Chapter 20Chapter 20

Induced Voltages and Induced Voltages and InductanceInductance

Induced emfInduced emf

A current can be produced by a changing A current can be produced by a changing magnetic fieldmagnetic field First shown in an experiment by Michael First shown in an experiment by Michael

FaradayFaraday A primary coil is connected to a batteryA primary coil is connected to a battery A secondary coil is connected to an ammeterA secondary coil is connected to an ammeter

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 detect current that might be produced by the magnetic fieldthe magnetic 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 When there is a steady current in the primary circuit, the ammeter reads zeroprimary circuit, the ammeter reads zero

Faraday’s ConclusionsFaraday’s Conclusions

An electrical current is produced An electrical current is produced by a by a changingchanging magnetic field magnetic field

The secondary circuit acts as if a The secondary circuit acts as if a source of emf were connected to it source of emf were connected to it for a short timefor a short time

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 changing secondary circuit by the changing magnetic fieldmagnetic 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 loop of wire and the through the plane of a loop of 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 a The wire is in a

uniform magnetic field uniform magnetic field BB

The loop has an area AThe loop has an area A The flux is defined asThe flux is defined as

ΦΦBB = B = BA = B A cos θA = B A cos θ θ is the angle between θ is the angle between

B and the normal to the B and the normal to the planeplane

Magnetic Flux, 3Magnetic Flux, 3

When the field is perpendicular to the plane of the When the field is perpendicular to the plane of the loop, as in a, loop, as in a, θ = 0 and Φθ = 0 and ΦBB = Φ = ΦB, maxB, max = BA = BA

When When the field is parallel to the plane of the loop, as in the field is parallel to the plane of the loop, as in b, b, θ = 90° and Φθ = 90° and ΦBB = 0 = 0 The flux can be negative, for example if The flux can be negative, for example if θ = 180°θ = 180°

SI units of flux are T m² = Wb (Weber)SI units of flux are T m² = 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 loopthe total number of lines passing through the loop 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 lines When the area is parallel to the lines, no lines pass through the area and the flux is 0pass through the area and the flux is 0

Electromagnetic Induction Electromagnetic Induction ––An ExperimentAn Experiment

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

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

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

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

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 is it produced by because is it 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 N tightly wound If a circuit contains N tightly wound loops and the flux changes by loops and the flux changes by ΔΦ ΔΦ during a time interval Δt, the average during a time interval Δt, the average emf induced is given by emf induced is given by Faraday’s Faraday’s Law:Law:

tN B

Faraday’s Law and Lenz’ Faraday’s Law and Lenz’ LawLaw

The change in the flux, The change in the flux, ΔΦ, can be ΔΦ, can be produced by a change in B, A or produced by a change in B, A or θθ Since Since ΦΦBB = B A cos θ = B A cos θ

The negative sign in Faraday’s Law is The negative sign in Faraday’s Law is included to indicate the polarity of the included to indicate the polarity of the induced emf, which is found by induced emf, which is found by Lenz’ Law Lenz’ Law The polarity of the induced emf is such that it The polarity of the induced emf is such that it

produces a current whose magnetic field opposes produces a current whose magnetic field opposes the change in magnetic flux through the loopthe change in magnetic flux through the loop

That is, the induced current tends to maintain That is, the induced current tends to maintain the original flux through the circuitthe original flux through the circuit

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

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

outlet to the applianceoutlet to the appliance Wire 2 leads from the Wire 2 leads from the

appliance back to the wall appliance back to the wall outletoutlet

The iron ring confines the The iron ring confines the magnetic field, which is magnetic field, which is generally 0generally 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 inside the coil magnetizes a portion of magnetizes a portion of the string nearest the coilthe string nearest 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

Application of Faraday’s Application of Faraday’s Law – Motional emfLaw – Motional emf

A straight conductor of A straight conductor of length length ℓ moves ℓ 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 conductor experience a magnetic forcea magnetic force F = q v BF = q 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

QUICK QUIZ 20.1

The figure below is a graph of magnitude B versus time t for a magnetic field that passes through a fixed loop and is oriented perpendicular to the plane of the loop. Rank the magnitudes of the emf generated in the loop at the three instants indicated (a, b, c), from largest to smallest.

QUICK QUIZ 20.1 ANSWER

(b), (c), (a). At each instant, the magnitude of the induced emf is proportional to the rate of change of the magnetic field (hence, proportional to the slope of the curve shown on the graph).

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

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, contMotional emf, cont

The potential difference between the The potential difference between the ends of the conductor can be found byends of the conductor can be found by ΔV = B ℓ vΔV = B ℓ v The upper end is at a higher potential than The upper end is at a higher potential than

the lower endthe lower end A potential difference is maintained A potential difference is maintained

across the conductor as long as there is across the conductor as long as there is motion through the fieldmotion through the field If the motion is reversed, the polarity of the If the motion is reversed, the polarity of the

potential difference is also reversedpotential difference is also reversed

Motional emf in a CircuitMotional emf in a Circuit Assume the moving bar Assume the moving bar

has zero resistancehas zero resistance As the bar is pulled to As the bar is pulled to

the right with velocity v the right with velocity v under the influence of under the influence of an applied force, F, the an applied force, F, the free charges experience free charges experience a magnetic force along a magnetic force along the length of the barthe length of the bar

This force sets up an This force sets up an induced current induced current because the charges because the charges are free to move in the are free to move in the closed pathclosed path

Motional emf in a Circuit, Motional emf in a Circuit, contcont

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

R

vBIandvB

QUICK QUIZ 20.2

As an airplane flies due north from Los Angeles to Seattle, it cuts through Earth's magnetic field. As a result, an emf is developed between the wing tips. Which wing tip is positively charged?

QUICK QUIZ 20.2 ANSWER

The left wingtip on the west side of the airplane. The magnetic field of the Earth has a downward component in the northern hemisphere. As the airplane flies northward, the right-hand rule indicates that positive charge experiences a force to the left side of the airplane. Thus, the left wingtip becomes positively charged and the right wingtip negatively charged.

QUICK QUIZ 20.3You wish to move a rectangular loop of wire into a region of uniform magnetic field at a given speed so as to induce an emf in the loop. The plane of the loop must remain perpendicular to the magnetic field lines. In which orientation should you hold the loop while you move it into the region of magnetic field in order to generate the largest emf? (a) With the long dimension of the loop parallel to the velocity vector; (b) With the short dimension of the loop parallel to the velocity vector. (c) Either way—the emf is the same regardless of orientation.

QUICK QUIZ 20.3 ANSWER(b). According to Equation 20.3, because B

and v are constant, the emf depends only on the length of the wire moving in the magnetic field. Thus, you want the long dimension moving through the magnetic field lines so that it is perpendicular to the velocity vector. In this case, the short dimension is parallel to the velocity vector. From a more conceptual point of view, you want the rate of change of area in the magnetic field to be the largest, which you do by thrusting the long dimension into the field.

Lenz’ Law Revisited – Lenz’ Law Revisited – Moving Bar ExampleMoving Bar Example

As the bar moves to As the bar moves to the right, the magnetic the right, the magnetic flux through the circuit flux through the circuit increases with time increases with time because the area of because the area of the loop increasesthe loop increases

The induced current The induced current must in a direction must in a direction such that it opposes such that 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 in The flux due to the external field in 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 bar magnet is moved to the right toward a A bar magnet is moved to the right toward a stationary loop of wire (a)stationary loop of wire (a) As the magnet moves, the magnetic flux increases As the magnet moves, the magnetic flux increases

with timewith time The induced current produces a flux to the left, The induced current produces a flux to the left,

so the current is in the direction shown (b)so the current is in the direction shown (b)

Lenz’ Law, Final NoteLenz’ Law, Final Note

When applying Lenz’ Law, there When applying Lenz’ Law, there are are twotwo magnetic fields to consider magnetic fields to consider The external changing magnetic field The external changing magnetic field

that induces the current in the loopthat induces the current in the loop The magnetic field produced by the The magnetic field produced by the

current in the loopcurrent in the loop

QUICK QUIZ 20.4A bar magnet is falling through a loop of wire with constant velocity with the north pole entering first. Viewed from the same side of the loop as the magnet, as the north pole approaches the loop, the induced current will be in what direction? (a) clockwise (b) zero (c ) counterclockwise (d) along the length of the magnet

QUICK QUIZ 20.4 ANSWER

(c). In order to oppose the approach of the north pole, the magnetic field generated by the induced current must be directed upward. An induced current directed counterclockwise around the loop will produce a field with this orientation along the axis of the loop.

Application – Tape Application – Tape RecorderRecorder

A magnetic tape moves past A magnetic tape moves past a recording and playback a recording and playback headhead The tape is a plastic ribbon The tape is a plastic ribbon

coated with iron oxide or coated with iron oxide or chromium oxidechromium oxide

To record, the sound is To record, the sound is converted to an electrical converted to an electrical signal which passes to an signal which passes to an electromagnet that electromagnet that magnetizes the tape in a magnetizes the tape in a particular patternparticular pattern

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

GeneratorsGenerators

Alternating Current (AC) generatorAlternating Current (AC) generator Converts mechanical energy to Converts mechanical energy to

electrical energyelectrical energy Consists of a wire loop rotated by Consists of a wire loop rotated by

some external meanssome external means There are a variety of sources that There are a variety of sources that

can supply the energy to rotate the can supply the energy to rotate the looploop

These may include falling water, heat by These may include falling water, heat by burning coal to produce steamburning coal to produce steam

AC Generators, contAC Generators, cont Basic operation of the Basic operation of the

generatorgenerator As the loop rotates, the As the loop rotates, the

magnetic flux through it magnetic flux through it changes with timechanges with time

This induces an emf and This induces an emf and a current in the external a current in the external circuitcircuit

The ends of the loop are The ends of the loop are connected to slip rings connected to slip rings that rotate with the loopthat rotate with the loop

Connections to the Connections to the external circuit are made external circuit are made by stationary brushed in by stationary brushed in contact with the slip ringscontact with the slip rings

AC Generators, finalAC Generators, final The emf generated by the The emf generated by the

rotating loop can be found rotating loop can be found bybyε =2 B ℓ vε =2 B ℓ v=2 B ℓ sin θ=2 B ℓ sin θ

If the loop rotates with a If the loop rotates with a constant angular speed, constant angular speed, ω, and N turnsω, and N turnsε = N B A ω sin ω tε = N B A ω sin ω t

ε = εε = εmaxmax when loop is when loop is parallel to the fieldparallel to the field

ε = 0 when when the loop ε = 0 when when the loop is perpendicular to the is perpendicular to the field field

DC GeneratorsDC Generators

Components are Components are essentially the essentially the same as that of an same as that of an ac generatorac generator

The major The major difference is the difference is the contacts to the contacts to the rotating loop are rotating loop are made by a split made by a split ring, or commutatorring, or commutator

DC Generators, contDC Generators, cont The output voltage The output voltage

always has the same always has the same polaritypolarity

The current is a pulsing The current is a pulsing currentcurrent

To produce a steady To produce a steady current, many loops current, many loops and commutators and commutators around the axis of around the axis of rotation are usedrotation are used The multiple outputs The multiple outputs

are superimposed and are superimposed and the output is almost the output is almost free of fluctuationsfree of fluctuations

MotorsMotors

Motors are devices that convert Motors are devices that convert electrical energy into mechanical electrical energy into mechanical energyenergy A motor is a generator run in reverseA motor is a generator run in reverse

A motor can perform useful A motor can perform useful mechanical work when a shaft mechanical work when a shaft connected to its rotating coil is connected to its rotating coil is attached to some external deviceattached to some external device

Motors and Back emfMotors and Back emf The phrase The phrase back emfback emf

is used for an emf is used for an emf that tends to reduce that tends to reduce the applied currentthe applied current

When a motor is When a motor is turned on, there is turned on, there is no back emf initiallyno back emf initially

The current is very The current is very large because it is large because it is limited only by the limited only by the resistance of the coilresistance of the coil

Motors and Back emf, contMotors and Back emf, cont

As the coil begins to rotate, the As the coil begins to rotate, the induced back emf opposes the applied induced back emf opposes the applied voltagevoltage

The current in the coil is reducedThe current in the coil is reduced The power requirements for starting a The power requirements for starting a

motor and for running it under heavy motor and for running it under heavy loads are greater than those for loads are greater than those for running the motor under average running the motor under average loadsloads

Self-inductanceSelf-inductance Self-inductanceSelf-inductance occurs when the changing occurs when the changing

flux through a circuit arises from the circuit flux through a circuit arises from the circuit itselfitself As the current increases, the magnetic flux As the current increases, the magnetic flux

through a loop due to this current also increasesthrough a loop due to this current also increases The increasing flux induces an emf that opposes The increasing flux induces an emf that opposes

the currentthe current As the magnitude of the current increases, the As the magnitude of the current increases, the

rate of increase lessens and the induced emf rate of increase lessens and the induced emf decreasesdecreases

This opposing emf results in a gradual increase This opposing emf results in a gradual increase of the currentof the current

Self-inductance contSelf-inductance cont

The self-induced emf must be The self-induced emf must be proportional to the time rate of change of proportional to the time rate of change of the currentthe current

L is a proportionality constant called the L is a proportionality constant called the inductanceinductance of the device of the device

The negative sign indicates that a changing The negative sign indicates that a changing current induces an emf in opposition to that current induces an emf in opposition to that changechange

t

IL

Self-inductance, finalSelf-inductance, final

The inductance of a coil depends The inductance of a coil depends on geometric factorson geometric factors

The SI unit of self-inductance is the The SI unit of self-inductance is the HenryHenry 1 H = 1 (V 1 H = 1 (V · s) / A· s) / A

You can determine an equation for You can determine an equation for LL

I

NB

INL B

Inductor in a CircuitInductor in a Circuit

Inductance can be interpreted as a Inductance can be interpreted as a measure of opposition to the rate of measure of opposition to the rate of change in the currentchange in the current Remember resistance R is a measure of Remember resistance R is a measure of

opposition to the currentopposition to the current As a circuit is completed, the current As a circuit is completed, the current

begins to increase, but the inductor begins to increase, but the inductor produces an emf that opposes the produces an emf that opposes the increasing currentincreasing current Therefore, the current doesn’t change from 0 Therefore, the current doesn’t change from 0

to its maximum instantaneouslyto its maximum instantaneously

RL CircuitRL Circuit

When the current When the current reaches its reaches its maximum, the rate of maximum, the rate of change and the back change and the back emf are zeroemf are zero

The time constant, The time constant, , , for an RL circuit is the for an RL circuit is the time required for the time required for the current in the circuit current in the circuit to reach 63.2% of its to reach 63.2% of its final valuefinal value

RL Circuit, contRL Circuit, cont

The time constant depends on R The time constant depends on R and Land L

The current at any time can be The current at any time can be found byfound by

R

L

/te1R

I

QUICK QUIZ 20.5The switch in the circuit shown in the figure below is closed and the lightbulb glows steadily. The inductor is a simple air-core solenoid. An iron rod is inserted into the interior of the solenoid, which increases the magnitude of the magnetic field in the solenoid. As the rod is inserted into the solenoid, the brightness of the lightbulb (a) increases, (b) decreases, or (c) remains the same.

QUICK QUIZ 20.5 ANSWER

(b). When the iron rod is inserted into the solenoid, the inductance of the coil increases. As a result, more potential difference appears across the coil than before. Consequently, less potential difference appears across the bulb and its brightness decreases.

Energy Stored in a Energy Stored in a Magnetic FieldMagnetic Field

The emf induced by an inductor The emf induced by an inductor prevents a battery from establishing prevents a battery from establishing an instantaneous current in a circuitan instantaneous current in a circuit

The battery has to do work to produce The battery has to do work to produce a currenta current This work can be thought of as energy This work can be thought of as energy

stored by the inductor in its magnetic stored by the inductor in its magnetic fieldfield

PEPELL = = ½ L I½ L I22