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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

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

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 LoopAnalogous to Electric Flux (Gauss’ Law)

(1) Uniform

(2) Non-Uniform

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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

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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

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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)

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

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Minus Sign? Lenz’s Law

Induced EMF is in direction that opposes the change in flux that caused it

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Sign Conventions: Right Hand Rule

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

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

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

Answer: 1. Induced current is clockwise

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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. Clockwise2. Counterclockwise 3. Neither, the current is zero

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Concept Q. Answer: Moving Loop

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)

Answer: 1. Induced current is clockwise

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. Current clockwise; force up

2. Current counterclockwise; force up

3. Current clockwise; force down

4. Current counterclockwise; force down

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Concept Question Answer: Faraday’s Law: Loop

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).

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).

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Faraday’s LawProblem Solving

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Ways to Induce EMF

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 AreaConducting rod pulled along two conducting rails in a uniform magnetic field B at constant velocity v

1. Find the direction of induced current.

2. Find the direction of resultant force.

3. What is the magnitude of EMF?

4. What is the magnitude of current?

5. 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 InductionIf C is a stationary closed curve and S is a surface spanning C then

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