Chapter 23

Faraday’s Law

and

Induction

Michael Faraday 1791 – 1867 Great experimental

physicist Contributions to early

electricity include Invention of motor,

generator, and transformer

Electromagnetic induction

Laws of electrolysis

Induction An induced current is produced by a

changing magnetic field There is an induced emf associated with

the induced current A current can be produced without a

battery present in the circuit Faraday’s Law of Induction describes

the induced emf

EMF Produced by a Changing Magnetic Field, 1

A loop of wire is connected to a sensitive ammeter When a magnet is moved toward the loop, the

ammeter deflects The deflection indicates a current induced in the wire

EMF Produced by a Changing Magnetic Field, 2

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

Therefore, there is no induced current Even though the magnet is inside the loop

EMF Produced by a Changing Magnetic Field, 3

The magnet is moved away from the loop The ammeter deflects in the opposite

direction

EMF Produced by a Changing Magnetic Field, Summary The ammeter deflects when the magnet is

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

moved toward or away from the magnet An electric current is set up in the coil as long

as relative motion occurs between the magnet and the coil This is the induced current that is produced by an

induced emf

Faraday’s Experiment – Set Up

A primary coil is connected to a switch and a battery

The wire is wrapped around an iron ring

A secondary coil is also wrapped around the iron ring

There is no battery present in the secondary coil

The secondary coil is not electrically connected to the primary coil

Faraday’s 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

Faraday’s Experiment – Conclusions

An electric current can be produced by a time-varying magnetic field This would be the current in the secondary circuit of this

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

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

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

to produce the induced emf, the field must be changing

Magnetic Flux To express

Faraday’s finding mathematically, the magnetic flux is used

The flux depends on the magnetic field and the area:

B d B A

Flux and Induced emf An emf is induced in a circuit when the

magnetic flux through the surface bounded by the circuit changes with time This summarizes Faraday’s experimental

results

Faraday’s Law – Statements Faraday’s Law of Induction states that

the emf induced in a circuit is equal to the time rate of change of the magnetic flux through the circuit

Mathematically,

Bd

dt

Faraday’s Law – Statements, cont If the circuit consists of N identical and

concentric loops, and if the field lines pass through all loops, the induced emf becomes

The loops are in series, so the emfs in the individual loops add to give the total emf

BdN

dt

Faraday’s Law – Example Assume a loop

enclosing an area A lies in a uniform magnetic field

The magnetic flux through the loop is B = B A cos

The induced emf is cos

dBA

dt

Ways of Inducing an emf The magnitude of the field can change

with time The area enclosed by the loop can

change with time The angle between the field and the

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

Applications of Faraday’s Law – Pickup Coil

The pickup coil of an electric guitar uses Faraday’s Law

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

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

The induced emf is fed to an amplifier

Motional emf A motional emf is one

induced in a conductor moving through a magnetic field

The electrons in the conductor experience a force that is directed along l

B q F v B

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 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

Motional emf, final For equilibrium, q E = q v B or E = v B A potential difference is maintained

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

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

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 resistor R

Sliding Conducting Bar, cont The induced emf is

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

Bd dxB B v

dt dt

B vI

R R

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

appF v I B vR

AC Generators Electric generators take

in energy by work and transfer it out by electrical transmission

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

Induced emf In an AC Generator The induced emf in

the loops is

This is sinusoidal, with max = N A B

sin

BdN

dtNAB t

Lenz’ Law Faraday’s Law indicates the induced

emf and the change in flux have opposite algebraic signs

This has a physical interpretation that has come to be known as Lenz’ Law

It was developed by a German physicist, Heinrich Lenz

Lenz’ Law, cont Lenz’ Law states the polarity of the

induced emf in a loop is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop

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

Lenz’ Law – Example 1

When the magnet is moved toward the stationary loop, a current is induced as shown in a

This induced current produces its own magnetic field that is directed as shown in b to counteract the increasing external flux

Lenz’ Law – Example 2

When the magnet is moved away the stationary loop, a current is induced as shown in c

This induced current produces its own magnetic field that is directed as shown in d to counteract the decreasing external flux

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 an electric field in empty space

This induced electric field has different properties than a field produced by stationary charges

Induced emf and Electric Fields, cont The emf for any closed path can be

expressed as the line integral of over the path

Faraday’s Law can be written in a general form

Bdd

dt

E s

dE s

Induced emf and Electric 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 line integral would be zero and it isn’t

Self-Induction When the switch is

closed, the current does not immediately reach its maximum value

Faraday’s Law can be used to describe the effect

Self-Induction, 2 As the current increases with time, the

magnetic flux through the circuit loop due to this current also increases with time

The corresponding flux due to this current also increases

This increasing flux creates an induced emf in the circuit

Self-Inductance, 3 The direction of the induced emf is such that

it would cause an induced current in the loop which would establish a magnetic field opposing the change in the original magnetic field

The direction of the induced emf is opposite the direction of the emf of the battery Sometimes called a back emf

This results in a gradual increase in the current to its final equilibrium value

Self-Induction, 4 This effect is called self-inductance

Because the changing flux through the circuit and the resultant induced emf arise from the circuit itself

The emf L is called a self-induced emf

Self-Inductance, Equations An induced emf is always proportional

to the time rate of change of the current

L is a constant of proportionality called the inductance of the coil It depends on the geometry of the coil and

other physical characteristics

L

dIL

dt

Inductance of a Coil A closely spaced coil of N turns carrying

current I has an inductance of

The inductance is a measure of the opposition to a change in current Compared to resistance which was

opposition to the current

BNL

I dI dt

Inductance Units The SI unit of inductance is a Henry (H)

Named for Joseph Henry

AsV

1H1

Joseph Henry 1797 – 1878 Improved the design of

the electromagnet Constructed one of the

first motors Discovered the

phenomena of self-inductance

Inductance of a Solenoid Assume a uniformly wound solenoid

having N turns and length l Assume l is much greater than the radius

of the solenoid The interior magnetic field is

o o

NB nI I

Inductance of a Solenoid, cont The magnetic flux through each turn is

Therefore, the inductance is

This shows that L depends on the geometry of the object

B o

NABA I

2oB N AN

LI

RL Circuit, Introduction A circuit element that has a large self-

inductance is called an inductor The circuit symbol is We assume the self-inductance of the

rest of the circuit is negligible compared to the inductor However, even without a coil, a circuit will

have some self-inductance

RL Circuit, Analysis An RL circuit contains

an inductor and a resistor

When the switch is closed (at time t=0), the current begins to increase

At the same time, a back emf is induced in the inductor that opposes the original increasing current

RL Circuit, Analysis, cont Applying Kirchhoff’s Loop Rule to the

previous circuit gives

Looking at the current, we find

0dI

IR Ldt

1 Rt LI eR

RL Circuit, Analysis, Final The inductor affects the current

exponentially The current does not instantly increase

to its final equilibrium value If there is no inductor, the exponential

term goes to zero and the current would instantaneously reach its maximum value as expected

RL Circuit, Time Constant The expression for the current can also be

expressed in terms of the time constant, , of the circuit

where = L / R Physically, is the time required for the

current to reach 63.2% of its maximum value

1 tI t eR

RL Circuit, Current-Time Graph, 1

The equilibrium value of the current is /R and is reached as t approaches infinity

The current initially increases very rapidly

The current then gradually approaches the equilibrium value

RL Circuit, Current-Time Graph, 2 The time rate of

change of the current is a maximum at t = 0

It falls off exponentially as t approaches infinity

In general, tdI

edt L

Energy in a Magnetic Field In a circuit with an inductor, the battery

must supply more energy than in a circuit without an inductor

Part of the energy supplied by the battery appears as internal energy in the resistor

The remaining energy is stored in the magnetic field of the inductor

Energy in a Magnetic Field, cont Looking at this energy (in terms of rate)

I is the rate at which energy is being supplied by the battery

I2R is the rate at which the energy is being delivered to the resistor

Therefore, LI dI/dt must be the rate at which the energy is being delivered to the inductor

2 dII I R LI

dt

Energy in a Magnetic Field, final Let U denote the energy stored in the

inductor at any time The rate at which the energy is stored is

To find the total energy, integrate and UB = ½ L I2

BdU dILI

dt dt

Energy Density of a Magnetic Field Given U = ½ L I2,

Since Al is the volume of the solenoid, the magnetic energy density, uB is

This applies to any region in which a magnetic field exists not just the solenoid

2 221

2 2oo o

B BU n A A

n

2

2Bo

U Bu

V

Inductance Example – Coaxial Cable Calculate L and

energy for the cable The total flux is

Therefore, L is

The total energy is

ln2 2

bo o

B a

I I bBdA dr

r a

ln2

oB bL

I a

221

ln2 4

o I bU LI

a

Magnetic Levitation – Repulsive Model A second major model for magnetic

levitation is the EDS (electrodynamic system) model

The system uses superconducting magnets

This results in improved energy effieciency

Magnetic Levitation – Repulsive Model, 2 The vehicle carries a magnet As the magnet passes over a metal plate that

runs along the center of the track, currents are induced in the plate

The result is a repulsive force This force tends to lift the vehicle

There is a large amount of metal required Makes it very expensive

Japan’s Maglev Vehicle The current is

induced by magnets passing by coils located on the side of the railway chamber

EDS Advantages Includes a natural stabilizing feature

If the vehicle drops, the repulsion becomes stronger, pushing the vehicle back up

If the vehicle rises, the force decreases and it drops back down

Larger separation than EMS About 10 cm compared to 10 mm

EDS Disadvantages Levitation only exists while the train is in

motion Depends on a change in the magnetic flux Must include landing wheels for stopping and

starting The induced currents produce a drag force as

well as a lift force High speeds minimize the drag Significant drag at low speeds must be overcome

every time the vehicle starts up