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Chapter 31

Faradays Law

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Michael Faraday Great experimental

physicist

1791 1867 Contributions to early

electricity include: Invention of motor,

generator, and

transformer Electromagnetic

induction

Laws of electrolysis

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Induction An induced currentis produced by a

changing magnetic field

There is an induced emfassociatedwith the induced current

A current can be produced without a

battery present in the circuit Faradays law of induction describes the

induced emf

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EMF Produced by a Changing

Magnetic Field, 1 A loop of wire is

connected to a

sensitive ammeter When a magnet ismoved toward theloop, the ammeter

deflects The direction waschosen to be towardthe right arbitrarily

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Magnetic Field, 2 When the magnet is

held stationary,

there is nodeflection of the

ammeter

Therefore, there is

no induced current Even though the

magnet is in the loop

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Magnetic Field, 3 The magnet is moved

away from the loop

The ammeter deflects in

the opposite direction

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Active Figure 31.1

(SLIDESHOW MODE ONLY)

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Magnetic Field, Summary The ammeter deflects when the magnet is

moving toward or away from the loop

The ammeter also deflects when the loop ismoved toward or away from the magnet Therefore, the loop detects that the magnet is

moving relative to it

We relate this detection to a change in themagnetic field

This is the induced current that is produced by aninduced emf

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Faradays Experiment

Set Up A primary coil is connected to

a switch and a battery

The wire is wrapped aroundan iron ring

A secondary coil is also

wrapped around the iron ring

There is no battery present inthe secondary coil

The secondary coil is not

directly connected to the

primary coil

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Active Figure 31.2


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Faradays Experiment

Findings At the instant the switch is closed, the

galvanometer (ammeter) needle deflects in

one direction and then returns to zero When the switch is opened, the galvanometer

needle deflects in the opposite direction and

then returns to zero

The galvanometer reads zero when there is a

steady current or when there is no current in

the primary circuit

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Faradays Experiment

Conclusions An electric current can be induced in a circuit

by a changing magnetic field

This would be the current in the secondary circuitof this experimental set-up

The induced current exists only for a shorttime while the magnetic field is changing

This is generally expressed as: an inducedemf is produced in the secondary circuitby the changing magnetic field The actual existence of the magnetic flux is not

sufficient to produce the induced emf, the fluxmust be changing

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Faradays Law Statements Faradays law of induction states that

the emf induced in a circuit is directly

proportional to the time rate of changeof the magnetic flux through the circuit

Mathematically,

Bddt

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Faradays Law Statements,

cont RememberB is the magnetic flux

through the circuit and is found by

If the circuit consists ofNloops, all of

the same area, and ifB

is the flux

through one loop, an emf is induced in

every loop and Faradays law becomes

Bd B A

Bd

N dt

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Faradays Law Example Assume a loop

enclosing an areaAlies in a uniformmagnetic field B

The magnetic fluxthrough the loop is

B= BA cos

The induced emf is

= - d/dt(BA cos )

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Ways of Inducing an emf The magnitude ofB can change with

time

The area enclosed by the loop can

change with time

The angle between B and the normal

to the loop can change with time Any combination of the above can

occur

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Applications of Faradays Law

GFI A GFI (ground fault indicator)

protects users of electrical

appliances against electric

shock When the currents in the

wires are in opposite

directions, the flux is zero

When the return current inwire 2 changes, the flux is no

longer zero

The resulting induced emf

can be used to trigger a

circuit breaker

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Applications of Faradays Law

Pickup Coil The pickup coil of an electric

guitar uses Faradays law

The coil is placed near the

vibrating string and causes aportion of the string to become

magnetized

When the string vibrates at the

same frequency, themagnetized segment produces

a changing flux through the coil

The induced emf is fed to an

amplifier

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Motional emf A motional emfis

one induced in aconductor movingthrough a constantmagnetic field

The electrons in theconductorexperience a force,FB = qv x B that is

directed along

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Motional emf, cont. Under the influence of the force, the electrons

move to the lower end of the conductor and

accumulate there As a result of the charge separation, an

electric field E is produced inside the

conductor

The charges accumulate at both ends of the

conductor until they are in equilibrium with

regard to the electric and magnetic forces

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Motional emf, final For equilibrium, qE= qvB orE= vB A potential difference is maintained

between the ends of the conductor aslong as the conductor continues tomove through the uniform magneticfield

If the direction of the motion is reversed,the polarity of the potential difference isalso reversed

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Sliding Conducting Bar

A bar moving through a uniform field and the

equivalent circuit diagram Assume the bar has zero resistance

The work done by the applied force appears as

internal energy in the resistorR

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Active Figure 31.10


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Sliding Conducting Bar, cont. The induced emf is

Since the resistance in the circuit is R,

the current is

Bd dx

B B v dt dt

l l

I B v

R R

l

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Sliding Conducting Bar,

Energy Considerations The applied force does work on the conducting

bar

This moves the charges through a magnetic field The change in energy of the system during some

time interval must be equal to the transfer of

energy into the system by work

The power input is equal to the rate at which

energy is delivered to the resistor

2

appI

F v B v

R

l

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Lenzs Law Faradays law indicates that the induced

emf and the change in flux have

opposite algebraic signs This has a physical interpretation that

has come to be known as Lenzs law

Developed by German physicistHeinrich Lenz

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Lenzs Law, cont. Lenzs law: the induced current in a

loop is in the direction that creates a

magnetic field that opposes the changein magnetic flux through the area

enclosed by the loop

The induced current tends to keep theoriginal magnetic flux through the circuit

from changing

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Induced emf and Electric

Fields An electric field is created in the conductor

as a result of the changing magnetic flux

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

This induced electric field is

nonconservative Unlike the electric field produced by stationary

charges

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Fields, cont. The emf for any closed path can be

expressed as the line integral ofE.ds

over the path Faradays law can be written in a

general form:

Bd

ddt

E s

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Fields, final The induced electric field is a

nonconservative field that is generated

by a changing magnetic field The field cannot be an electrostatic field

because if the field were electrostatic,

and hence conservative, the lineintegral ofE.ds would be zero and it

isnt

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Generators Electric generators

take in energy by work

and transfer it out byelectrical transmission

The AC generator

consists of a loop of

wire rotated by someexternal means in a

magnetic field

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Rotating Loop Assume a loop with

Nturns, all of the

same area rotatingin a magnetic field

The flux through the

loop at any time tis

B = BA cos =

BA cos t

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Induced emf in a Rotating

Loop The induced emf in

the loop is

This is sinusoidal,with max = NAB

sin

Bd Ndt

NAB t

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Active Figure 31.21


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Induced emf in a Rotating

Loop, cont. max occurs when t= 90

o or 270o

This occurs when the magnetic field is in

the plane of the coil and the time rate ofchange of flux is a maximum

= 0 when t= 0o or 180o

This occurs when B is perpendicular to theplane of the coil and the time rate of

change of flux is zero

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DC Generators The DC (direct

current) generatorhas essentially thesame componentsas the AC generator

The main differenceis that the contactsto the rotating loopare made using asplit ring called acommutator

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DC Generators, cont. In this configuration, the

output voltage always has

the same polarity

It also pulsates with time

To obtain a steady DC

current, commercial

generators use many coils

and commutatorsdistributed so the pulses

are out of phase

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Active Figure 31.23


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Motors Motors are devices into which energy is

transferred by electrical transmission

while energy is transferred out by work A motor is a generator operating in

reverse

A current is supplied to the coil by abattery and the torque acting on thecurrent-carrying coil causes it to rotate

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Motors, cont. Useful mechanical work can be done by

attaching the rotating coil to some

external device However, as the coil rotates in a

magnetic field, an emf is induced This induced emf always acts to reduce the

current in the coil The back emf increases in magnitude as

the rotational speed of the coil increases

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Motors, final The current in the rotating coil is limited

by the back emf

The term back emfis commonly used toindicate an emf that tends to reduce thesupplied current

The induced emf explains why the

power requirements for starting a motorand for running it are greater for heavyloads than for light ones

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Eddy Currents Circulating currents called

eddy currents are induced in

bulk pieces of metal moving

through a magnetic field The eddy currents are in

opposite directions as the plate

enters or leaves the field

Eddy currents are often

undesirable because they

represent a transformation of

mechanical energy into internal

energy

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Active Figure 31.26


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Maxwells Equations,

Introduction Maxwells equations are regarded as

the basis of all electrical and magnetic

phenomena Maxwells equations represent the laws

of electricity and magnetism that have

already been discussed, but they haveadditional important consequences

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Maxwells Equations,

Statement

Gauss's law electric

0 Gauss's law in magnetism

Faraday's law

Ampere-Maxwell lawI

oS

S

B

Eo o o

qd

d

dd

dtd

d dt

E A

B A

E s

B s

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Maxwells Equations, Details Gausss law (electrical):

The total electric flux through any

closed surface equals the net chargeinside that surface divided by o

This relates an electric field to the

charge distribution that creates it

oS

qd

E A

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Maxwells Equations, Details 2 Gausss law (magnetism): The total magnetic flux through any closed

surface is zero This says the number of field lines that entera closed volume must equal the number thatleave that volume

This implies the magnetic field lines cannotbegin or end at any point Isolated magnetic monopoles have not been

observed in nature

0S

d B A

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Maxwells Equations, Details 3 Faradays law of Induction: This describes the creation of an electric field

by a changing magnetic flux The law states that the emf, which is the lineintegral of the electric field around any closedpath, equals the rate of change of the

magnetic flux through any surface boundedby that path One consequence is the current induced in a

conducting loop placed in a time-varying B

Bdddt

E s

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Maxwells Equations, Details 4 The Ampere-Maxwell law is a generalization

of Amperes law

It describes the creation of a magnetic field

by an electric field and electric currents

The line integral of the magnetic field around

any closed path is the given sum

I Eo o o dd dt B s

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The Lorentz Force Law Once the electric and magnetic fields are

known at some point in space, the forceacting on a particle of charge q can be

calculated F = qE + qv x B This relationship is called the Lorentz force

law Maxwells equations, together with this force

law, completely describe all classicalelectromagnetic interactions

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Maxwells Equations,

Symmetry The two Gausss laws are symmetrical, apart

from the absence of the term for magnetic

monopoles in Gausss law for magnetism Faradays law and the Ampere-Maxwell law

are symmetrical in that the line integrals ofE

and B around a closed path are related to the

rate of change of the respective fluxes Maxwells equations are of fundamental

importance to all of science

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