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Faraday’s Law and Inductance
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Faraday’s Law• A moving magnet can exert a force on a
stationary charge.
• Faraday’s Law of Induction
• Induced emf is directly proportional to the change in the magnetic flux.
dt
dN m
AdBm
Remember:
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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
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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.
w
lv
B l = 20 cmw = 20 cmv = 5 m/sB = 0.75 Td = 60 cmR = 1 d
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Motional EMF• For a moving conductor in a magnetic
field
or
With a resistor attached the current is
qvBqE
lvBV
R
lvBI
R vl
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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.
vI
As the loop enters the fieldthe downward flux increases.Therefore, the current flowsin the loop to generate anupward flux to cancel thechange.
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Induced EMF• A changing magnetic field always generates
an electric field.
• This electric field is non-conservative and time varying.
dt
dsdE m
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Generators & Motors
• AC Generator - A spinning loop in a magnetic field generates an AC voltage.
• Eddy Currents - Currents generated within a conductor due to changes in the magnetic field.
tBAm cos tNBA sin
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Induced EMF• Self induced emf is due to a
change in current.
• Inductance, L
• Units, henry (H) which is a V·s/A
dt
dIL
dt
dN m
I
NL m
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Solenoid• From Ampere’s Law
• If the cross sectional area of the solenoid is A, then
• Inductance is then
dsl
INlBsdB 0
l
IANABm
0
l
AN
I
NL m 0
2
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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?
)05.0(
)1026.1(/104)500( 232720
2
m
mAN
l
ANL
2322 1026.1)02.0( mmrA
HL 3109.7
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RL Circuits• After the switch is closed,
Kirchoff’s rules gives
• Solution is0
dt
dILIR
/0
/ 11)( tLRt eIeR
tI
R
L
• If power supply is shorted out, then/
0)( teItI
R
L
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RL Circuit Current
• How much current is flowing in a 10 0.1H RL circuit 15 ms after it is hooked up to a 10V power supply?
0
0.5I0
1.0I0
Current Flow
msH
1010
1.0
AeV
tI msms 78.0110
10)( 10/15
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Energy in the Magnetic Field
• Power is voltage times current, but also the rate at which work is done.
dt
dIILIP
dt
dU
dt
dWP m
dt
dIIL
dt
dUm 221 LIUm
• Energy Density of a solenoid:
0
2
2B
lA
Uu m
m
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Important E & M Equations
• Maxwell’s Equations
• Lorentz Force
0
QAdE
0AdB
dt
dIsdB e 000
dt
dsdE m
BvqEqF