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Transcript
Faraday’s Law and Inductance
Faraday’s Law
• A moving magnet can exert a force on a
stationary charge.
Remember:
• Faraday’s Law of Induction
d m
  N
dt
 
 m   B dA
• Induced emf is directly proportional to the
change in the magnetic flux.
Collapsing Field
• Ex. What is the induced emf in a 100 turn
coil with an area of 0.035 m² when the
magnetic field decreases linearly from 2.5 T
to 0 T in 10 seconds?
B
Moving Loop
• Ex. For a rectangular loop moving through
a magnetic field, plot the total flux, emf,
and force on the loop with respect to time.
v
l
w
d
B l = 20 cm
w = 20 cm
v = 5 m/s
B = 0.75 T
d = 60 cm
R=1W
Motional EMF
• For a moving conductor in a
magnetic field
qE  qvB
or
R
V  lvB
With a resistor attached the
current is
lvB
I
R
l
v
Lenz’s Law
• Polarity of the induced emf is such that it
produces a current that will create a
magnetic flux to oppose the change in
magnetic flux through the loop.
As the loop enters the field
the downward flux increases.
Therefore, the current flows I
in the loop to generate an
upward flux to cancel the
change.
v
Induced EMF
• A changing magnetic field always generates
an electric field.
 
d m
 E  ds   dt
• This electric field is non-conservative and
time varying.
Generators &
Motors
• AC Generator - A spinning loop in a magnetic
field generates an AC voltage.
 m  BA cost 
  NBA sin t 
• Eddy Currents - Currents generated within a
conductor due to changes in the magnetic field.
Induced EMF
• Self induced emf is due to a
change in current.
d m
dI
  N
 L
dt
dt
• Inductance, L
N m
L
I
• Units, henry (H) which is a
V·s/A
Solenoid
• From Ampere’s Law
 
 B  ds  B  l  N0 I
• If the cross sectional area
of the solenoid is A, then
N 0 IA
m  B  A 
• Inductance is then
l
N m N 2  0 A
L

I
l
ds
l
Solenoid (cont.)
• Ex. What is the inductance of a solenoid of
500 turns where the length is 0.05 m and the
diameter of the coil is 0.04 m?
A  r   (0.02m)  1.26 10 m
2
2

3
2

N 2 0 A (500) 2 4 107 N / A2 (1.26 103 m2 )
L

l
(0.05m)
L  7.9 10 3 H
RL Circuits
• After the switch is closed,
Kirchoff’s rules gives
R

dI
  IR  L  0
dt
• Solution is
I (t ) 


1 e
R
 Rt / L
  I 1  e 
t / 
0
• If power supply is shorted out, then
I (t )  I 0 e
t /
L
L

R
RL Circuit Current
• How much current is
flowing in a 10W 0.1H 1.0I0
RL circuit 15 ms after
0.5I0
it is hooked up to a
10V power supply?
0.1H

 10ms
10W

Current Flow
0

10V
15ms / 10 ms
I (t ) 
1 e
 0.78 A
10W

2
3
4
Energy in the
Magnetic Field
• Power is voltage times current, but also the
rate at which work is done.
dI
P  I     IL
dt
dU m
dW
P

dt
dt
dU m
dI
 IL
dt
dt
• Energy Density of a solenoid:
Um
B2
um 

A  l 2 0
U m  LI
1
2
2
Important E & M
Equations
• Maxwell’s Equations
  Q
 E  dA 
0
 
B

d
A

0

• Lorentz Force
 
d m
 E  ds   dt
 
d e
 B  ds  0 I   0 0 dt


 
F  qE  qv  B