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General Physics (PHY 2140)

Lecture 16Lecture 16Electricity and Magnetism

Induced voltages and inductionMagnetic flux and induced emfFaraday’s law

Chapter 20

http://www.physics.wayne.edu/~apetrov/PHY2140/

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Lightning ReviewLightning Review

Last lecture:

1.1. MagnetismMagnetismAmpere’s law and applicationsAmpere’s law and applications

2.2. Induced voltages and inductionInduced voltages and inductionMagnetic flux

0B nIµ=

0

2IBr

µπ

=

Magnetic fluxcosBA θΦ =

Review Problem: A sphere of radius R is placed near a long, straight wire that carries a steady current I. The magnetic field generated by the current is B. The total magnetic flux passing through the sphere is

1. µoI.2. µoI /(4 πR2).3. 4 πR2 µoI4. zero.5. need more information

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Recall: right hand rule IIRecall: right hand rule II

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20.1 Induced EMF and magnetic flux20.1 Induced EMF and magnetic fluxFaraday’s experiment

Two circuits are not connected: Two circuits are not connected: no current?no current?However, However, closing the switchclosing the switchwe see that the compass’ we see that the compass’ needle needle movesmoves and then goes and then goes back to its previous positionback to its previous positionNothing happensNothing happens when the when the currentcurrent in the primary coil is in the primary coil is steadysteadyBut But same thingsame thing happens when happens when the the switch is openedswitch is opened, except , except for the for the needle going in the needle going in the opposite directionopposite direction……

Picture © Molecular Expressions

What is going on?

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20.2 Faraday’s law of induction20.2 Faraday’s law of inductionInduced current

I NSv

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20.2 Faraday’s law of induction20.2 Faraday’s law of induction

I

v

v

B

B

I

I

NS

A current is set up in the circuit as long as there is relative motion between the magnet and the loop.

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Does there have to be motion?Does there have to be motion?

AC Delco- +

1 volt

II(induced)

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Does there have to be motion?Does there have to be motion?

AC Delco- +

1 volt

I

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Does there have to be motion?Does there have to be motion?

AC Delco- +

1 volt

I(induced)

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NO!!Does there have to be motion?Does there have to be motion?

AC Delco- +

1 volt

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Maybe the BMaybe the B--field needs to change…..field needs to change…..

Bv

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Maybe the BMaybe the B--field needs to change…..field needs to change…..

Bv

I

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Maybe the BMaybe the B--field needs to change…..field needs to change…..

Bv

I

I

I

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Faraday’s law of magnetic inductionFaraday’s law of magnetic induction

In all of those experiment induced EMF is caused by a change in In all of those experiment induced EMF is caused by a change in the the number of field lines through a loop. In other words,number of field lines through a loop. In other words,The instantaneous EMF induced in a circuit equals the The instantaneous EMF induced in a circuit equals the rate of change rate of change of magnetic fluxof magnetic flux through the circuit.through the circuit.

Lenz’s Law: Lenz’s Law: The polarity of the induced The polarity of the induced emfemf is such that it produces a is such that it produces a current whose magnetic field current whose magnetic field opposes the opposes the changechange in magnetic fluxin magnetic fluxthrough the loop. That is, the induced current tends to maintainthrough the loop. That is, the induced current tends to maintain the the original flux through the circuit.

Nt

∆Φ=

∆E

The number of loops mattersLenz’s law

original flux through the circuit.

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Applications:Applications:

Ground fault interrupterGround fault interrupterElectric guitarElectric guitarSIDS monitorSIDS monitorMetal detectorMetal detector……

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Example 1: EMF in a loopExample 1: EMF in a loop

A wire loop of radius 0.30m lies so that an external magnetic fiA wire loop of radius 0.30m lies so that an external magnetic field eld of strength +0.30T is perpendicular to the loop. The field changof strength +0.30T is perpendicular to the loop. The field changes es to to --0.20T in 1.5s. (The plus and minus signs here refer to opposite 0.20T in 1.5s. (The plus and minus signs here refer to opposite directions through the loop.) Find the magnitude of the average directions through the loop.) Find the magnitude of the average induced induced emfemf in the loop during this time. in the loop during this time.

B

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A wire loop of radius 0.30m lies so that an external magnetic fiA wire loop of radius 0.30m lies so that an external magnetic field of strength eld of strength +0.30T is perpendicular to the loop. The field changes to +0.30T is perpendicular to the loop. The field changes to --0.20T in 1.5s. (The 0.20T in 1.5s. (The plus and minus signs here refer to opposite directions through tplus and minus signs here refer to opposite directions through the loop.) Find he loop.) Find the magnitude of the average induced the magnitude of the average induced emfemf in the loop during this time.in the loop during this time.

The loop is always perpendicular to the field, so the normal to the loop is parallel to the field, so cosθ = 1. The flux is then

2BA B rπΦ = =

Given:

r = 0.30 mBi = 0.30 T Bf = -0.20 T∆t = 1.5 s

Find:

EMF=?

Initially the flux is

( ) ( )2 20.30 0.30m =0.085 T mi T πΦ = ⋅

and after the field changes the flux is

( ) ( )2 20.20 0.30m =-0.057 T mf T πΦ = − ⋅

The magnitude of the average induced emf is: 2 20.085 T m -0.057 T m 0.095

1.5f iemf V

t t sΦ −Φ∆Φ ⋅ ⋅

= = = =∆ ∆

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Example 2: EMF of a flexible loopExample 2: EMF of a flexible loop

The flexible loop in figure below has a radius of 12cm and is inThe flexible loop in figure below has a radius of 12cm and is in a magnetic a magnetic field of strength 0.15T. The loop is grasped at points A and B afield of strength 0.15T. The loop is grasped at points A and B and nd stretched until it closes. If it takes 0.20s to close the loop, stretched until it closes. If it takes 0.20s to close the loop, find the find the magnitude of the average induced magnitude of the average induced emfemf in it during this time. in it during this time.

X X X X

X X X X

X X X X

X X X X

A

B

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20.3 Motional EMF20.3 Motional EMF

vl

B F

Let's consider a conducting bar moving perpendicular to a uniforLet's consider a conducting bar moving perpendicular to a uniform m magnetic field with constant velocity v. magnetic field with constant velocity v.

sinF qvB θ=This force will act on free charges in the conductor. It will tend to move negative charge to one end, and leave the other end of the bar with a net positive charge.

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Motional EMFMotional EMFThe separated charges will create an electric field which The separated charges will create an electric field which will tend to pull the charges back together. will tend to pull the charges back together. When equilibrium exists, the magnetic force,When equilibrium exists, the magnetic force, F=F=qvBqvB, will , will balance the electric force, balance the electric force, F=F=qEqE, such that a free charge , such that a free charge in the bar will feel no net force. in the bar will feel no net force. Thus, at equilibrium, Thus, at equilibrium, E = E = vBvB. The potential difference . The potential difference across the ends of the bar is given by across the ends of the bar is given by ∆∆V=ElV=El oror

A potential difference is maintained across the conductor A potential difference is maintained across the conductor as long as there is motion through the field. If the motion as long as there is motion through the field. If the motion is reversed, the polarity of the potential difference is also is reversed, the polarity of the potential difference is also reversed.

V El Blv∆ = =

reversed.

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Motional EMF Motional EMF –– conducting railsconducting rails

v∆xB

We can apply Faraday's law to the complete loop. The change of fWe can apply Faraday's law to the complete loop. The change of flux through lux through the loop is proportional to the change of area from the motion othe loop is proportional to the change of area from the motion of the bar:f the bar:

BA Bl x∆Φ = = ∆

R

xBl Blvt t

∆Φ ∆= = =∆ ∆

Eor (Faraday’s law)

BlvIR R

= =E

current Motional EMF

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Example: wire in the magnetic fieldExample: wire in the magnetic field

Over a region where the vertical component of the Earth's magnetOver a region where the vertical component of the Earth's magnetic field is ic field is 40.0µT directed downward, a 5.00 m length of wire is held in an 40.0µT directed downward, a 5.00 m length of wire is held in an easteast--west west direction and moved horizontally to the north with a speed of 10direction and moved horizontally to the north with a speed of 10.0 .0 m/sm/s. . Calculate the potential difference between the ends of the wire,Calculate the potential difference between the ends of the wire, and and determine which end is positive. determine which end is positive.

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20.4 Lenz’s law revisited20.4 Lenz’s law revisitedApplication of Lenz's law will tell Application of Lenz's law will tell

us the direction of induced us the direction of induced currents, the direction of currents, the direction of applied or produced forces, applied or produced forces, and the polarity of induced and the polarity of induced emf'semf's..

Lenz's law says that the induced current will produce Lenz's law says that the induced current will produce magnetic flux magnetic flux opposing this changeopposing this change. To oppose an . To oppose an increase into the page, it generates magnetic field which increase into the page, it generates magnetic field which points out of the page, at least in the interior of the loop. points out of the page, at least in the interior of the loop. Such a magnetic field is produced by a Such a magnetic field is produced by a counterclockwisecounterclockwisecurrent (use the right hand rule to verify). current (use the right hand rule to verify).

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Lenz’s law: energy conservationLenz’s law: energy conservation

We arrive at the same conclusion from We arrive at the same conclusion from energy conservation point of viewenergy conservation point of viewThe preceding analysis found that the The preceding analysis found that the current is moving current is moving ccwccw. Suppose that this . Suppose that this is not so.is not so.

If the current If the current I I is is cwcw, the direction of the , the direction of the magnetic force, magnetic force, BlIBlI, on the sliding bar , on the sliding bar would be would be rightright..This would accelerate the bar to the right, This would accelerate the bar to the right, increasing the area of the loop even more.increasing the area of the loop even more.This would produce even greater force This would produce even greater force and so on.and so on.In effect, this would generate energy out of In effect, this would generate energy out of nothing violating the law of conservation of nothing violating the law of conservation of energy.

Our original Our original assertion that the assertion that the current is current is cwcw is is not right, so the not right, so the current is current is ccwccw!energy. !

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N

SN

SS SThe induced

flux seeks to counteract the change.v v

N

change change

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Example: direction of the currentExample: direction of the current

Find the direction of the current induced in the resistor at the instant the switch is closed.

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Applications of Magnetic InductionApplications of Magnetic Induction

Tape / Hard Drive / ZIP ReadoutTape / Hard Drive / ZIP ReadoutTiny coil responds to change in flux as the magnetic domains (enTiny coil responds to change in flux as the magnetic domains (encoding coding 0’s or 1’s) go by.0’s or 1’s) go by.

Question: How can your VCR display an image while paused?Question: How can your VCR display an image while paused?

Credit Card ReaderCredit Card ReaderMust swipe cardMust swipe card

generates changing fluxgenerates changing flux–– Faster swipe Faster swipe bigger signalbigger signal