Faraday's Law

dS

B B

SdBΦB

dt

dΦε B

General Review• Electrostatics

» motion of “q” in external E-field» E-field generated by qi

• Magnetostatics» motion of “q” and “I” in external B-field» B-field generated by “I”

• Electrodynamics» time dependent B-field generates E-field

• AC circuits, inductors, transformers, etc.» time dependent E-field generates B-field

• electromagnetic radiation - light!

Today...

• Induction Effects

• Faraday’s Law (Lenz’ Law)• Energy Conservation with induced

currents?

• Faraday’s Law in terms of Electric Fields

• Cool Applications

1

Lecture 16, Act 1 Inside the shaded region, there is

a magnetic field into the board.– If the loop is stationary, the Lorentz force (on the electrons in the wire) predicts:

1A

1B

(a) A Clockwise Current (b) A Counterclockwise Current (c) No Current

Now the loop is pulled to the right at a velocity v.– The Lorentz force will now give rise to:

×

B

(a) A Clockwise Current

(b) A Counterclockwise Current

(c) No Current, due to Symmetry

×

B v

Other Examples of Induction1831: Michael Faraday did the previous experiment, and a few others:

Move the magnet, not the loop. Here there is no Lorentz force but there was still an identical current.

v B×

B-v I

×

B

Decreasing B↓

IDecrease the strength of B. Now nothing is moving, but M.F. still saw a current:

This “coincidence” bothered Einstein, and eventually led him to the Special Theory of Relativity (all that matters is relative motion).

dBI

dt

Induction Effectsfrom Currents

• Switch closed (or opened)

current induced in coil b

• Steady state current in coil a

no current induced in coil b

ab

• Conclusion:

A current is induced in a loop when:

• there is a change in magnetic field through it

• this can happen many different ways

• How can we quantify this?

Faraday's Law

• Define the flux of the magnetic field through an open surface as:

• Faraday's Law:

The emf induced in a circuit is determined by the time rate of change of the magnetic flux through that circuit.

The minus sign indicates direction of induced current (given by Lenz's Law).

dt

dΦε B

dS

B B SdBΦB

So what is this emf??

Electro-Motive Force or emf

A magnetic field, increasing in time, passes through the blue loop

An electric field is generated “ringing” the increasing magnetic field

Circulating E-field will drive currents, just like a voltage difference

Loop integral of E-field is the “emf”:

time

Bdε E dl

dt

˜

Note: The loop does not have to be a wire—the emf exists even in vacuum! When we put a wire there, the electrons respond to the emf, producing a current.

Disclaimer: In Lect. 5 we claimed , so we could define a potential independent of path. This holds only for charges at rest (electrostatics). Forces from changing magnetic fields are nonconservative, and no potential can be defined!

0 ldE

Escher depiction of nonconservative emf

Lenz's Law• Lenz's Law:

The induced current will appear in such a direction that it opposes the change in flux that produced it.

• Conservation of energy considerations:

Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated.

» Why???

• If current reinforced the change, then the change would get bigger and that would in turn induce a larger current which would increase the change, etc..

v

BS N

v

BN S

2

Lecture 16, Act 2• A conducting rectangular loop moves with

constant velocity v in the +x direction through a region of constant magnetic field B in the -z direction as shown.

• What is the direction of the induced current in the loop?

(a) ccw (b) cw (c) no induced current

• A conducting rectangular loop moves with constant velocity v in the -y direction and a constant current I flows in the +x direction as shown.

• What is the direction of the induced current in the loop?

2A

X X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X X

v

x

y

(a) ccw (b) cw (c) no induced current

2B v

I

x

y

Lecture 16, Act 2

(a) ccw (b) cw (c) no induced current

1A

• There is a non-zero flux B passing through the loop since B is perpendicular to the area of the loop.

• Since the velocity of the loop and the magnetic field are CONSTANT, however, this flux DOES NOT CHANGE IN TIME.

• Therefore, there is NO emf induced in the loop; NO current will flow!!

• A conducting rectangular loop moves with constant velocity v in the +x direction through a region of constant magnetic field B in the -z direction as shown.– What is the direction of the induced current in

the loop?2A

X X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X X

v

x

y

Induced Current – quantitative Suppose we pull with velocity v a coil of resistance R through a region of constant magnetic field B. – Direction of induced current?

vw

x

I

– What is the magnitude?

R

wBv

R

εI

BΦ B Area Bwx » Magnetic Flux:

dxBw Bwv

dt

dt

dΦε B» Faraday’s Law:

x x x x x x

x x x x x x

x x x x x x

x x x x x x » Lenz’s Law clockwise!

We must supply a constant force to move the loop (until it is completely out of the B-field region). The work we do is exactly equal to the energy dissipated in the resistor, I2R (see Appendix).

DemoE-M Cannon v

• Connect solenoid to a source of alternating voltage.

F

F

B

B

B

top view

• The flux through the area to axis of solenoid therefore changes in time.

• A conducting ring placed on top of the solenoid will have a current induced in it opposing this change.

• There will then be a force on the ring since it contains a current which is circulating in the presence of a magnetic field.

~

side view

3

Lecture 16, Act 3• For this act, we will predict the results of variants of the

electromagnetic cannon demo which you just observed.– Suppose two aluminum rings are used in the demo;

Ring 2 is identical to Ring 1 except that it has a small slit as shown. Let F1 be the force on Ring 1; F2 be the force on Ring 2.

3B – Suppose two identically shaped rings are used in the demo. Ring 1 is made of copper (resistivity = 1.7X10-8 -m); Ring 2 is made of aluminum (resistivity = 2.8X10-8 -m). Let F1 be the force on Ring 1; F2 be the force on Ring 2.

(a) F2 < F1 (b) F2 = F1 (c) F2 > F1

3ARing 1

Ring 2

(a) F2 < F1 (b) F2 = F1 (c) F2 > F1

Applications of Magnetic Induction• AC Generator

– Water turns wheel rotates magnet changes flux induces emf drives current

• “Dynamic” Microphones(E.g., some telephones)

– Sound oscillating pressure waves oscillating [diaphragm + coil] oscillating magnetic flux oscillating induced emf oscillating current in wire

Question: Do dynamic microphones need a battery?

More Applications of Magnetic Induction

• Tape / Hard Drive / ZIP Readout– Tiny coil responds to change in flux as the magnetic domains

(encoding 0’s or 1’s) go by.

Question: How can your VCR display an image while paused?

• Credit Card Reader– Must swipe card generates changing flux– Faster swipe bigger signal

More Applications of Magnetic Induction• Magnetic Levitation (Maglev) Trains

– Induced surface (“eddy”) currents produce field in opposite direction

Repels magnet

Levitates train

– Maglev trains today can travel up to 310 mph Twice the speed of Amtrak’s fastest conventional train!

– May eventually use superconducting loops to produce B-field No power dissipation in resistance of wires!

N

S

rails“eddy” current

4

Lecture 16, Act 4• The magnetic field in a region of space of

radius 2R is aligned with the z-direction and changes in time as shown in the plot.

– What is the direction of the induced electric field around a ring of radius R at time t=t1?

4A

4B– What is the relation between the magnitudes of the induced electric fields ER at radius R and E2R at radius 2R?

(a) E2R = ER(b) E2R = 2ER (c) E2R = 4ER

x x x x x x x x x x x x x x x x x x x x x x x x x x x x

t

Bz

t1

x

y

R

(a) E is ccw (b) E = 0 (c) E is cw

Summary• Faraday’s Law (Lenz’s Law)

– a changing magnetic flux through a loop induces a current in that loop

dt

dΦε B

dt

dldE B

• Faraday’s Law in terms of Electric Fields

negative sign indicates that the induced EMF opposes the change in flux

SdBΦB

Appendix: Energy Conservation?

• Energy is dissipated in circuit at rate P'

• The induced current gives rise to a net magnetic force ( ) on the loop which opposes the motion.

F

R

vBwwB

R

wBvIwBF

22

RvBw

FvP222

P = P' !R

vBwR

R

wBvRIP

22222

• Agent must exert equal but

opposite force to move the loop with velocity v; therefore, agent does work at rate P, where

x x x x x x

x x x x x x

x x x x x x

x x x x x x

vw

x

I

F

F '

F '

R

wBvI