Chapter 20Chapter 20Induced Voltages and Induced Voltages and

InductanceInductance

20.1 Induced emf20.1 Induced emf A current can be produced A current can be produced

by a by a changingchanging magnetic magnetic field [field [BB==f f ((tt)], i.e., )], i.e., BB varies varies over timeover time First shown in an First shown in an

experiment by Michael experiment by Michael FaradayFaraday

A primary coil is A primary coil is connected to a batteryconnected to a battery

A secondary coil is A secondary coil is connected to an connected to an ammeterammeter

Michael FaradayMichael Faraday Faraday is often regarded Faraday is often regarded

as the greatest as the greatest experimental scientist of experimental scientist of the 1800s. His the 1800s. His contributions to the study contributions to the study of electricity include the of electricity include the invention of the electric invention of the electric motor, generator, and motor, generator, and transformer. transformer.

Faraday’s ExperimentFaraday’s Experiment The purpose of the secondary circuit is to The purpose of the secondary circuit is to

detect current that might be produced by the detect current that might be produced by the magnetic fieldmagnetic field

When the switch is closed, the ammeter When the switch is closed, the ammeter deflects in one direction and then returns to deflects in one direction and then returns to zerozero

When the switch is opened, the ammeter When the switch is opened, the ammeter deflects in the opposite direction and then deflects in the opposite direction and then returns to zeroreturns to zero

When there is a steady current in the primary When there is a steady current in the primary circuit, the ammeter reads zerocircuit, the ammeter reads zero

Faraday’s ConclusionsFaraday’s ConclusionsAn electrical current is An electrical current is

produced by a produced by a changingchanging magnetic fieldmagnetic field

It is customary to say that It is customary to say that an an induced emf is produced in the induced emf is produced in the secondary circuit by the secondary circuit by the changing magnetic fieldchanging magnetic field

Magnetic FluxMagnetic Flux The emf is actually induced by a change in The emf is actually induced by a change in

the quantity called the the quantity called the magnetic fluxmagnetic flux rather rather than simply by a change in the magnetic than simply by a change in the magnetic fieldfield

Magnetic flux is defined in a manner similar Magnetic flux is defined in a manner similar to that of electrical fluxto that of electrical flux

Magnetic flux is proportional to both the Magnetic flux is proportional to both the strength of the magnetic field passing strength of the magnetic field passing through the plane of a wire loop wire and the through the plane of a wire loop wire and the area of the looparea of the loop

Magnetic Flux, 2Magnetic Flux, 2 You are given a loop of You are given a loop of

wirewire The wire is in an uniform The wire is in an uniform

magnetic field magnetic field BB The loop has an area The loop has an area AA The flux is defined asThe flux is defined as

ΦΦBB = = BBAA = = B AB A cos cos θθ θθ is the angle is the angle

between between BB and the and the normal to the planenormal to the plane

Magnetic Flux, 3Magnetic Flux, 3 (a) When the field is (a) When the field is

perpendicular to the perpendicular to the plane of the loop, plane of the loop, θθ = 0 = 0 and and ΦΦBB = = ΦΦBB, max, max = = BABA

(b) When (b) When the field is the field is parallel to the plane of parallel to the plane of the loop, the loop, θθ = 90° = 90° and and ΦΦBB = 0= 0 The flux can be negative, The flux can be negative,

for example if for example if θθ = 180° = 180° SI units of flux are T m² SI units of flux are T m²

= Wb (Weber)= Wb (Weber)

Magnetic Flux, finalMagnetic Flux, final The flux can be visualized with respect to The flux can be visualized with respect to

magnetic field linesmagnetic field lines The value of the magnetic flux is proportional to The value of the magnetic flux is proportional to

the total number of lines passing through the the total number of lines passing through the looploop

When the area is perpendicular to the lines, When the area is perpendicular to the lines, the maximum number of lines pass through the maximum number of lines pass through the area and the flux is a maximumthe area and the flux is a maximum

When the area is parallel to the lines, no When the area is parallel to the lines, no lines pass through the area and the flux is 0lines pass through the area and the flux is 0

20.2 Electromagnetic 20.2 Electromagnetic InductionInduction When a magnet moves When a magnet moves

toward a loop of wire, the toward a loop of wire, the ammeter shows the presence ammeter shows the presence of a current (a)of a current (a)

When the magnet is held When the magnet is held stationary, there is no current stationary, there is no current (b)(b)

When the magnet moves When the magnet moves away from the loop, the away from the loop, the ammeter shows a current in ammeter shows a current in the opposite direction (c)the opposite direction (c)

If the loop is moved instead of If the loop is moved instead of the magnet, a current is also the magnet, a current is also detecteddetected

Electromagnetic Induction Electromagnetic Induction – Results of the – Results of the ExperimentExperiment

A current is set up in the circuit as A current is set up in the circuit as long as there is long as there is relative motionrelative motion between the magnet and the loopbetween the magnet and the loop The same experimental results are The same experimental results are

found whether the loop moves or the found whether the loop moves or the magnet movesmagnet moves

The current is called an The current is called an induced induced currentcurrent because it is produced by because it is produced by an induced emfan induced emf

Faraday’s Law and Faraday’s Law and Electromagnetic InductionElectromagnetic Induction

The instantaneous emf induced in a The instantaneous emf induced in a circuit equals the time rate of change circuit equals the time rate of change of magnetic flux through the circuitof magnetic flux through the circuit

If a circuit contains If a circuit contains NN tightly wound tightly wound loops and the flux through each loop loops and the flux through each loop changes by changes by ΔΔΦΦ during an interval Δ during an interval Δtt, , the average emf induced is given by the average emf induced is given by Faraday’s Law:Faraday’s Law:

tN B

Faraday’s Law and Lenz’ Faraday’s Law and Lenz’ LawLaw The minus signThe minus sign is included is included

because of the polarity of the because of the polarity of the emf. The induced emf in the coil emf. The induced emf in the coil gives rise to a current whose gives rise to a current whose magnetic field magnetic field OPPOSESOPPOSES ( ( Lenz’s lawLenz’s law) the change in ) the change in magnetic flux that produced it magnetic flux that produced it

There are three There are three possibilities to produce an possibilities to produce an emf emf

1) Time-varying magnetic field1) Time-varying magnetic field=-=-NN[[((A A coscos)()(BB//tt)+)+

2) Time-varying loop area2) Time-varying loop area    +(+(B B cos cos )()(AA//tt)+ )+ 

3) Turning of the loop 3) Turning of the loop (generator)(generator)  

++BABA(([[coscos]]//tt))]]

Applications of Faraday’s Applications of Faraday’s Law – Ground Fault Law – Ground Fault InterruptersInterrupters

The The ground fault interrupter (GFI)ground fault interrupter (GFI) is a is a safety device that protects against safety device that protects against electrical shockelectrical shock Wire 1 leads from the wall outlet to Wire 1 leads from the wall outlet to

the appliancethe appliance Wire 2 leads from the appliance Wire 2 leads from the appliance

back to the wall outletback to the wall outlet The iron ring confines the magnetic The iron ring confines the magnetic

field, which is generally 0field, which is generally 0 If a leakage occurs, the field If a leakage occurs, the field

is no longer 0 and the is no longer 0 and the induced voltage triggers a induced voltage triggers a circuit breaker shutting off circuit breaker shutting off the currentthe current

Applications of Faraday’s Applications of Faraday’s Law – Electric GuitarLaw – Electric Guitar

A vibrating string induces A vibrating string induces an emf in a coilan emf in a coil

A permanent magnet A permanent magnet inside the coil magnetizes inside the coil magnetizes a portion of the string a portion of the string nearest the coilnearest the coil

As the string vibrates at As the string vibrates at some frequency, its some frequency, its magnetized segment magnetized segment produces a changing flux produces a changing flux through the pickup coilthrough the pickup coil

The changing flux The changing flux produces an induced emf produces an induced emf that is fed to an amplifierthat is fed to an amplifier

Applications of Faraday’s Applications of Faraday’s Law – Apnea MonitorLaw – Apnea Monitor

The coil of wire The coil of wire attached to the chest attached to the chest carries an alternating carries an alternating currentcurrent

An induced emf An induced emf produced by the produced by the varying field passes varying field passes through a pick up coilthrough a pick up coil

When breathing stops, When breathing stops, the pattern of induced the pattern of induced voltages stabilizes and voltages stabilizes and external monitors external monitors sound an alertsound an alert

20.3 Application of 20.3 Application of Faraday’s Law – Motional Faraday’s Law – Motional emfemf A straight conductor of A straight conductor of

length length ℓℓ moves moves perpendicularly with perpendicularly with constant velocity constant velocity through a uniform fieldthrough a uniform field

The electrons in the The electrons in the conductor experience a conductor experience a magnetic forcemagnetic force FF = = q v Bq v B

The electrons tend to The electrons tend to move to the lower end move to the lower end of the conductorof the conductor

ℓℓ

Motional emfMotional emf As the negative charges accumulate at the As the negative charges accumulate at the

base, a net positive charge exists at the base, a net positive charge exists at the upper end of the conductorupper end of the conductor

As a result of this charge separation, an As a result of this charge separation, an electric field is produced in the conductorelectric field is produced in the conductor

Charges build up at the ends of the conductor Charges build up at the ends of the conductor until the downward magnetic force is until the downward magnetic force is balanced by the upward electric forcebalanced by the upward electric force

There is a potential difference between the There is a potential difference between the upper and lower ends of the conductorupper and lower ends of the conductor

Motional emf, cont.Motional emf, cont.VV = =EEℓℓ F=qvB   

VV==BBℓℓv, voltage across the conductorv, voltage across the conductor    If the motion is reversed, the If the motion is reversed, the

polarity of the potential difference polarity of the potential difference is also reversedis also reversed

FF==qEqE==qq ( (VV//ℓℓ ) ) ==qvBqvB

Magnitude of the Motional Magnitude of the Motional emfemf

vBtxB

t

xBAB

B

B

Motional emf in a CircuitMotional emf in a Circuit A conducting bar sliding A conducting bar sliding

with with vv along two along two conducting rails under conducting rails under the action of an applied the action of an applied force force FFappapp. The magnetic . The magnetic force force FFmm opposes the opposes the motion, and a motion, and a counterclockwise current counterclockwise current is induced. is induced.

Motional emf in a Circuit, Motional emf in a Circuit, cont.cont. The changing The changing

magnetic flux through magnetic flux through the loop and the the loop and the corresponding corresponding induced emf in the induced emf in the bar result from the bar result from the change in areachange in area of the of the looploop

The induced, motional The induced, motional emf, acts like a emf, acts like a battery in the circuitbattery in the circuit

RvBIvB

and

Example: Operating a light bulbExample: Operating a light bulbRod and rail have negligible resistance but the bulb has a resistance of 96 B=0.80 T, v=5.0 m/s and ℓℓ =1.6 m.

Calculate (a) emf in the rod, (b) induced current (c) power delivered to the bulb and (d) the energy used by the bulb in 60 s.(a) =vBℓ =(5.0 m/s)(0.80 T)(1.6 m)=6.4 V(b) I=/RI=(6.4V)/(96 )=0.067 A(c) P=IP=I=(6.4 V)(0.067 A)=0.43 W(d) E=PtE=(0.43 W)(60 s)=26 J (=26 Ws)

20.4 Lenz’ Law 20.4 Lenz’ Law Revisited – Moving Bar Revisited – Moving Bar ExampleExample As the bar moves to As the bar moves to

the right, the the right, the magnetic flux through magnetic flux through the circuit increases the circuit increases with time because the with time because the area of the loop area of the loop increasesincreases

The induced current The induced current must be in a direction must be in a direction such that such that it opposes it opposes the change in the the change in the external magnetic fluxexternal magnetic flux

Lenz’ Law, Bar Example, Lenz’ Law, Bar Example, contcont

The flux due to the external field is The flux due to the external field is increasing into the pageincreasing into the page

The flux due to the induced current The flux due to the induced current must be out of the pagemust be out of the page

Therefore the current must be Therefore the current must be counterclockwise when the bar moves counterclockwise when the bar moves to the rightto the right

Lenz’ Law, Bar Example, Lenz’ Law, Bar Example, finalfinal

The bar is moving The bar is moving toward the lefttoward the left

The magnetic flux The magnetic flux through the loop is through the loop is decreasing with timedecreasing with time

The induced current The induced current must be clockwise to must be clockwise to to produce its own to produce its own flux into the pageflux into the page

Lenz’ Law Revisited, Lenz’ Law Revisited, Conservation of EnergyConservation of Energy

Assume the bar is moving to the rightAssume the bar is moving to the right Assume the induced current is clockwiseAssume the induced current is clockwise

The magnetic force on the bar would be to the The magnetic force on the bar would be to the rightright

The force would cause an acceleration and the The force would cause an acceleration and the velocity would increasevelocity would increase

This would cause the flux to increase and the This would cause the flux to increase and the current to increase and the velocity to increase…current to increase and the velocity to increase…

This would violate Conservation of Energy This would violate Conservation of Energy and so therefore, the current must be and so therefore, the current must be counterclockwisecounterclockwise

Lenz’ Law, Moving Magnet Lenz’ Law, Moving Magnet ExampleExample

(a) A bar magnet is moved to the right toward a (a) A bar magnet is moved to the right toward a stationary loop of wire. As the magnet moves, the stationary loop of wire. As the magnet moves, the magnetic flux increases with timemagnetic flux increases with time

(b) The induced current produces a flux to the left (b) The induced current produces a flux to the left to counteract the increasing external flux to the to counteract the increasing external flux to the rightright

Lenz’ Law, Final NoteLenz’ Law, Final Note When applying Lenz’ Law, there When applying Lenz’ Law, there

are are twotwo magnetic fields to magnetic fields to considerconsiderThe external changing The external changing

magnetic field that induces the magnetic field that induces the current in the loopcurrent in the loop

The magnetic field produced The magnetic field produced by the current in the loopby the current in the loop

Application – Tape Application – Tape RecorderRecorder

A magnetic tape A magnetic tape moves past a moves past a recording and recording and playback headplayback head The tape is a The tape is a

plastic ribbon plastic ribbon coated with iron coated with iron oxide or oxide or chromium oxidechromium oxide

x

Application – Tape Recorder, Application – Tape Recorder, cont. cont. To record, the sound is To record, the sound is

converted to an converted to an electrical signal which electrical signal which passes to an passes to an electromagnet that electromagnet that magnetizes the tape in a magnetizes the tape in a particular patternparticular pattern

To playback, the To playback, the magnetized pattern is magnetized pattern is converted back into an converted back into an induced current driving a induced current driving a speakerspeaker