LECTURE 17
Reminder: How to Change Magnetic Flux in a
Coil
•  Motional EMF
•  Eddy Currents
•  Self Inductance

 
dΦ m dB
1. B changes:
=
NA cos B, A
dt
dt
 
dΦ m
dA
2. A changes:
= NB
cos B, A
dt
dt
 
d ⎡⎣ cos B, A ⎤⎦
 
dΦ m
3. B, A changes:
= NBA
dt
dt
 
dΦ m dN
4. N changes (unlikely):
=
BA cos B, A
dt
dt
(
(
(
)
)
)
(
)
(
)
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Example Problem: Solenoid within a Coil
2
Example Problem: Solenoid within a Coil
120 turn coil of
radius 2.4 cm and
resistance 5.3 "
solenoid with
radius 1.6 cm and
n = 220 turns/cm
Initial current in the solenoid is 1.5 A.
Current is reduced to zero in 25 ms.
What is the current in the coil while the
current is being reduced?
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1
Motional EMF
Calculation
Motional EMF is any emf induced by the motion of a
conductor in a magnetic field.
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Energy Conservation
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Eddy Currents
*Relative motion between a B field and a
conductor induces a current in the conductor.
•  Rate of work by applied force:
•  The induced current gives rise
to a net magnetic force F in the
loop which opposes the motion:
*The induced current give rises to a net magnetic
force, FM, which opposes the motion.
•  Energy is dissipated in circuit at rate P’:
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2
Eddy Currents
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Demonstration
9
Demonstration: Eddy Currents
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Eddy Currents
• During the last lecture we considered currents
confined to a loop or a coil.
• Electrical equipment contains other metal part
which are often located in changing B fields.
Currents in such parts are called Eddy currents
Induced current
clockwise
Induced current
anti-clockwise
x
x
x
F
x
v
x
Flux
increasing
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x x
x x
x x
x x
x x
x
x
x
x
x
x
x
x
x
x
x x
x x
x x
x Fx
x x
v
B field into
the page
Flux
decreasing
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3
Reduce Eddy Currents
Inductors & Inductance
*An inductor can be used to produce a desired B field.
DEMO: Lenz’s Law
Symbol for inductor
Metal strips with
insulating glue
*Inductance:
L=
Cut slots into
the metal.
Units of L:
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Self Inductance
An induced emf, L, appears in any
coil in which the current is changing.
NΦ M
i
1 Henry = 1H = 1
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Tm 2
A
14
Self Inductance in a Coil
X XX X
X XX XX X
dI/dt
X XX X
a
b
Self-Induction: Changing current through a loop induces
an opposing voltag in that same loop.
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4
Magnetic Energy in an Inductor
Magnetic Energy in an Inductor
Upon closing switch, S, apply
Kirchoff’s loop rule:
ε − IR − L
If U m = energy in the inductor,
dU m
dI
= LI ⇒ U m = LIdI
dt
dt
dI
=0
dt
If
U m = ∫ dU m = ∫ LIdI
Multiply through by I:
0
ε I = I 2 R + LI
power
delivered
by battery
power
dissipated
by resistor
dI
dt
Um =
1 2
LI f
2
energy stored in
an inductor
power
delivered
to inductor
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Magnetic Energy in a Solenoid
B = µ0 nI ⇒ I =
B
µ0 n
L = µ0 n 2 Al
2
⎛ B ⎞
1
1
B2
U m = LI 2 = µ0 n 2 Al ⎜
=
Al
2
2
2 µ0
⎝ µ0 n ⎟⎠
magnetic energy density um =
electric energy density ue =
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B2
2 µ0
1
ε0E 2
2
Both of
these are
general
results.
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