Lecture 25 induction. faradays law. lenz law

1. Lecture 25
Induction.
Faraday’s law. Lenz’s law.

2. ACT: Moving loop
A closed loop with a bulb (no battery) is half-way in a region with a uniform
magnetic field (shaded area).
If the loop is stationary, the magnetic force on the electrons in the wire
is… zero, right?
If the loop is pulled to the right, the
bulbs lights up because of a current
in the
+
F
A. CW direction
B. CCW direction
• Consider positive charge
C. Huh?
• Velocity to the right
• Force points down
v
I

3. Faraday’s experiment
Faraday tried the opposite (keep the loop stationary, move the
magnet) and obtained the same current!
v
I
But the charges in the wire are not moving now, so the magnetic
force on them should be zero!
Hm…
This “coincidence” bothered Einstein and eventually led to to the Special
Theory of Relativity.

4. Another induction example
Now nothing moves, but the magnitude of the magnetic field
decreases with time.
Again, CCW current observed!
I ~
dB
dt
More hmm…
Overall, it seems that current is induced when the magnetic
field through the loop changes…
I

5. Faraday’s Law
It turns out it’s not exactly when B changes…
r r
Magnetic flux through a surface: ΦB = ∫ B ×
da
The emf induced in a circuit is determined by the time rate of
change of the magnetic field flux through that circuit:
d ΦB
ε =−
dt
Minus sign = Lenz’s law

6. EMF?
Remember the EMF of a battery?
It’s what produces a current in a circuit.
Now we have an EMF without a battery.
This should be surprising because:
• Battery’s emf = potential difference
between ends. And here??
• Since we don’t have +/- plates, the
electric field that drives the charges in
the wire is a closed line??
Back to this later...
I =
ε
R
ε
ε
I =
R
ε =−
ε
d ΦB
dt
DEMO:
B –field
through loop
R
R

7. Lenz’s law
The induced current will appear in such a direction that
is opposes the change in flux that produced it.
v
v
Bin
Bin
Ring moving left
Ring moving right
Flux decreasing
Flux increasing
To compensate, Binduced
points left
To compensate, Binduced
points right

8. ACT: Loop near a wire
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?
A. CCW
B. CW
C. No induced current
I
y
v
x
• The flux through this loop DOES change in time since the loop is moving
from a region of higher magnetic field to a region of lower field.
• Therefore, by Lenz’ Law, an emf will be induced which will oppose the
change in flux.
• Current is induced in the clockwise direction to restore the flux.

9. Back to the moving loop
v
Loop (dimensions w × L) moves to the
right at constant v. Assume right
edge of B-field region and right wire
of loop are aligned at t = 0. Bulb has
resistance R.
Uniform B
1. Flux through loop:
ΦB = Bw ( L − vt )
2. EMF and current:
Bin

L
0 <t < ÷

v

d ΦB
ε =−
= Bwv
dt
Iin =
ε Bwv
=
R
R
3. Direction: Use Lenz’s law
Flux is decreasing ⇒ Bin
to compensate ⇒ Iin is CCW
Iin
L
w

10. ACT: Lenz’s law for the moving loop
The external magnetic field exerts a
force on the loop because of the
induced current. The net force on the
loop points:
A. To the right
B. To the left
C. Is zero.
F
Iin
F
Uniform B
v
L
F
Moving to the right decreases the flux.
The induced current is against this change, and resultant
force thus opposed motion by pulling to the left (this is
another way of seeing Lenz’ law).
w

11. Energy conservation check
Force by external B on loop pulls
left

ε Bwv 
B 2w 2v
 Iin = =
÷
F = wIinB =
R
R 

R
Another force (same magnitude)
needs to be applied to the right to
keep the loop moving at constant v
B 2w 2v
F =
R
Work per unit time by this force:
Iin
Uniform B
B 2w 2v 2
P = Fv =
R
2
Power dissipated in this circuit:
 Bwv 
B 2w 2v 2
2
P =I R =
÷R =
R 
R

w
L
ok

12. In-class example: Induction in square loop
A square loop (0.1 m on a side) is in a region of space that has a
magnetic field increasing from B = 0 to B = 10 T into the page over 5
seconds. The loop has R = 3 Ω. During this time, what is the size and
direction of the current?
B
L
ΦB = L2B
d ΦB
dB
ε =
= L2
dt
dt
ε
A.
B.
C.
D.
E.
0
0. 67 A CW
0.67 A CCW
6.7×10−3 A CW
6.7×10−3 A CCW
L2 dB
I =
=
R R dt
dB 10 T
=
= 2 T/s
dt
5s
=
( 0.1 m )
3Ω
2
( 2 T/s ) = 6.7 × 10
Direction: Applied B is increasing, so
induced B is out of the page, ie,
current is CCW
−3
A

13. Applications: AC generators
AC Generator
Water turns wheel
 rotates magnet
 changes flux
 induces emf
 drives current
DEMO:
Loop rotating between
electromagnets

14. Applications: Microphones
“Dynamic” Microphones
(E.g., some telephones)
Incoming sound
 oscillating pressure waves
 oscillating [diaphragm + coil]
 oscillating magnetic flux
 oscillating induced emf
 oscillating current in wire

15. Application: Credit card reader
Tiny coils in the reader respond to changes in flux as the
magnetic domains (encoding 0’s and 1’s) go by.
Must swipe card
 generates changing flux
Faster swipe  bigger signal