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. Previously: Next: Changing magnetic flux induces EMFs
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’s Law
Biot-Savart Law
Current loops are elementary dipoles
B due to long straight wire carrying a current i:
B due to circular loop carrying a current i :
B inside a solenoid: 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

3. Magnetic Flux: r B defined analogously to flux of electric field
Electrostatic Gauss Law
Magnetic Gauss Law
over surface (open or closed) B r
Flux Unit: 1 Weber = 1 T.m 2 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
Induced current stops when relative motion stops (case b).
Faster motion produces a larger current.
Induced current direction reverses when magnet motion reverses direction (case c versus case a)
Any relative motion that changes the flux works
EMF/current is induced in the loop whenever magnetic flux through the loop is changing. (We mean flux rather than just field)
Induced current creates it’s own induced B field and flux, opposing the changing flux FB (Lenz’ Law)

5. CHANGING magnetic flux induces EMFs and currents in wires
Key Concepts:
Magnetic flux
Faraday’s Law of Induction
Lenz’s Law
Generator principle:
Loops rotating in B field generate EMF and current.
Lenz’s Law  Applied torque is needed (energy conservation)

6. Motional EMF: Lorentz Force on moving charges in conductors
Uniform magnetic field points away from the viewer.
Wire of length L moves with constant velocity v perpendicular to the field
Electrons feel a magnetic force and migrate to the lower end of the wire. Upper end becomes positive.
Result is an induced electric field Eind 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 the wire continues to move through the magnetic field. +
FIELD
BOUNDARY L v -
FBD of free charge q in a wire v + - Fe Fm
Eind
Induced EMF is created without batteries.
Induced current flows only if the circuit is completed.

7. Flux approach to Motional EMF in a moving wire
via Lorentz force
The rate of flux change = field B x rate of sweeping out area (B uniform)
Connection with Flux:
Flux arguments apply generally, for example:
E is induced directly in 15 turn coil by changing flux in solenoid
inside solenoid
B = 0 outside, at location of charges moved
in the loop.
Lorentz force doesn’t explain induced current
Changing magnetic flux through coil creates
electric field that drives induced current in
loop that creates induced B field
Flux is proportional to solenoid’s
cross-section area (not the coil’s)

8. A rectangular loop is moving in a uniform B field+ - a b c d
DOES CURRENT FLOW?
Segments a-b & d-c create equal but opposed EMFs in circuit
No EMF from b-c & a-d or
FLUX is constant
No current flows
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
FLUX is DECREASING v + - a b c d
Which way does current flow?
What is different when loop is entering field?

9. Direction of induced fields and currents
Replace magnet with circuit below
Flux change is enough to cause induction
Do not need actual motion
Let current i1 be changing -> changing flux.
Flux is constant if current is constant
Current i2 flows only while i1 (flux F1) 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

10. 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
Change Flux by:
changing |B| through a coil
changing area of a coil or loop
changing 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. Find the induced EMF
Current iind creates field Bind that opposes increase in B
uniform
dB/dt
front

11. + iind - x Slidewire Generator
Slider moves, increasing loop area (flux) + - x
iind L
Induced EMF:
Slider: constant speed v to right
Uniform field into the page
Loop area & FLUX are increasing
Slider acts like a battery
Lenz’s Law says induced field Bind
is out of page.
RH rule says current is CCW
due to the induced current
opposes the motion of the wire
FEXT (an external force) is needed to keep wire from slowing down
FM DRAG FORCE on slider
FEXT needed is opposed to Fm
Mechanical power supplied & dissipated:

12. 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 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 !!!

13. 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 from the far left to right
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 from left to right B

14. The induced current and EMF create induced magnetic flux that opposes the change in magnetic flux that created them
Lenz’s Law
front of loop
induced dipole
In same loop
induced dipole
in same loop
The induced current try to keep the original magnetic flux through the loop from changing.
front of loop
front of loop

15. Lenz’s Law Example: A loop crossing 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
B = 0 v v v v v

16. 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

17. Rotation axis is out of slide
Generator Effect (AC)
Changing flux through a rotating current loop
angular velocity w = 2pf in B field:
Rotation axis is out of slide q
peak flux magnitude when wt = 0, p, etc.
Eind has sinusoidal behavior - alternating polarity over a cycle
maxima when wt = +/- p/2
EMF induced is the time derivative of the flux

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

19. AC Generator, continued
No reversal of output E
DC Generator, continued
Reversal of output E

20. Flat coil with N turns of wire
Numerical
Example
Flat coil with N turns of wire 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 EMF induced in the coil by it’s own changing flux
Flux change due to external B field produces induced field Bind
INDUCED field/flux produces its own EMF.
The BACK EMF opposes current change – analogous to inertia

21. Transformer Principle
Primary current is changing after switch closes
 Changing flux in primary coil
....which links to....
changing flux through secondary coil
 changing secondary current & EMF
PRIMARY
SECONDARY
Iron ring strengthens flux linkage

22. conducting loop, resistance R
Changing magnetic flux directly induces electric fields
A thin solenoid, cross section A, n turns/unit length
zero B field outside solenoid
inside solenoid:
conducting loop, resistance R
Bind
Flux through a conducting loop:
Current I varies with time  flux varies  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 there so it’s not the Lorentz force
An induced electric field Eind along the loop causes current to flow
It is caused directly by dF/dt within the loop path
E-field is there even without the conductor (no current flowing)
Electric field lines here are loops that don’t terminate on charge.
E-field is a non-conservative (non-electrostatic) field as the line
integral around a closed path is not zero
What makes the induced current I’ flow, outside solenoid?
Generalized Faradays’ Law
(hold loop path constant)

23. Example: Find the electric field induced by changing magnetic flux
Find the magnitude E of the induced electric field at points within and outside the magnetic field.
Assume: dB/dt = constant within the circular shaded area. E must be tangential: Gauss’ law says any normal component of E would require charge enclosed. |E| is constant on the circular integration path due to symmetry.
For r > R:
For r < R:
The magnitude of induced electric field grows linearly with r, then falls off as 1/r for r>R

24. Example: EMF generated by Faraday Disk Dynamo
Conducting disk, radius R, rotates at rate w in uniform constant field B,
FLUX ARGUMENT:
areal velocity
w, q +
dA = area swept out by radius vector in dq
= fraction of full circle in dq x area of circle
USING MOTIONAL EMF FORMULA:
Emf induced across conductor length ds
Moving conductor sees vXB as electric field E’
For points on rotating disk: v = wr, vXB = E’ is radially outward, ds = dr
current flows
radially out
(Equation 29.7)