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W09D2:
Faraday’s Law:
The Best Law in the Entire
Universe
Today’s Reading Assignment Course Notes: Sections 10.1-10.4
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Announcements
Math Review Week Ten Tuesday from 9-11 pm in 26-152
PS 7 due Week Ten Tuesday at 9 pm in boxes outside 32-082 or
26-152
Next Reading Assignment:
W09D3 Course Notes: Sections 9.10.2; 10.7, 10.9-10
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Outline
Faraday’s Law
Applications of Faraday’s Law
Problem Solving
Experiment 3: Faraday’s Law
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Faraday’s Law of Induction
dF B
dt
µ I induced
Changing magnetic flux
induces a current
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Demonstration:
Induction
and Simulation of Induction
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H%203&show=0
http://peter-edx.99k.org/FaradaysLaw.html
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Demo: Electromagnetic Induction
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H%203&show=0
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Demonstration: Magnet Falling
Through Plastic Tube and
Aluminum Tube
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H%2016&show=0
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Demonstration: Jumping Ring
An aluminum ring
jumps into the air
when the solenoid
beneath it is
energized
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H%2022&show=0
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What is Going On?
This is a dramatic example of
Faraday’s Law and Lenz’s
Law: When current is turned on
through the solenoid the created
magnetic field tries to permeate
the conducting aluminum ring,
currents are induced in the ring
to try to keep this from
happening, and the ring is
repelled upwards.
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Magnetic Flux Thru Wire Loop
Analogous to Electric Flux (Gauss’ Law)
(1) Uniform B
F B = B^ A = B Ac o sq = B × A
(2) Non-Uniform B
F B = òò B × d A
S
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Concept Question: Loop in Uniform Field
While a rectangular wire loop is
pulled upward though a uniform
magnetic field B field penetrating its
bottom half, as shown, there is
1. a current in the loop.
2. no current in the loop.
3. I do not understand the concepts of current and
magnetic field.
4. I understand the concepts of current and magnetic field
but am not sure of the answer.
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Concept Q. Ans.: Loop in Uniform Field
Answer: 1. The motion changes the magnetic flux through the
loop. The magnetic flux is decreasing in time as more of the
loop enters a region of zero magnetic field. According to
Faraday’s Law there is an induced current through the loop.
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Concept Q.: Loop in Uniform Field
While a rectangular wire loop is
pulled sideways though a uniform
magnetic field B field penetrating its
bottom half, as shown, there is
1. a current in the loop.
2. no current in the loop.
3. I do not understand the concepts of current and
magnetic field.
4. I understand the concepts of current and magnetic field
but am not sure of the answer.
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Concept Q. Ans.: Loop in Uniform Field
Answer: 2. The motion does not change the magnetic flux
through the loop. The magnetic flux is constant in time.
According to Faraday’s Law there is no induced current
through the loop.
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Electromotive Force (EMF)
e= ò
E×ds
closd path
Looks like electric potential. It’s a “driving force” for
current
If a conducting closed path is present for charge
carriers then the electric field exerts forces on
charge carriers producing and induced current
e = IR
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Minus Sign? Lenz’s Law
e=-
dF B
dt
Induced EMF is in direction that opposes
the change in flux that caused it
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Sign Conventions: Right Hand Rule
d
E×ds = B×dA
ò
òò
dt open surface
closed path
Integration direction
clockwise for line
integral requires that unit
normal points into page
for open surface integral
Magnetic flux positive
into page, negative out
of page
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Sign Conventions: Right Hand Rule
d
E×ds = B×dA
ò
òò
dt open surface
closed path
Integration direction
counterclockwise for line
integral requires that unit
normal points out of
page for open surface
integral
Magnetic flux positive
out of page, negative
into page
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Concept Question: Loop
The magnetic field through
a wire loop is pointed
upwards and increasing
with time. The induced
current in the coil is
1. Clockwise as seen from the top
2. Counterclockwise
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Concept Question Answer: Loop
Answer: 1. Induced current
is clockwise
This produces an “induced”
B field pointing down over
the area of the loop.
The “induced” B field
opposes the increasing flux
through the loop – Lenz’s
Law
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Concept Question: Moving Loop
A circuit in the form of a
rectangular piece of wire is
pulled away from a long wire
carrying current I in the
direction shown in the sketch.
The induced current in the
rectangular circuit is
1. Clockwise
2. Counterclockwise
3. Neither, the current is zero
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Concept Q. Answer: Moving Loop
Answer: 1. Induced current is
clockwise
B due to I is into page; the flux
through the circuit due to that field
decreases as the circuit moves
away. So the induced current is
clockwise (to make a B into the
page)
Note: Iind dl x B force is left on the left segment and
right on the right, but the force on the left is bigger.
So the net force on the rectangular circuit is to the
left, again trying to keep the flux from decreasing by
slowing the circuit’s motion
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Concept Question: Faraday’s Law:
Loop
A coil moves up from
underneath a magnet
with its north pole
pointing upward. The
current in the coil and
the force on the coil:
1.
2.
3.
4.
Current clockwise; force up
Current counterclockwise; force up
Current clockwise; force down
Current counterclockwise; force down
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Concept Question Answer:
Faraday’s Law: Loop
Answer: 3. Current is clockwise; force is down
The clockwise current creates
a self-field downward, trying
to offset the increase of
magnetic flux through the coil
as it moves upward into
stronger fields (Lenz’s Law).
The I dl x B force on the coil is a force which is trying
to keep the flux through the coil from increasing by
slowing it down (Lenz’s Law again).
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Faraday’s Law
Problem Solving
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Ways to Induce EMF
e
d
=BAcosq
dt
(
)
Quantities which can vary with time:
• Magnitude of B
• Area A enclosed by the loop
• Angle between B and normal vector to loop
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Problem: Changing Area
Conducting rod pulled along two conducting rails in a
uniform magnetic field B at constant velocity v
1.
2.
3.
4.
5.
Find the direction of induced current.
Find the direction of resultant force.
What is the magnitude of EMF?
What is the magnitude of current?
What is the external power supplied to
move at constant v? That is, calculate
Fext dot v.
6. What is the Joule heating rate in the
circuit, I2R, and how does it relate to
the answer in (5)?
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Faraday’s Law of Induction
If C is a stationary closed curve and S is a
surface spanning C then
d
E
×
d
s
=
òC
dt
B
×
d
A
òò
S
The changing magnetic flux through S
induces a non-electrostatic electric field
whose line integral around C is non-zero
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Problem: Calculating Induced
Electric Field
Consider a uniform magnetic field
which points into the page and is
confined to a circular region with
radius R. Suppose the magnitude
increases with time, i.e. dB/dt > 0.
Find the magnitude and direction of
the induced electric field in the
regions (i) r < R, and (ii) r > R. (iii)
Plot the magnitude of the electric
field as a function r.
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Experiment 3:
Faraday’s Law of Induction
http://peter-edx.99k.org/FaradaysLaw.html
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