Chapter 31Chapter 31

Faraday’s LawFaraday’s Law

Michael FaradayMichael Faraday Great experimental Great experimental

physicistphysicist 1791 – 18671791 – 1867 Contributions to early Contributions to early

electricity include: electricity include: Invention of motor, Invention of motor,

generator, and generator, and transformertransformer

Electromagnetic Electromagnetic inductioninduction

Laws of electrolysisLaws of electrolysis

InductionInduction An An induced currentinduced current is produced by a is produced by a

changing magnetic fieldchanging magnetic field There is an There is an induced emfinduced emf associated with the associated with the

induced currentinduced current A current can be produced without a battery A current can be produced without a battery

present in the circuitpresent in the circuit Faraday’s law of induction describes the Faraday’s law of induction describes the

induced emfinduced emf

EMF Produced by a Changing EMF Produced by a Changing Magnetic FieldMagnetic Field

A loop of wire is A loop of wire is connected to a connected to a sensitive ammetersensitive ammeter

When a magnet is When a magnet is moved toward the loop, moved toward the loop, the ammeter deflectsthe ammeter deflects The direction was The direction was

chosen to be toward the chosen to be toward the right arbitrarilyright arbitrarily

EMF Produced by a Changing EMF Produced by a Changing Magnetic FieldMagnetic Field

When the magnet is When the magnet is held stationary, held stationary, there is no there is no deflection of the deflection of the ammeterammeter

Therefore, there is Therefore, there is no induced currentno induced current Even though the Even though the

magnet is in the loopmagnet is in the loop

EMF Produced by a Changing EMF Produced by a Changing Magnetic FieldMagnetic Field

The magnet is moved The magnet is moved away from the loopaway from the loop

The ammeter deflects in The ammeter deflects in the opposite directionthe opposite direction

Active Figure 31.1

(SLIDESHOW MODE ONLY)

EMF Produced by a Changing EMF Produced by a Changing Magnetic Field, SummaryMagnetic Field, Summary

The ammeter deflects when the magnet is The ammeter deflects when the magnet is moving toward or away from the loopmoving toward or away from the loop

The ammeter also deflects when the loop is The ammeter also deflects when the loop is moved toward or away from the magnetmoved toward or away from the magnet

Therefore, the loop detects that the magnet is Therefore, the loop detects that the magnet is moving relative to itmoving relative to it We relate this detection to a change in the We relate this detection to a change in the

magnetic fieldmagnetic field This is the induced current that is produced by an This is the induced current that is produced by an

induced emfinduced emf

Faraday’s Experiment – Faraday’s Experiment – Set UpSet Up

A primary coil is connected to A primary coil is connected to a switch and a batterya switch and a battery

The wire is wrapped around The wire is wrapped around an iron ringan iron ring

A secondary coil is also A secondary coil is also wrapped around the iron ringwrapped around the iron ring

There is no battery present in There is no battery present in the secondary coilthe secondary coil

The secondary coil is not The secondary coil is not directly connected to the directly connected to the primary coilprimary coil

Active Figure 31.2

(SLIDESHOW MODE ONLY)

Faraday’s Experiment – Findings Faraday’s Experiment – Findings

At the instant the switch is closed, the At the instant the switch is closed, the galvanometer (ammeter) needle deflects in galvanometer (ammeter) needle deflects in one direction and then returns to zeroone direction and then returns to zero

When the switch is opened, the galvanometer When the switch is opened, the galvanometer needle deflects in the opposite direction and needle deflects in the opposite direction and then returns to zerothen returns to zero

The galvanometer reads zero when there is a The galvanometer reads zero when there is a steady current or when there is no current in steady current or when there is no current in the primary circuitthe primary circuit

Faraday’s Experiment – Conclusions Faraday’s Experiment – Conclusions

An electric current can be induced in a circuit by a An electric current can be induced in a circuit by a changing magnetic fieldchanging magnetic field This would be the current in the secondary circuit of this This would be the current in the secondary circuit of this

experimental set-upexperimental set-up The induced current exists only for a short time while the The induced current exists only for a short time while the

magnetic field is changing magnetic field is changing This is generally expressed as: This is generally expressed as: an induced emf is an induced emf is

produced in the secondary circuit by the changing produced in the secondary circuit by the changing magnetic fieldmagnetic field The actual existence of the magnetic flux is not sufficient to The actual existence of the magnetic flux is not sufficient to

produce the induced emf, the flux must be produce the induced emf, the flux must be changingchanging

Faraday’s Law – Statements Faraday’s Law – Statements

Faraday’s law of induction states that “the Faraday’s law of induction states that “the emfemf induced in a circuit is directly proportional induced in a circuit is directly proportional to the time rate of change of the magnetic flux to the time rate of change of the magnetic flux through the circuit”through the circuit”

Mathematically,Mathematically,

Bdε

dt

Faraday’s Law – StatementsFaraday’s Law – Statements

Remember Remember BB is the magnetic flux through the circuit is the magnetic flux through the circuit

and is found byand is found by

If the circuit consists of If the circuit consists of NN loops, all of the same area, loops, all of the same area, and if and if BB is the flux through one loop, an emf is is the flux through one loop, an emf is

induced in every loop and Faraday’s law becomesinduced in every loop and Faraday’s law becomes

B d B A

Bdε N

dt

Faraday’s Law – Example Faraday’s Law – Example

Assume a loop enclosing Assume a loop enclosing an area an area AA lies in a lies in a uniform magnetic field uniform magnetic field BB

The magnetic flux The magnetic flux through the loop is through the loop is

The induced emf isThe induced emf is

cosBAB

cosBAdt

d

Ways of Inducing an emfWays of Inducing an emf

The magnitude of The magnitude of BB can change with time can change with time The area enclosed by the loop can The area enclosed by the loop can

change with timechange with time The angle The angle between between BB and the normal to and the normal to

the loop can change with timethe loop can change with time Any combination of the above can occurAny combination of the above can occur

Applications of Faraday’s Law – Applications of Faraday’s Law – GFI GFI

A A GFIGFI (ground fault indicator) (ground fault indicator) protects users of electrical protects users of electrical appliances against electric shockappliances against electric shock

When the currents in the wires When the currents in the wires are in opposite directions, the flux are in opposite directions, the flux is zerois zero

When the return current in wire When the return current in wire 2 2 changes, the flux is no longer changes, the flux is no longer zerozero

The resulting induced The resulting induced emfemf can be can be used to trigger a circuit breakerused to trigger a circuit breaker

Applications of Faraday’s Law – Applications of Faraday’s Law – Pickup CoilPickup Coil

The pickup coil of an electric The pickup coil of an electric guitar uses Faraday’s lawguitar uses Faraday’s law

The coil is placed near the The coil is placed near the vibrating string and causes a vibrating string and causes a portion of the string to become portion of the string to become magnetizedmagnetized

When the string vibrates at the When the string vibrates at the same frequency, the same frequency, the magnetized segment produces magnetized segment produces a changing flux through the coila changing flux through the coil

The induced The induced emfemf is fed to an is fed to an amplifieramplifier

Motional Motional emfemf

A A motional motional emfemf is is one induced in a one induced in a conductor moving conductor moving through a constant through a constant magnetic fieldmagnetic field

The electrons in the The electrons in the conductor conductor experience a force, experience a force, FFBB = = qqv x B v x B that is that is directed along directed along ℓℓ

Motional emfMotional emf

Under the influence of the force, the electrons Under the influence of the force, the electrons move to the lower end of the conductor and move to the lower end of the conductor and accumulate thereaccumulate there

As a result of the charge separation, an electric As a result of the charge separation, an electric field field EE is produced inside the conductor is produced inside the conductor

The charges accumulate at both ends of the The charges accumulate at both ends of the conductor until they are in equilibrium with regard conductor until they are in equilibrium with regard to the electric and magnetic forcesto the electric and magnetic forces

Motional emfMotional emf

For equilibrium, For equilibrium, qEqE = = qvBqvB or or EE = = vBvB A potential difference is maintained A potential difference is maintained

between the ends of the conductor as long between the ends of the conductor as long as the conductor continues to move as the conductor continues to move through the uniform magnetic fieldthrough the uniform magnetic field

If the direction of the motion is reversed, If the direction of the motion is reversed, the polarity of the potential difference is the polarity of the potential difference is also reversedalso reversed

Sliding Conducting BarSliding Conducting Bar

A bar moving through a uniform field and the equivalent A bar moving through a uniform field and the equivalent circuit diagramcircuit diagram

Assume the bar has zero resistanceAssume the bar has zero resistance The work done by the applied force appears as internal The work done by the applied force appears as internal

energy in the resistor energy in the resistor RR

Active Figure 31.10

(SLIDESHOW MODE ONLY)

Sliding Conducting BarSliding Conducting Bar

The induced emf isThe induced emf is

Since the resistance in the circuit is Since the resistance in the circuit is RR, the , the current is current is

Bd dxε B B v

dt dt

Iε B v

R R

Sliding Conducting Bar, Energy Sliding Conducting Bar, Energy ConsiderationsConsiderations

The applied force does work on the conducting barThe applied force does work on the conducting bar This moves the charges through a magnetic fieldThis moves the charges through a magnetic field The change in energy of the system during some time The change in energy of the system during some time

interval must be equal to the transfer of energy into the interval must be equal to the transfer of energy into the system by worksystem by work

The power input is equal to the rate at which energy is The power input is equal to the rate at which energy is delivered to the resistordelivered to the resistor

2

app Iε

F v B vR

Lenz’s LawLenz’s Law

Faraday’s law indicates that the induced Faraday’s law indicates that the induced emf emf and the change in flux have opposite and the change in flux have opposite algebraic signsalgebraic signs

This has a physical interpretation that has This has a physical interpretation that has come to be known as come to be known as Lenz’s lawLenz’s law

Developed by German physicist Heinrich Developed by German physicist Heinrich LenzLenz

Lenz’s LawLenz’s Law

Lenz’s lawLenz’s law: : the induced current in a loop is in the induced current in a loop is in the direction that creates a magnetic field that the direction that creates a magnetic field that opposes the change in magnetic flux through opposes the change in magnetic flux through the area enclosed by the loopthe area enclosed by the loop

The induced current tends to keep the The induced current tends to keep the original magnetic flux through the circuit from original magnetic flux through the circuit from changingchanging

Induced emf and Electric FieldsInduced emf and Electric Fields

An electric field is created in the conductor as An electric field is created in the conductor as a result of the changing magnetic fluxa result of the changing magnetic flux

Even in the absence of a conducting loop, a Even in the absence of a conducting loop, a changing magnetic field will generate an changing magnetic field will generate an electric field in empty spaceelectric field in empty space

This induced electric field is This induced electric field is nonconservativenonconservative Unlike the electric field produced by stationary Unlike the electric field produced by stationary

chargescharges

Induced emf and Electric FieldsInduced emf and Electric Fields

The The emfemf for any closed path can be for any closed path can be expressed as the line integral of expressed as the line integral of EE..ddss over the over the pathpath

Faraday’s law can be written in a general Faraday’s law can be written in a general form:form:

dt

ddsE B

Induced Induced emfemf and Electric Fields and Electric Fields

The induced electric field is a The induced electric field is a nonconservativenonconservative field that is generated by a field that is generated by a changing magnetic fieldchanging magnetic field

The field cannot be an electrostatic field The field cannot be an electrostatic field because if the field were electrostatic, and because if the field were electrostatic, and hence conservative, the line integral of hence conservative, the line integral of EE..ddss would be zero and it isn’twould be zero and it isn’t

GeneratorsGenerators

Electric generators Electric generators take in energy by work take in energy by work and transfer it out by and transfer it out by electrical transmissionelectrical transmission

The The ACAC generator generator consists of a loop of consists of a loop of wire rotated by some wire rotated by some external means in a external means in a magnetic fieldmagnetic field

Rotating LoopRotating Loop

Assume a loop with Assume a loop with NN turns, all of the same area turns, all of the same area rotating in a magnetic fieldrotating in a magnetic field

The flux through the loop The flux through the loop at any time at any time tt is is

BB = = BABA cos cos = = BABA cos costt

Induced Induced emfemf in a Rotating Loop in a Rotating Loop

The induced emf in The induced emf in the loop is the loop is

This is sinusoidal, This is sinusoidal, with with maxmax = = NABNAB

sin

Bdε N

dtNABω ωt

Active Figure 31.21

(SLIDESHOW MODE ONLY)

Induced emf in a Rotating LoopInduced emf in a Rotating Loop

maxmax occurs when occurs when tt = 90 = 90oo or or 270270oo

This occurs when the magnetic field is in This occurs when the magnetic field is in the plane of the coil and the time rate of the plane of the coil and the time rate of change of flux is a maximumchange of flux is a maximum

= 0 = 0 when when tt = 0 = 0oo or or 180180oo

This occurs when This occurs when BB is perpendicular to the is perpendicular to the plane of the coil and the time rate of plane of the coil and the time rate of change of flux is zerochange of flux is zero

DC GeneratorsDC Generators

The The DC DC (direct current) (direct current) generator has generator has essentially the same essentially the same components as the components as the ACAC generatorgenerator

The main difference is The main difference is that the contacts to the that the contacts to the rotating loop are made rotating loop are made using a split ring called using a split ring called a a commutatorcommutator

DC GeneratorsDC Generators

In this configuration, the In this configuration, the output voltage always has output voltage always has the same polaritythe same polarity

It also pulsates with timeIt also pulsates with time To obtain a steady DC To obtain a steady DC

current, commercial current, commercial generators use many coils generators use many coils and commutators and commutators distributed so the pulses distributed so the pulses are out of phaseare out of phase

Active Figure 31.23

(SLIDESHOW MODE ONLY)

Motors Motors are devices into which energy is Motors are devices into which energy is

transferred by electrical transmission while transferred by electrical transmission while energy is transferred out by workenergy is transferred out by work

A motor is a generator operating in reverseA motor is a generator operating in reverse

A current is supplied to the coil by a battery A current is supplied to the coil by a battery and the torque acting on the current-carrying and the torque acting on the current-carrying coil causes it to rotatecoil causes it to rotate

MotorsMotors

Useful mechanical work can be done by Useful mechanical work can be done by attaching the rotating coil to some external attaching the rotating coil to some external devicedevice

However, as the coil rotates in a magnetic However, as the coil rotates in a magnetic field, an field, an emfemf is induced is induced This induced emf always acts to reduce the This induced emf always acts to reduce the

current in the coilcurrent in the coil The back The back emfemf increases in magnitude as the increases in magnitude as the

rotational speed of the coil increasesrotational speed of the coil increases

MotorsMotors

The current in the rotating coil is limited by The current in the rotating coil is limited by the back the back emfemf The term The term back emfback emf is commonly used to indicate is commonly used to indicate

an an emf emf that tends to reduce the supplied currentthat tends to reduce the supplied current The induced The induced emfemf explains why the power explains why the power

requirements for starting a motor and for requirements for starting a motor and for running it are greater for heavy loads than for running it are greater for heavy loads than for light oneslight ones

Eddy Currents Eddy Currents Circulating currents called Circulating currents called

eddy currentseddy currents are induced in are induced in bulk pieces of metal moving bulk pieces of metal moving through a magnetic fieldthrough a magnetic field

The eddy currents are in The eddy currents are in opposite directions as the plate opposite directions as the plate enters or leaves the fieldenters or leaves the field

Eddy currents are often Eddy currents are often undesirable because they undesirable because they represent a transformation of represent a transformation of mechanical energy into internal mechanical energy into internal energyenergy

Active Figure 31.26

(SLIDESHOW MODE ONLY)

Maxwell’s Equations, IntroductionMaxwell’s Equations, Introduction

Maxwell’s equations are regarded as the Maxwell’s equations are regarded as the basis of all electrical and magnetic basis of all electrical and magnetic phenomenaphenomena

Maxwell’s equations represent the laws of Maxwell’s equations represent the laws of electricity and magnetism that have already electricity and magnetism that have already been discussed, but they have additional been discussed, but they have additional important consequencesimportant consequences

Maxwell’s EquationsMaxwell’s Equations

LawMaxwellAmperedt

dIsdB

LawsFaradaydt

dsdE

magnetisminLawsGaussAdB

electricLawsGaussq

AdE

E

B

000

0

'

'0

)('

Maxwell’s EquationsMaxwell’s Equations

Gauss’s law (electrical): Gauss’s law (electrical):

The total electric flux through any closed The total electric flux through any closed surface equals the net charge inside that surface equals the net charge inside that surface divided by surface divided by oo

This relates an electric field to the charge This relates an electric field to the charge distribution that creates itdistribution that creates it

0q

AdE

Maxwell’s EquationsMaxwell’s Equations

Gauss’s law (magnetism): Gauss’s law (magnetism): The total magnetic flux through any closed The total magnetic flux through any closed

surface is zerosurface is zero This says the number of field lines that enter This says the number of field lines that enter

a closed volume must equal the number that a closed volume must equal the number that leave that volumeleave that volume

This implies the magnetic field lines cannot This implies the magnetic field lines cannot begin or end at any pointbegin or end at any point

Isolated magnetic monopoles have not been Isolated magnetic monopoles have not been observed in natureobserved in nature

0 AdB

Maxwell’s EquationsMaxwell’s Equations

Faraday’s law of Induction: Faraday’s law of Induction: This describes the creation of an electric field This describes the creation of an electric field

by a changing magnetic fluxby a changing magnetic flux The law states that the The law states that the emfemf, which is the line , which is the line

integral of the electric field around any closed integral of the electric field around any closed path, equals the rate of change of the path, equals the rate of change of the magnetic flux through any surface bounded magnetic flux through any surface bounded by that pathby that path

One consequence is the current induced in a One consequence is the current induced in a conducting loop placed in a time-varying conducting loop placed in a time-varying BB

dt

dsdE B

Maxwell’s EquationsMaxwell’s Equations

The Ampere-Maxwell law is a generalization The Ampere-Maxwell law is a generalization of Ampere’s lawof Ampere’s law

It describes the creation of a magnetic field It describes the creation of a magnetic field by an electric field and electric currentsby an electric field and electric currents

The line integral of the magnetic field around The line integral of the magnetic field around any closed path is the given sumany closed path is the given sum

dt

dIsdB E

000

The Lorentz Force LawThe Lorentz Force Law

Once the electric and magnetic fields are Once the electric and magnetic fields are known at some point in space, the force known at some point in space, the force acting on a particle of charge acting on a particle of charge qq can be can be calculatedcalculated

F = F = qqE + E + qqv x Bv x B This relationship is called the This relationship is called the Lorentz force Lorentz force

lawlaw Maxwell’s equations, together with this force Maxwell’s equations, together with this force

law, completely describe all classical law, completely describe all classical electromagnetic interactionselectromagnetic interactions

Maxwell’s Equations, SymmetryMaxwell’s Equations, Symmetry

The two Gauss’s laws are symmetrical, apart The two Gauss’s laws are symmetrical, apart from the absence of the term for magnetic from the absence of the term for magnetic monopoles in Gauss’s law for magnetismmonopoles in Gauss’s law for magnetism

Faraday’s law and the Ampere-Maxwell law Faraday’s law and the Ampere-Maxwell law are symmetrical in that the line integrals of are symmetrical in that the line integrals of EE and and BB around a closed path are related to the around a closed path are related to the rate of change of the respective fluxesrate of change of the respective fluxes

Maxwell’s equations are of fundamental Maxwell’s equations are of fundamental importance to all of scienceimportance to all of science