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Electromagnetic Induction
Induced EMF We already know that moving charge (=current)
causes magnetic field
It also works the other way around: changing magnetic field (e.g. moving permanent magnet) causes current
Electromagnetic Induction 2
•it’s called induced current •since there is a current, there is electromotive force called induced emf
Induction Experiments Stationary permanent magnet does not
induce current, the moving one does
Stationary current in the first coil does not induce the current in the second coil, moving the coil in and out or switching the current on and off does
Electromagnetic Induction 3
What matters is change of magnetic field
Magnetic Flux
Electromagnetic Induction 4
B
AAB
if B is not perpendicular to surface then
AB
AB
cos
cos
if Bcosϕ is the same everywhere then
cosBA
A is like a vector: you need to pick one of the two possible directions
Faraday’s Law
Electromagnetic Induction 5
t
E
induced emf
rate of flux change
for a coil of N turns,
tN
E
[Φ] = weber, 1 Wb = 1 T∙m2=1 V∙s
notice the “−” sign in the formula
A Slide-Wire Generator
Electromagnetic Induction 6
B
v
tv
BLv
t
AB
tLvA
constB
E
L
Lenz’s Law The direction of induced current is such as to oppose
the direction of the phenomenon causing it
Electromagnetic Induction 7
Lenz’s law is a particular case of Le Chatelier's principle: in a stable equilibrium, any deviation would cause a force which drives the system back to the equilibrium (that’s why it’s stable!)
Lenz’s Law and a Slide-Wire rod
Electromagnetic Induction 8
B
v
I induced current
F
direction of induced current is such that force F acting on it due to B tries to slow down the moving rod
induced field is directed opposite to B
otherwise we would get infinite acceleration without external force
inducedB
Motional Electromotive Force
Electromagnetic Induction 9
B
v
I
separate from wires (no current)
qvBFB
qEFE
charges keep accumulating until electric force compensates the magnetic force
vBLV
vBE
ab
a
b
Generator =device which converts mechanical energy to
electricity
Electromagnetic Induction 10
tABt
tAB
t
sin
cos
E
need calculus to derive it
Alternating and Same Sign emf
Electromagnetic Induction 11
brush
slip ring
brush
commutator
Mutual Inductance =coupling between two coils
Electromagnetic Induction 12
•when current through the main coil changes, the secondary coil gets induced emf •the question is, how large is it?
tN
2
22E
Φ2 is proportional to the current in the main coil:
||1
121
2
2 iMN
for convenience
t
iM
1
212E
Mutual Inductance
Electromagnetic Induction 13
t
iM
1
212Et
iM
2
121E2112 MMM
it turns out that
even for different coils
[M]= henry, 1 H = 1 Wb/A=1 V∙s/A=1 Ω∙s
mutual inductance M depends on coils’ geometry and magnetic properties of the material
Transformers =two coils that share magnetic flux
Electromagnetic Induction 14
tN
11E
tN
22E
1
2
1
2
N
N
E
E
Self-Inductance =mutual inductance applied to the main coil itself
Electromagnetic Induction 15
t
iL
E
L
Capacitors vs inductors:
t
iLv
t
vCi
DC: i=0 (R=∞) DC: v=0 (R=0) Inductor:
Magnetic Field Energy
Electromagnetic Induction 16
t
iLv
Power supplied during time Δt:
iLitPU
t
iLiviP
i
i
2
2
1LIU
Capacitors vs Inductors
Electromagnetic Induction 17
2
2CVU
2
2LIU
2
2
0
Eu
0
2
2
Bu
Energy
Energy density
Mechanical analog
spring (potential energy)
mass (kinetic energy)
Resistance-Inductance Circuits
Electromagnetic Induction 18
S
E
R L
a b cI
0
0
bcab VV
I
Resistance-Inductance Circuits
Electromagnetic Induction 19
S
E
R L
a b cI
t
iLiR
E Switch closed )0( t
Lt
i
V
i
initial
ab
E
0
0
Resistance-Inductance Circuits
Electromagnetic Induction 20
S
E
R L
a b cI
t
iLiR
E Switch closed )0( t
iL
R
Lt
i
V
i
ab
E
0
0
current increases current change rate decreases
Resistance-Inductance Circuits
Electromagnetic Induction 21
S
E
R L
a b cI
t
iLiR
E Switch closed )( t
Ri
t
i
final
final
E
0
Current vs Time
Current, Resistance, and Direct-Current Circuits 22
– need calculus to derive it )1( / LtReR
i E
i 71828.2e
t
)/11( ei final
R
L
finali
L/R = time constant
Current Decay in LR Circuit
Electromagnetic Induction 23
SE
R L
a b cI
0i
R
L
i
t
LtReii /
0
The L-C Circuit
Electromagnetic Induction 24
E
L
a b cI
C
22
22 LICVEtotal LC
1
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