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Induction: DEMONSTRATION
Lecture 12
• Electrostatics
*Current flows only if there is
relative motion between the
loop and the magnet
• motion of “q” in external E-field
• E-field generated by q
*Current disappears when the
relative motion ceases
• Magnetostatics
• motion of “q” and “I” in external B-field
• B-field generated by “I”
*Faster motion produces a greater
current
• Electrodynamics
• time dependent B-field generates E-field
– ac circuits, inductors, transformers, etc
– time dependent E-field generates B-field
Caution – this picture is not an
example of right hand rule!
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Induction Effects from Currents
• When the switch is closed (or
opened)
b
⇒current induced in coil b
An emf is induced in a loop when the number of
magnetic field lines that pass through the loop is
changing. Only components perpendicular to loop
A
Define the flux of the magnetic
a
B
⇒no current induced in coil b
B
field through an open surface A
as:
Φ B = BA cosθ
Unit: 1 Weber=1Wb=1 Tm2
• Conclusion:
A current is induced in a loop when: there is a
change in magnetic field through it.
This can happen many different ways.
• How can we quantify this?
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Faraday’s Law
• Steady state current in coil a
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ammeter
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Lenz’ Law (to determine direction of induced I in loop)
Faraday’s Law Restated
The magnitude of the emf ε induced in a conducting
loop is equal to the rate at which the magnetic flux ΦB
through the loop changes.
A
B
B
An induced current has a direction such that the
magnetic field due to the current opposes the change
in the magnetic flux that induces the current.
Binduced always opposes the change in
Opposition to Flux:
the flux of B, but does not always point
opposite it!!!
ΔΦ B
ε=−
Δt
B
v
B
Direction of induced current is opposed to
that from B field.
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N
S
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v
Direction of I must be to oppose change...otherwise violate
conservation of energy.
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Magnetic Flux through a Changing Area
How to Change
Magnetic Flux in
a Coil

ΔΦ B
ΔB
1. B changes:
=N
A cosθ
Δt
Δt
ΔΦ B
ΔA
2. A changes:
= NB
cosθ
Δt
Δt
Δ [ cosθ ]
ΔΦ B
3. θ changes:
= NBA
Δt
Δt
ΔΦ B ΔN
4. N changes (unlikely):
=
BA cosθ
Δt
Δt
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S
Bi
The minus sign indicates opposition
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N
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Energy Conservation
Demo: E&M Cannon
• Rate of work by applied force:
P = Fv
• The induced current gives rise
to a net magnetic force F in the
loop which opposes the motion:
FL = ILB points to left
• Energy is dissipated in circuit at rate P’:
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Inductors & Inductance
Symbol for inductor
Induced current opposes change of in flux
from change in current.
This called self induced emf.
Loop at right: close switch at b, current starts
to flow. While dI/dt is not equal to zero, εL is
induced.
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side view
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An induced emf, εL, appears in any
coil in which the current is changing.
A long solenoid can be used to produce a desired B field
NΦ B
L=
i
1 Henry = 1H = 1
~
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Self Inductance
*An inductor can be used to produce a desired B field.
Units of L:
B
F = ILB
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*Inductance:
v
• Connect solenoid to a source of
alternating voltage.
• The flux through the area to axis of
solenoid therefore changes in time
• A conducting ring placed on top of
the solenoid will have a current
induced in it opposing this change.
• There will then be a force on the ring
since it contains a current which is
circulating in the presence of a
magnetic field.
• Note that it’s the off-axis component
of B (the “fringe field”) that flings the
ring.
εL = −
ΔNΦ B
Δi
= −L
Δt
Δt
direction of εL opposes change with time in
current.
Tm 2
A
X XX X
X XX XX X
dI/dt
X XX X
a
b
Energy Stored in an
Inductor
U=
1 2
LI
2
Self-Induction: Changing current through a loop induces
an opposing voltage in that same loop.
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Calculation of Inductance
RL Circuits
*Long solenoid with N turns, radius r, length L:
l
r
N turns
only depends on
geometry
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Initially, an inductor acts to oppose changes in current through
it. A long time later, it acts like an ordinary connecting wire.
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RL Circuits (ε on)
Current
I=
ε
ε
1 − e− Rt / L = 1 − e−t /τ RL
R
R
(
)
(
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RL Circuits
• Why does τRL increase for
larger L?
)
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a
I
R
b
ε
L
• Why does τRL decrease for
larger R?
Voltage on L
VL = ε e− Rt / L = ε e−t /τ RL
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I
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