a charge (+e) and an
intrinsic angular momentum (“spin”).
If we (naively) imagine the proton’s charge
circulating in a loop, it should have a magnetic
dipole moment μ. And indeed it does.
In an external B-field:
Classically: There will be torques unless m
is aligned along B or against it.
QM: The proton spin has only 2
projections onto B.
Aligned: U1  m B
Anti-aligned: U 2  m B
Energy Difference: U  U 2  U1  2m B
In magnetic resonance imaging, this energy difference is used to
determine the local ``environment’’ of protons in, say, tissue
using strong magnetic fields and high-frequency electromagnetic waves.
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Magnetic Resonance Imaging
As mentioned previously, the behavior of
the intrinsic spins and magnetic moments of
nuclei in a magnetic field allows the spatial
imaging of the positions of specific nuclei,
which can produce a high-resolution image of
the interior of the human body and other
objects. This is called magnetic resonance
imaging or MRI.
The technique requires a very strong and
homogeneous magnetic field. Large solenoids,
often superconducting, are used for this
purpose. The magnetic fields generated
range up to a few tesla. The B-field is
“swept” by auxiliary coils, so that the
conditions for resonance are met at
successive points in the volume of interest.
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Question
Which magnet configurations will
produce this induced magnetization?
(a) Magnets 1&2; (b) Magnets 1&3; (c) Magnets 1&4;
(c) Magnets 2&3; (e) Magnets 2&4;
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Chapter 32 - Summary (1)
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Chapter 32 - Summary (2)
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A Second Prelude to
Maxwell’s Equations
Suppose you come across a vector
field (flow, E, B) that looks
something like this.
What are the identifiable
structures in this field?
1. An “outflow” structure:
2. An “inflow” structure:
3. An “clockwise circulation”
structure:
4. An “counterclockwise
circulation” structure:
Maxwell’s Equations will tell us that the “flow” structures are charges
(+ and -) and the “circulation” structures are energy flows in the field.
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The History of Induction
In 1831, Joseph Henry, a Professor of Mathematics and
Natural Philosophy at the Albany Academy in New York,
discovered magnetic induction. In July, 1832 he published a
paper entitled “On the Production of Currents and Sparks of
Electricity from Magnetism” describing his work. Because
Henry published after Michael Faraday, his did not receive much
credit for this discovery, which actually preceded Faraday’s.
Michael Faraday's ideas about conservation of energy
led him to believe that since an electric current could
cause a magnetic field, a magnetic field should be able to
produce an electric current. He demonstrated this
principle of induction in 1831 and published his results
immediately. The principle of induction was a landmark in
applied science, for it made possible the dynamo, or
generator, which produces electricity by mechanical means.
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Joseph Henry
(1797-1878)
Michael Faraday
(1791-1867)
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Faraday’s Discovery
Faraday had wound two coils around the same iron
ring. He was using a current flow in one coil to
produce a magnetic field in the ring, and he hoped
that this field would produce a current in the other
coil. Like all previous attempts to use a static
magnetic field to produce a current, his attempt
failed to generate a current.
However, Faraday noticed something strange. In
the instant when he closed the switch to start the
current flow in the left circuit, the current meter in
the right circuit jumped ever so slightly. When he
broke the circuit by opening the switch, the meter
also jumped, but in the opposite direction. The
effect occurred when the current was stopping or
starting, but not when the current was steady.
Faraday had invented the picture of lines of
force, and he used this to conclude that the current
flowed only when lines of force cut through the coil.
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Faraday Investigates Induction
Was it necessary
to move the magnet?
Faraday replaced the
Faraday placed one coil
upper coil with a bar magnet. Faraday placed the
above the other, without
coil in the field of a
the iron ring. Again there He found that there was a
momentary current when the permanent magnet.
was a momentary current
when the switch opened or bar magnet was moved in or He found that there
was a momentary
out of the coil.
closed.
current when the
coil was moved.
Conclusion: There is a current in the coil if and only if
the magnetic field passing through the coil is changing.
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Motional EMF
Consider a length l of conductor moving to the right in a magnetic field
that is into the diagram. Positive charges in the conductor will experience
an upward force and negative charges a downward force. The net result is
that charges will “pile up” at the two ends of the conductor and create an
electric field E. When the force produced by E becomes large enough to
balance the magnetic force, the movement of charges will stop and the
system will be in equilibrium.
FB  qvB
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FE  qE
FB  FE
Physics 122B - Lecture 21
 E  vB
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This is also true ``locally’’
Separating Charge and EMF
l
l
0
0
V  Vtop  Vbottom    E y dy    (vB )dy  vlB
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E  vlB
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Question
The square conductor moves upward through a uniform
magnetic field that is directed out of the diagram.
Which of the figures shows the correct distribution of
charges on the conductor?
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Example: A Battery Substitute
A 6.0 cm long flashlight battery has an EMF of 1.5 V.
With what speed must a 6.0 cm wire move through a 0.10 T
magnetic field to create the same EMF?
E  vlB
E
(1.5 V)
v 
 250 m/s
lB (0.06 m)(0.10 T)
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Example: Potential Difference
along a Rotating Bar
A metal bar of length l rotates with
angular velocity w about a pivot at one
end. A uniform magnetic field B is
perpendicular to the plane of rotation.
What is the potential difference
between the ends of the bar?
E  Bv  Bw r
v wr
l
V  Vtip  Vpivot    Er dr
0
l
l
0
0
   ( Bw r )dr  Bw  rdr  12 Bwl 2
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Induced Current in a Circuit
The figure shows a conducting wire sliding
with speed v along a U-shaped conducting rail.
The induced emf E will create a current I
around the loop.
E  vlB
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E vlB
I 
R
R
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Force and Induction
We have assumed that the sliding
conductor moves with a constant speed v.
It turns out that a current carrying wire in
a magnetic field experiences a force Fmag,
so we must supply a counter-force Fpull to
make this happen.
Fpull  Fmag
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vl 2 B 2
 vlB 
 IlB  
 lB 
R
 R 
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Energy Considerations
Ppull
v2 l 2 B2
 Fpull v 
R
2
Pdissipated
v2 l 2 B2
 vlB 
2
 I R
 R
R
 R 
Therefore, the work done in moving the
conductor is equal to the energy dissipated in the
resistance. Energy is conserved.
Whether the wire is moved to the right or to
the left, a force opposing the motion is observed.
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Example: Lighting A Bulb
The figure shows a circuit including a
3 V 1.5 W light bulb connected by ideal
wires with no resistance. The right wire
is pulled with constant speed v through a
perpendicular 0.10 T magnetic field.
(a) What speed must the wire have
to light the bulb to full brightness?
(b) What force is needed to keep
the wire moving?
E
(3.0 V)
v 
 300 m/s
lB (0.10 m)(0.10 T)
P
(1.5 W)
I

 0.50 A
V (3.0 V)
R
V
(3.0 V)

 6.0 
I
(0.50 A)
Fpull
vl 2 B 2 (300 m/s)(0.10 m) 2 (0.10 T) 2


R
(6.0 )
 5.0 103 N
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Eddy Currents
Suppose that a rigid square copper loop is between the poles of a
magnet. If the loop moves, as long as no conductors are in the field of the
magnet there will be no current and no forces. But when one side of the
loop enters the magnetic field, a current flow will be induced and a force
will be produced. Therefore, a force will be required to pull the loop out
of the magnetic field, even though copper is not a magnetic material.
However, if we cut the loop, there will be no force.
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Eddy Currents (2)
Another way of looking at
the system is to consider the
magnetic field produced by
the current in the loop. The
current loop is effectively a
dipole magnet with a S pole
near the N pole of the
magnet, and vice versa.
The attractive forces
between these poles must be
overcome by an external
force to pull the loop out of
the magnet.
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Eddy Currents (3)
Now consider a sheet of conductor
pulled through a magnetic field. There
will be induced current, just as with the
wire, but there are now no well-defined
current paths.
As a consequence, two “whirlpools” of
current will circulate in the conductor.
These are called eddy currents.
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A magnetic braking system.
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Question
What is the ranking of the forces in the figure?
(a) F1=F2=F3=F4;
(d) F1=F4<F2=F3;
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(b) F1<F2=F3>F4;
(e) F1<F2<F3=F4;
Physics 122B - Lecture 21
(c) F1=F3<F2=F4;
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Air Flow and Flux
The amount of air flow through the loop depends on the
orientation of the loop with respect to the air-flow direction.
Aeff  ab cos   A cos 
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Magnetic Flux
The number of arrows
passing through the loop
depends on two factors:
(1) The density of arrows,
which is proportional to B
(2) The effective area
Aeff = A cos  of the loop
We use these ideas to
define the magnetic flux:
Flux :  m  Aeff B  AB cos 
Flux units : 1 weber = 1 Wb = 1 Tm 2
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Area Vector
Define the area vector A of a loop such that it has the loop area
as its magnitude and is perpendicular to the plane of the loop. If a
current is present, the area vector points in the direction given by
the thumb of the right hand when the fingers curl in the direction of
current flow. If the area is part of a closed surface, the area vector
points outside the enclosed volume. With this definition:
m  Aeff B  AB cos  A  B
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Example: A Circular Loop
Rotating in a Magnetic Field
The figure shows a 10 cm
diameter loop rotating in a
uniform 0.050 T magnetic field.
What is the magnitude of
the flux through the loop when
the angle is =00, 300, 600, and
900?
A   R 2   (0.005 m)2  7.85 103 m2
 3.93 104 Wb for   0

4
3.40 10 Wb for   30
 m  AB cos   
4
1.96

10
Wb for   60


0 Wb for   90
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Magnetic Flux
in a Nonuniform Field
So far, we have assumed that
the loop is in a uniform field. What
if that is not the case?
The solution is to break up the
area into infinitesimal pieces, each
so small that the field within it is
essentially constant. Then:
d m  B  dA
m 

B  dA
area of loop
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Example: Magnetic Flux from
a Long Straight Wire
The near edge of a 1.0 cm x 4.0 cm
rectangular loop is 1.0 cm from a long
straight wire that carries a current of
1.0 A, as shown in the figure.
What is the magnetic flux through
the loop?
dA  b dx
B
m0 2 I
4 x
d  m  B  dA 
m0
dx
2 Ib
4
x
ca
m0
m
dx m0
ca
ca
m 
2 Ib 

2 Ib ln x c  0 2 Ib ln

4
x 4
4
c
c
 m  5.55 109 Wb
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Lenz’s Law (1)
Heinrich Friedrich Emil Lenz
(1804-1865)
In 1834, Heinrich Lenz announced a rule for determining the direction
of an induced current, which has come to be known as Lenz’s Law.
Here is the statement of Lenz’s Law:
There is an induced current in a closed conducting loop if and only if
the magnetic flux through the loop is changing. The direction of the
induced current is such that the induced magnetic field opposes the
change in the flux.
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Lenz’s Law (2)
If the field of the bar magnet is already in
the loop and the bar magnet is removed, the
induced current is in the direction that tries
to keep the field constant.
Superconducting
loop
If the loop is a superconductor, a persistent
standing current is induced in the loop, and the
field remains constant.
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Six Induced Current Scenarios
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Example: Lenz’s Law 1
-
+
- +
The switch in the circuit shown has been closed for a long time.
What happens to the lower loop when the switch is opened?
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Example: Lenz’s Law 2
+
-
The figure shows two solenoids facing each other.
When the switch for coil 1 is closed, does the current in coil 2
flow from right to left or from left to right?
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Example: A Rotating Loop
A loop of wire is initially in
the xy plane in a uniform
magnetic field in the x
direction. It is suddenly
rotated 900 about the y axis,
until it is in the yz plane.
In what direction will be the
induced current in the loop?
Initially there is no flux through the coil. After rotation the coil will
be threaded by magnetic flux in the x direction. The induced current in
the coil will oppose this change by producing flux in the –x direction. Let
your thumb point on the –x direction, and your fingers will curl clockwise.
Therefore, the induced current will be clockwise, as shown in the figure.
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End of Lecture 21
 Before the next lecture, read Knight,
sections 33.5 through 33.7.
 Lecture HW is due tonight at 10 PM.
.
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