1. Faraday's Law dS B

2. General Review Electrostatics Magnetostatics Electrodynamics
motion of “q” in external E-field
E-field generated by Sqi
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!

3. Today... Induction Effects Faraday’s Law (Lenz’ Law)
Energy Conservation with induced currents?
Faraday’s Law in terms of Electric Fields
Cool Applications

4. Lecture 16, Act 1 1A
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: × B
(a) A Clockwise Current (b) A Counterclockwise Current (c) No Current 1B
Now the loop is pulled to the right at a velocity v.
The Lorentz force will now give rise to: × B v
(a) A Clockwise Current
(b) A Counterclockwise Current
(c) No Current, due to Symmetry

5. Other Examples of Induction
1831: Michael Faraday did the previous experiment, and a few others: × B -v I
Move the magnet, not the loop. Here there is no Lorentz force but there was still an identical current.
This “coincidence” bothered Einstein, and eventually led him to the Special Theory of Relativity (all that matters is relative motion).
Decrease the strength of B. Now nothing is moving, but M.F. still saw a current: × B
Decreasing B↓ I

6. Induction Effects from Currentsa b
Switch closed (or opened)
Þ current induced in coil b
Steady state current in coil a
Þ no current induced in coil b
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?

7. Faraday's Law
Define the flux of the magnetic field through an open surface as: dS B
Faraday's Law:
The emf e induced in a circuit is determined by the time rate of change of the magnetic flux through that circuit.
So what is
this emf??
The minus sign indicates direction of induced current (given by Lenz's Law).

8. Electro-Motive Force or emf
time
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”:
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!

9. Escher depiction of nonconservative emf

10. Lenz's Law B B v v Lenz's Law:
The induced current will appear in such a direction that it opposes the change in flux that produced it. v B S N v B N S
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..

11. Lecture 16, Act 2 (a) ccw (b) cw (c) no induced current (a) ccw (b) cw
X X X X X X X X X X X X v x y
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
(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? y I 2B v x
(a) ccw
(b) cw
(c) no induced current

12. Lecture 16, Act 2 (a) ccw (b) cw (c) no induced current
X X X X X X X X X X X X v x y
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 1A
(a) ccw
(b) cw
(c) no induced current
There is a non-zero flux FB 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!!

13. 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?
x x x x x x I v w x
Lenz’s Law  clockwise!
What is the magnitude? Þ
Magnetic Flux:
Faraday’s Law:
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).

14. Demo E-M Cannon v ~ side view · · top view
Connect solenoid to a source of alternating voltage. ~
side 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. B
top view F
There will then be a force on the ring since it contains a current which is circulating in the presence of a magnetic field.

15. Lecture 16, Act 3 (a) F2 < F1 (b) F2 = F1 (c) F2 > F1
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. 3A
(a) F2 < F1
(b) F2 = F1
(c) F2 > F1 3B
Suppose two identically shaped rings are used in the demo. Ring 1 is made of copper (resistivity = 1.7X10-8 W-m); Ring 2 is made of aluminum (resistivity = 2.8X10-8 W-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

16. 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?

17. 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

18. 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! 4 N S
rails
“eddy” current

19. Lecture 16, Act 4 (c) E is cw (a) E is ccw (b) E = 0 (a) E2R = ER
x x x x
x x x x x x
x x x x x x x x t B z 1 x y R
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
(a) E is ccw
(b) E = 0
(c) E is cw
What is the relation between the magnitudes of the induced electric fields ER at radius R and E2R at radius 2R? 4B
(a) E2R = ER
(b) E2R = 2ER
(c) E2R = 4ER

20. Faraday’s Law in terms of Electric Fields
Summary
Faraday’s Law (Lenz’s Law)
a changing magnetic flux through a loop induces a current in that loop
negative sign indicates that the induced EMF opposes the change in flux
Faraday’s Law in terms of Electric Fields

21. Appendix: Energy Conservation?
x x x x x x v w x I F '
The induced current gives rise to a net magnetic force ( ) on the loop which opposes the motion. F R v B w wB
wBv
IwB F 2 = ÷ ø ö ç è æ
Agent must exert equal but
opposite force to move the loop with velocity v; therefore, agent does work at rate P, where R v B w Fv P 2 =
Energy is dissipated in circuit at rate P' R v B w
wBv I P 2 = ÷ ø ö ç è æ Þ
P = P' !