Chapter 31

Faraday’s Law

Michael Faraday

Great experimental physicist and chemist 1791 – 1867 Contributions to early electricity include:

Invention of motor, generator, and transformer Electromagnetic induction Laws of electrolysis: A method of separating

bonded elements and compounds by passing an electric current through them

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

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 in 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

Faraday’s Law – Statements Faraday’s law of induction states that

“the emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit”

Mathematically,

Bdε

dt

Faraday’s Law – Statements, cont

Remember B is the magnetic flux through the circuit and is found by

If the circuit consists of N loops, all of the same area, and if B is the flux through one loop, an emf is induced in every loop and Faraday’s law becomes

B d B A

Bdε N

dt

Faraday’s Law – Example Assume a loop

enclosing an area A lies in a uniform magnetic field B

The magnetic flux through the loop is B = BA cos

The induced emf is = - d/dt (BA cos )

Ways of Inducing an emf

The magnitude of B 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

Lenz’s Law

Faraday’s 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 Lenz’s law

Developed by German physicist Heinrich Lenz

Lenz’s Law, cont. Lenz’s law: the induced current in a

loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop

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

A loop of wire in the shape of a rectangle of width w and length L and a long, straight wire carrying a current I lie on a tabletop. Determine the magnetic flux through the loop due to the current I. Suppose the current is changing with time according to I = a + bt, where a and b are constants. Determine the emf that is induced. What is the direction of the induced current in the rectangle?

Problem 1

Problem 2The square loop in the figure is made of wires with a total resistance of 20.0  . It is placed in a uniform 0.100 T magnetic field directed perpendicular into the plane of the paper. The loop, which is hinged at each corner, is pulled as shown until the separation between points A and B is 3.00 m. What is the magnitude and direction of the induced current. [0.0301] A, clockwise

Problem 3

A 39 turn circular coil of radius 4.80 cm and resistance 1.00 is placed in a magnetic field directed perpendicular to the plane of the coil. The magnitude of the magnetic field varies in time according to the expression B = 0.0100t + 0.0400t2, where t is in seconds and B is in tesla. Calculate the induced emf in the coil at t = 5.60 s. [129 mV]

Applications of Faraday’s Law – Electric Generator

http://www.wvic.com/how-gen-works.htm

An electric conductor, like a copper wire, is moved through a magnetic field, which causes an electric current to flow (be induced) in the conductor

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

Applications of Faraday’s Law – Pickup Coil

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

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

The induced emf is fed to an amplifier

An alternating current in one winding creates a time-varying magnetic flux in the core, which induces a voltage in the other windings

Applications of Faraday’s Law – Transformers

Is this a step-up or a step- down transformer?

Yamanashi maglev The magnetized coil running along the track, called a guideway, repels the large magnets on the train's undercarriage, allowing the train to levitate between 0.39 and 3.93 inches (1 to 10 cm) above the guideway.

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 dxε B B v

dt dt

Iε B v

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

app Iε

F v B vR

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 is nonconservative Unlike the electric 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 E.ds over the path

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

dt

dds.E B

Maxwell’s Equations, Statement

dt

dds.E B Faraday’s Law

Ampere-Maxwell Law

s

dA.B 0 Gauss’s Law in Magnetism

s

qdA.E

0 Gauss’s Law in Electricity

dt

dIdl.B e

000

Maxwell’s Equations, Details Gauss’s law (electrical): The total electric flux through any

closed surface equals the net charge inside that surface divided by o

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

s

qdA.E

0

Maxwell’s Equations, Details 2 Gauss’s law (magnetism): The total magnetic flux through any closed

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

a closed volume must equal the number that leave that volume

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

Isolated magnetic monopoles have not been observed in nature

s

dA.B 0

Maxwell’s Equations, Details 3 Faraday’s 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 line

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

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

dt

dds.E B

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

of Ampere’s 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

dt

dIdl.B e

000

The Lorentz Force Law Once the electric and magnetic fields are

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

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

law Maxwell’s equations, together with this force

law, completely describe all classical electromagnetic interactions

Maxwell’s Equations, Symmetry The two Gauss’s laws are symmetrical, apart

from the absence of the term for magnetic monopoles in Gauss’s law for magnetism

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

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