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Chapter 29
Electromagnetic Induction
Induced current
You mean you can
generate electricity this way??!
For my next magic act…
Note: No moving parts
Summary
Faraday’s Law of Induction
An emf is induced when the number of magnetic field
lines that pass through the loop changes
Magnetic Flux
Similar to electric flux
Unit: Weber
1Wb = 1Tm 2
Magnetic Flux
Faraday’s Law (restated)
Emf is induced whenever FB changes
dF
x=dt
The minus sign will be explained later
What if you have a coil?
dF1
dF N
x = -N
= (Coil of N turns)
dt
dt
where
F1 : flux of one turn
F N = NF1 : flux of N turns
EMF induced in a solenoid
A=1m2, N=2000 turns
An external magnetic field of B = 1mT is removed
suddenly in 1s. What is the emf generated?
Solution
A=1m2, N=2000 turns
An external magnetic field of B = 1mT is removed
suddenly in 1s. What is the emf generated?
What are Fi and F f for one turn?
(initial and final flux)
Fi = Bi A = (10 -3 T )(1m 2 ) = 10 -3Wb
F f = B f A = (0T )(1m 2 ) = 0Wb
dF B
DF B
x = -N
» -N
dt
Dt
F f - Fi
(0 - 10 -3 )Wb
Þ x » -N
= -(2000)
Dt
1s
Þ x » 2V
Lenz’s Law
An induced current has a direction such that
the B field due to the current opposes the
change in the magnetic flux
Lenz’ Law – Example 1
When the magnet is moved toward the stationary
loop, a current is induced as shown in a
This induced current produces its own magnetic
field that is directed as shown in b to counteract the
increasing external flux
The Logic
Bext: 
Bext: increasing
BI:  (to oppose the increase)
I: counterclockwise (view from left)
Lenz’ Law – Example 2
When the magnet is moved away the stationary
loop, a current is induced as shown in c
This induced current produces its own magnetic
field that is directed as shown in d to counteract
the decreasing external flux
The Logic
Bext: 
Bext: decreasing
BI:  (to slow down the decrease)
I: clockwise (view from left)
Summary
Direction of current
What is the direction of current in B when the switch S
is closed?
I
Do it yourself!
Which way do the currents flow?
What is the current?
Resistance: R
dBA
dA
x== -B
dt
dt
dA
but
= -Lv
dt
Þ x = BLv
x
BLv
ÞI= =
R
R
What is the force?
Resistance: R
Displacement Current
There is something wrong with Ampere’s Law
Depending on the surface, Iencl could be either zero or
non-zero. Inside the capacitor there is no conduction
current.
Displacement Current
We need to account for the E field in Ampere’s Law.
Does it work?
Apply the generalized Ampere's Law to the bulging surface :
IC (bulge) = 0 on that surface, but ID is non - zero.
dF E
d(EA)
d s
dq
ID (bulge) = e 0
= e0
= e 0 ( A) =
dt
dt
dt e 0
dt
Þ ID (bulge) = IC (plane)
Displacement current density
ID
JD =
A
Example
What is the B field at point a given IC?
dF E
d(Epr 2 )
Iencl = ID = e 0
= e0
dt
dt
s
q
E= = 2
e 0 pR e 0
d r2 q
r 2 dq r 2
Þ ID = e 0 ( 2 ) = 2
= 2 IC
dt R e 0
R dt R
Ampere-Maxwell law
Assume the capacitor has radius R.
At distance r around the wire:
m0 I
Bw (2p r) = m 0 I Þ Bw =
2p r
The E field inside the capacitor:
s
q
q
E= =
Þ F E = EA =
e 0 Ae 0
e0
At distance R around the capacitor:
dF E
dq
Bc (2p R) = m0e 0
= m0
= m0 I
dt
dt
mI
Þ Bc = 0 = Bw
2p R
Isolated rod vs
closed circuit
Eddy Currents
Eddy currents want to stop
whatever you are doing!
Which one falls faster?
Movie
Potential is not well defined
dF
x=dt
Faraday’s Law (modern form)
Magnetic materials
Diamagnetism
Paramagnetism
Ferromagnetism
Diamagnetism
No net magnetic dipole for each atom when B=0.
When magnetic field is switched on, an induced magnetic
dipole points in the opposite direction to B due to Lenz’s
Law, this causes the object to be repelled.
Copper, lead, NaCl, water, superconductor
Paramagnetism
•Each atom already has a permanent dipole moment.
•This dipole will align with external B field.
•Forces points from weak field to strong (attraction).
Oxygen, aluminum, chromium, sodium
Movie
Liquid Oxygen
Ferromagnetism
•Each atom has a net magnetic dipole.
•Atoms arrange themselves into domains.
•External fields can affect the alignment of the
domains.
•Heat can destroy the domains.
B Field
•Magnets are made this way.
Insert Picture
Iron, Permalloy
Details
Picture
Applications of Faraday’s Law
Power plants
Flashlight with no battery
Toothbrush?
Transformers (a.c. versus d.c.)
The wonders of magnetic field
View from afar
Big magnetic field