1. Physics 121 - Magnetism Lecture 11 - Faraday’s Law of Induction Y&F Chapter 29, Sect. 1-5
Magnetic Flux
Motional EMF: moving wire in a B field
Two Magnetic Induction Experiments
Faraday’s Law of Induction
Lenz’s Law
Rotating Loops – Generator Principle
Concentric Coils – Transformer Principle
Induction and Energy Transfers
Induced Electric Fields
Summary

2. Quick summary of magnetism so far
Current-lengths in a magnetic field feel forces and torques q
Force on charge and wire carrying current
torque and potential energy of a dipole
Current-lengths (changing electric fields) produce magnetic fields
Ampere
Biot-Savart
Current loops are elementary dipoles
B outside a long straight wire carrying a current i:
B at center of circular loop carrying a current i :
Long Solenoid field: B inside a torus carrying
B outside = a current i :
B on the symmetry axis of a current loop (far field):
Next: Changing magnetic flux induces EMFs
and currents in wires (Faraday’ Law)

3. Magnetic Flux: r B defined analogously to flux of electric field
Electrostatic Gauss Law
Magnetic Gauss Law B r
Flux Unit: 1 Weber = 1 T.m 2
Surface Integral
over open or closed surface q

4. Changing magnetic flux induces EMFs
EMF / current is induced in a loop if there is relative motion between loop and magnet - the magnetic flux inside the loop is changing.
Faster motion produces a larger current.
Induced current stops when relative motion stops (case b).
Induced current direction reverses when magnet motion reverses direction (case c versus case a)
Any relative motion that changes the flux works
Motion is one of several ways to change flux
Changing magnetic flux causes an induced EMF (and possibly induced current) in a wire loop bounding the area. We really mean flux rather than just field is changing.
Induced current creates it’s own induced B field (and flux) that opposes the changing flux (Lenz’ Law)

5. CHANGING magnetic flux induces EMFs and currents in wires
Key Concepts:
Faraday’s Law of Induction
Magnetic flux
Area bounded
by a loop
Lenz’s Law
Basis of generator principle:
Loops rotating in B field generate EMF and current in load circuit.
Lenz’s Law implies that external torque must do work.

6. Motional EMF: Lorentz Force on moving charges in conductors
Uniform magnetic field points away from viewer.
Wire of length L moves with constant velocity v perpendicular to the field
Electrons in the wire feel a Lorentz force and migrate to the lower end of the wire. Upper end becomes positive.
Charge separation  an induced electric field Eind exists inside wire
Charges come to equilibrium when the forces on charges balance:
Electric field Eind in the wire corresponds to potential difference Eind across the ends of wire:
Potential difference Eind is maintained between the ends of the wire as long as it moves through the magnetic field. +
FIELD
BOUNDARY L v -
FBD of free charge q in a wire v + - Fe Fm
Eind
Induced EMF is created by the motion.
Induced current flows only if the circuit is completed.

7. Complementary Flux view for Motional EMF in a moving wire
FLUX F = BA B is uniform and constant.
The rate of flux change = [field B] x [ rate of sweeping out area ]

8. Flux change explains what happens even without motion
Simple transformer: changing current in solenoid causes current to flow in outer coil.
Field inside solenoid:
Field outside solenoid = 0 at location
of free charges in the 15 turn coil.
So: Lorentz force can’t cause current there.
Magnetic flux in solenoid also links to outer coil geometrically and is proportional to solenoid’s cross-section area (not the outer coil’s).
So what causes induced current to flow in outer coil?
Changing magnetic flux must create electric field at the outer coil
location that induces current in outer coil (like other currents).
Outer coil also creates its own field Bind, owing to induced current.

9. Examples: A rectangular loop is moving through a uniform B field
DOES CURRENT FLOW?
Segments a-b & d-c create equal but opposed motional EMFs in circuit.
No EMF from b-c & a-d or
FLUX in loop abcd remains constant
Net flux is constant
No current flows v + - a b c d
DOES CURRENT FLOW NOW?
B-field ends or is not uniform
Segment c-d now creates NO EMF
Segment a-b creates EMF as above
Un-balanced EMF drives current
just like a battery or
FLUX is DECREASING v + - a b c d
Which way does current flow? CW

10. Expanded Lenz’s Law Example:
As loop crosses a region of uniform magnetic field…
The induced current and EMF create induced magnetic flux that opposes the change in magnetic flux that created them
B = 0
B = uniform v

11. Directions of induced fields and currents
Replace moving magnet in sketch with circuit
No actual motion, switch closed
Current i1 creates field B1 (RH rule)
Flux is constant if current i1 is constant
Changing current i1 --> changing flux in loop 2.
Current i2 flows only while i1 is
changing (after switch S closes or opens)
Ammeter
Induced current i2 creates it’s own induced field B2 whose flux F2 opposes the change in F1 (Lenz’ Law)
Close S i2
Open S later
F1 = B1A
F2 = B2A
Induced Dipoles i1 i2

12. Faraday’s Law / Changing Flux
uniform B
Rate of flux change
(Webers/sec)
“-” sign for Lenz’s Law
Induced EMF
(Volts)
Insert N for multiple turns in loop. Loop of any shape
B(t)
EMF=0
slope  EMF
Several ways to change flux:
change |B| through a coil
change area of a coil or loop
change angle between B and coil
e.g., rotating coils  generator effect
Example:
B through a loop increases by 0.1 Tesla in 1 second: Loop area A = 10-3 m2 is constant & along field. Find the induced EMF
Current iind creates field Bind that opposes increase in B
uniform
dB/dt
front

13. + iind - x Slidewire Generator
Slider moves, increasing loop area (flux)
Induced EMF:
Slider: moving rightward at speed v
Uniform B field into the page
Loop area & FLUX are increasing + - x
iind L
Lenz’s Law says induced field Bind
is out of page.
RH rule says induced current is CCW
Slider acts like a battery (Lorentz)
What about i2R? Can v be constant?
Due to the induced current:
Opposes the motion of the wire
External force FEXT is needed to
keep wire from slowing down
Drag force FM must act on slider
FEXT needed for constant speed is opposed to Fm with equal magnitude
Mechanical power supplied & dissipated:

14. Slidewire Generator: Numerical Examplev + -
iind L
B = 0.35 T
L = 25 cm
v = 55 cm/s
a) Find the EMF generated:
DIRECTION: Bind is into slide, iind is clockwise
b) Find the induced current if resistance R for the whole loop = 18 W:
c) Find the thermal power dissipated:
d) Find the power needed to move slider at constant speed
!!! Power dissipated via R = Mechanical power !!!

15. Induced Current and Emf
11 – 1: A circular loop of wire is in a uniform magnetic field covering the area shown. The plane of the loop is perpendicular to the field lines.
Which of the following will not cause a current to be induced in the loop?
A. Sliding the loop into the field region from the far left
B. Rotating the loop about an axis perpendicular to the field lines.
C. Keeping the orientation of the loop fixed and moving it along
the field lines.
D. Crushing the loop.
E. Sliding the loop out of the field region from left to right B

16. induced dipole in loop opposes flux growth
Induced EMF drives induced current that creates induced magnetic flux opposing the change in magnetic flux that created them.
Lenz’s Law Again
front of loop
induced dipole in loop opposes flux growth
induced dipole in loop
opposes flux decrease
The induced current and field try to keep the original magnetic flux through the loop from changing.
front of loop

17. Direction of induced current
11-2: A circular loop of wire is falling toward a straight wire carrying a steady current to the left as shown.
What is the direction of the induced current
in the loop of wire?
Clockwise
Counterclockwise
Zero
Impossible to determine
I will agree with whatever the majority chooses v I
11-3: The loop continues falling until it is below the straight wire.
Now what is the direction of the induced current in the loop of wire?
Clockwise
Counterclockwise
Zero
Impossible to determine
I will oppose whatever the majority chooses

18. Rotation axis is out of slide
Generator Effect (Produce Sinusoidal AC)
Changing flux through a current loop rotating with
angular velocity w = 2pf in B field:
Rotation axis is out of slide q
peak flux magnitude when wt = 0, p, etc.
Induced EMF is the time derivative of the flux
Eind is sinusoidal – polarity alternates over a cycle
Maxima/minima when wt = +/- (2n+1)p/2
Eind peaks when dipole moment is perpendicular to Field.

19. No mechanical reversal of output E Mechanical Reversal of output E
AC Generator
No mechanical reversal of output E
DC Generator
Mechanical Reversal of output E

20. AC Generator DC Generator
Back-torque t=mxB in rotating loop, m ~ N.A.iind

21. Stationary flat coil with N turns in B field
Numerical
Example
Stationary flat coil with N turns in B field E
N turns
Each turn increases the flux and induced EMF
N = 1000 turns
B through coil decreases from +1.0 T to -1.0 T in 1/120 s.
Coil area A is 3 cm2 (one turn)
Find the “back EMF” induced in the coil by it’s own changing flux
Changing magnetic flux creates induced E field linking to wire turns
that creates “back EMF”.
, Back EMF induces current & magnetic field opposing
external field change.
The Lenz’s law effect is analogous to mechanical inertia.

22. Transformer Principle
Primary current is changing (for a short time) after switch closes.
 changing flux through primary coil that links to....
changing flux through secondary coil
 induced secondary current & EMF
PRIMARY
SECONDARY
Iron ring strengthens flux linkage

23. conducting loop, resistance R
Changing magnetic flux directly induces electric fields
conducting loop, resistance R
Bind
A thin solenoid, n turns/unit length, cross section A
Zero B field outside solenoid (it is ideal)
Field inside solenoid:
Flux linked to the wire loop:
Changing current I  changing flux 
EMF is induced in wire loop:
Current induced in the loop is:
If dI/dt is positive, B is growing, then Bind opposes change and I’ is counter-clockwise
B = 0 outside solenoid, so it’s not Lorentz force acting on charges
An induced electric field Eind along the loop causes current to flow
It is caused directly by dF/dt within the loop path
Eind is there even without the outer wire loop (no current flowing)
E-field field lines are loops that don’t terminate on charge.
E-field is a non-conservative (non-electrostatic) field inasmuch
as the line integral of E (potential) around a closed path is not zero
What induces current I’ to flow outside the solenoid?
Generalized Faradays’ Law
(hold loop path constant)

24. Example: Generalized Faraday’s Law Electric field induced by a region of changing magnetic flux
Find the magnitude Eind of the induced electric field at points outside and within the magnetic field region.
Assume:
dB/dt = uniform across the circular shaded area.
E must be tangential: Gauss’ law says any radial
component of E would require enclosed charge.
|E| is constant on any circular integration path due
to cylindrical symmetry.
For r > R (outside, circular integration path):
The magnitude of induced electric field grows linearly with r, then falls off as 1/r for r>R
For r < R (inside):

25. EXTRA CREDIT Example: EMF generated by Faraday Disk Dynamo
Conducting disk, radius R, rotates at rate w in uniform constant field B,
FLUX ARGUMENT:
evaluate areal velocity
w, q +
dA = area swept out by radius vector in dq
= fraction of full circle in dq x area of disk
MOTIONAL EMF FORMULA: (work/unit charge)
For points on rotating disk: v = wr, vXB = E’ is radially outward, so is ds = dr
current flows
radially out
EMF induced across length ds of moving conductor in B field
vXB looks like electric field to moving charge
(Equation 29.7)
Radial conductor length

26. Recap: Faraday’s Law of Induction
Magnetic Flux:
Faraday’s Law: A changing magnetic flux through a coil of wire induces an EMF in the wire, proportional also to the number of turns, N.
Bind & iind oppose changes in FB
Lenz’s Law: The current driven by an induced EMF creates an induced magnetic field that opposes the flux change.
Induction and energy transfer: The forces on the loop oppose the motion of the loop, and the power required to sustain motion provides electrical power to the loop. i1 i2
Transformer principle: changing current i1 in primary induces EMF and current i2 in secondary coil.
Generalized Faraday Law: A changing magnetic flux creates non-conservative electric field.