Physics 122B Electricity and Magnetism

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Physics 122B Electricity and Magnetism
Lecture 21 (Knight33.1-33.4) Electromagnetic
Induction

• Martin Savage

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Lecture 21 Announcements

• Lecture HW due tonight at 10 PM.

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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.
Partially magnetized
Saturated
Unmagnetized
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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 protons 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.
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 12 (b) Magnets 13 (c)
Magnets 14(c) Magnets 23 (e) Magnets
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Chapter 32 - Summary (1)
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Chapter 32 - Summary (2)
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A Second Prelude toMaxwells 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
Maxwells 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 Faradays.
Joseph Henry (1797-1878)
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.
Michael Faraday (1791-1867)
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Faradays 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 placed the coil in the field of a
permanent magnet. He found that there was a
momentary current when the coil was moved.
Faraday placed one coil above the other,
without the iron ring. Again there was a
momentary current when the switch opened or
closed.
Faraday replaced the upper coil with a bar
magnet. He found that there was a momentary
current when the bar magnet was moved in or out
of the coil.
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.
This is also true locally
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Separating Charge and EMF
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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?
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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?
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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.
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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.
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Energy Considerations
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?
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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.
A magnetic braking system.
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Question
What is the ranking of the forces in the figure?
(a) F1F2F3F4 (b) F1ltF2F3gtF4 (c)
F1F3ltF2F4 (d) F1F4ltF2F3 (e)
F1ltF2ltF3F4
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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.
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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
• The effective areaAeff A cos q of the loop
• We use these ideas to define the magnetic
  flux

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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
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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 q00, 300, 600, and 900?
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Magnetic Fluxin 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
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Example Magnetic Flux froma 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?
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Lenzs 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 Lenzs Law.
Here is the statement of Lenzs 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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Lenzs Law (2)
If the field of the bar magnet is already
inthe loop and the bar magnet is removed,
theinduced current is in the direction that
triesto keep the field constant. If the
loop is a superconductor, a persistentstanding
current is induced in the loop, and thefield
remains constant.
Superconductingloop
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Six Induced Current Scenarios
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Example Lenzs 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 Lenzs 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.
• .