Chapter 31

Faraday’s Law

Introduction

• This section we will focus on the last of the fundamental laws of electromagnetism, called Faraday’s Law of Induction

• Michael Faraday 1791-1867– Determined Laws of Electrolysis– Invented electric motor, generator, and transformer.

Introduction

• In this chapter we will look at the processes in which a magnetic field (more importantly, a change in the magnetic field) can induce an electric current.

31.1 Faraday’s Law of Induction

• An emf and therefore, a current can be induced in a circuit with the use of a magnet.

• The magnetic field by itself is not capable of inducing a current.

31.1

• A change in the magnetic field is necessary.– As the magnet is moved towards the current loop

a positive current is measured.

31.1

– As the magnet is moved away from loop a negative current is measured.

– Note that this also applies to stationary magnets and moving coils.

31.1

• Here is the basic setup of actual experiment conducted by Faraday to confirm this phenomenon.

31.1

• With the use of insulated wires, the first circuit and battery is completely isolated from the second circuit with the ammeter. – With the 1st circuit open, there is no reading in the

ammeter.– With the 1st circuit closed, there is no reading in the ammeter.

31.1

• The instant the switch is open, the ammeter needle deflects to one side and returns to zero.

• The instant the switch is closed the ammeter needle deflects to the opposite side and returns to zero.

31.1

• So its not the magnetic field that induces the current, but the change in magnetic field.

• Faraday’s Law of Induction– The emf induced in a circuit is directly

proportional to the time rate of change of magnetic flux through the circuit.

dt

d B

31.1

• If the circuit is a coil with N number of loops of the same area, then

• Assuming a uniform magnetic field the magnetic flux is equal to BAcosθ so

dt

dN B

cosBAdt

d

31.1

• So there are several things that change if there is going to be an induced current.– The magnitude of B can change with time. – The area enclosed by the loop can change with

time.– The angle , between B and the area vector can

change with time.– Any combination of the above.

31.1

• Quick quizzes p. 970-971

• Applications of Faraday’s Law– GFI- induced current in the coil trips the circuit breaker.

31.1

– Electric Guitar Pickups- the vibrating metal string induces a current in the coil.

31.1

• Example 31.1, 31.2

31.2 Motional EMFs

• Motional EMF- induced in a conductor moving through a constant magnetic field.

• Consider a conductor length ℓ, moving through a constant magnetic field B, with velocity v.

31.2

• The first thing we notice is that any free electrons (charge carriers) will feel a magnetic force as per FB = qv x B

• This will leave one end of the conductor with extra electrons, and the other with a deficit.

• This creates an electric field within the conductor which enacts a force on the electrons opposite of the magnetic force.

31.2

• The forces up and down will balance giving

• The electric field is associated with the potential difference and the length of the conductor

• This potential difference is maintained as long as the conductor continues to move with velocity v through the field.

qvBqE vBE

vBEV

31.2

• A more interesting example occurs when the conducting bar is part of a closed circuit.

• We assume zero resistancein the bar.• The rest of the circuit has resistance R.

31.2

• With the magnetic field present, and the conducting bar free to slide along the conducting rails, the same potential difference or EMF is produced, which drives a current through the circuit.

31.2

• This is another example of Faraday’s law where the induced current is proportional to the changing magnetic flux (increasing area).

• Because the area at any time is A = ℓx, the magnetic flux is given as

xBB

31.2

• From Faraday’s Law, the EMF will be

xBdt

d

dt

d B

dt

dxB

vB

31.2

• From this result and Ohm’s law, the induced current will be

• The source of the energy is the work done by the applied force.

R

vB

RI

RR

vBvBIvFP A

2222

31.2

• Quick Quizzes p 975• Ex 31.4, 31.5

31.3 Lenz’s Law

• Faraday’s Law indicates that the induced emf and the change in flux have opposite signs.

• This physical effect is known as 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.

31.3

• We will look at the sliding conductor example to illustrate.

• In this picture the magnetic flux is increasing.• Since the magnetic field is into the page, the current induced creates a magnetic field out of the page.

31.3

• If we switch the direction of travel for the bar, the flux through the loop is decreasing.

• The current is induced to oppose that change and creates additional magnetic field into the page.

31.3

• We can examine the bar magnet and loop example again.

31.3

31.3

• Quick Quizzes p. 979• Conceptual Example 31.6

• Induced Current– The instant the switch closes– After a few seconds– The instant the switch is opened.

31.4 Induced EMF and Electric Fields

• An E-field within a conductor is responsible for moving charges through circuit.

• Since Faraday’s law discusses induced currents, we can claim that the changing magnetic field creates an E-field within the conductor.

31.4

• In fact, a changing magnetic field generates an electric field even without a conducting loop.

• The E-field is however non-conservative unlike electrostatic fields.

• The work to move a charge around the loop is given as

rqEFdqW 2

31.4

• The electric field in the ring is given as

• Knowing this and the fact that

• We can apply Faraday’s Law to get

rE

2

dt

dBr

dt

d

rE B

22

1

BrBAB2

31.4

• So if we have B as a function of time, the induced current can easily be determined.

• The emf for any closed path can be given as the line integral of E.ds so Faraday’s Law is often given in the general form

dt

dd BsE

31.4

• The most important conclusion from this is the fact that a changing magnetic field, creates and electric field.

• Quick quiz p 982• Example 31.8

31.5 Generators and Motors

• Faraday’s Law has a primary application in Generators and Motors

• AC Generator-– Work is done to rotate a loop of wire in a

magnetic field.– The changing magnetic flux creates an emf that

alternates between positive and negative.

31.5

31.5

• If we look at our rotating loop, the flux through single turn is given as

• And assuming a constant rotational speed of ω,

• Where θ = 0 at t = 0.

cosBAB

t

tBAB cos

31.5

• If we have more than 1 loop, say N loops, then Faraday’s Law gives the emf produced as

dt

dN B

tdt

dNAB cos

tNAB sin

31.5

• The maximum emf produced is given as

• When ωt = 90o and 270o

• Omega is named the angular frequency and is given as ω = 2πf, where f is the frequency in Hz.

• Commercial generators in the US operate at f = 60 Hz.

NABmax

31.5

• Quick Quiz p. 984• Example 31.9

• DC Generators– Operation very similar two AC Generators– Instead of 2 rings, a DC generator uses one split

ring, called a commutator.

31.5

• Commutator flips the polarity of the brushes in sync with the rotating loop, ensuring all emf is of one sign.

• While the emf is always positive, it pulses with time.

31.5

• Pulsing DC current is not suitable for most applications, so multiple coil/commutator combos oriented at different angles are used simultaneously.

• By superimposing the emf pulses, we get a very nearly steady value.

31.5

• Motors- Make use of electrical energy to do work.

• Generator operating in reverse-– Current is supplied so a loop in a magnetic field.– The torque on the loop causes rotation which can be applied to work.

31.5

• The problem is we also have an emf induced because the magnetic flux changes as the loop rotates.

• From Lenz’s law this emf opposes the current running through the loop and is typically called a “Back emf”

31.5

• When the motor is initially turned on the back emf is zero.

• As it speeds up the back emf increases. • If a load is attached to the motor (to do work)

the speed will drop and therefore back emf will as well.

• This draws higher than normal current from the voltage source running the motor.

31.5

• If the load jams the motor, and it stops the motor can quickly burn out, from the increased current draw.

• Example 31.10

31.6 Eddy Currents

• Eddy Current- A circular current induced in a bulk piece of conductor moving through magnetic field.

31.6

• By Lenz’s law the induced current opposes the changing flux and therefore creates a magnetic field on the conductor, that opposes the source magnetic field.

• Because of this the passing conductor behaves like an opposing magnetic and the force is resistive.

31.6

31.6

• The concept is applied to mass transit braking systems which combine electromagnetic induction and Eddy currents to steadly slow subways/trains etc.

• Quick Quiz p. 987

31.7 Maxwell’s Equations

• James Clerk Maxwell developed a list of equations summarizing the fundamental nature of electricity and magnetism. – Gauss’s Law (Electric Fields)

• The total electric flux through a closed surface is proportional to the charge contained.

oS

qd

AE

31.7

– Gauss’s Law (Magnetic Fields)

• The total magnetic flux through a closed surface is zero.• Magnetic Monopoles have never been observed.

– Faraday’s Law of Induction

• Electric Fields are created by changing magnetic flux

0S

dAB

dt

dd B

sE

31.7

– Ampere-Maxwell Law

• Magnetic Fields are created by current• Magnetic Fields are created by changing electric flux.

dt

dId E

ooo

sE

31.7

• These 4 equations when joined with the Lorentz Force Law (below) completely describe all classical electromagnetic interactions.

• They are as fundamental to the understanding of the physical world as Newton’s Laws of Motion/Universal Gravitation

BvEF qq