LECTURE 16
General Review
•  Electrostatics
•  Faraday’s Law of Induction
•  motion of “q” in external E-field
•  E-field generated by qi
dA
•  Magnetostatics
•  motion of “q” and “I” in external B-field
•  B-field generated by “I”
•  Electrodynamics
Induction: DEMONSTRATION
Induction Effects
*Current flows only if there is
relative motion between the
loop and the magnet
Bar magnet moves through
coil: Current induced in coil
Change pole that enters:
Induced current changes sign
*Current disappears when the
relative motion ceases
Bar magnet stationary inside
coil: No current induced in coil
*Faster motion produces a greater
current
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dΦ M
dt


Φ M ≡ ∫ B • dA
•  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!
B
B
ε=−
Coil moves past fixed bar
magnet: Current induced in
coil
S
N
N
S
N
S
v
v
v
S
N
Change direction of motion in any of above examples 
ammeter
3
Induced current changes sign
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Induction Effects from Currents
• When the switch is closed (or
opened)
b
current induced in coil b
Induction Example
A wire loop falling into a B field (increasing)
a
Force acting on
moving charges
time
magnetic
Field B
• Steady state current in coil a
downward
Velocity v
no current induced in coil b
•  Conclusion:
A current is induced in a loop when: there is a
change in magnetic field through it.
This can happen many different ways.
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•  How can we quantify this?
N
5
Faraday’s Law
N
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Faraday’s Law Restated
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
dA
Define the flux of the magnetic
B
N
B
The magnitude of the emf  induced in a conducting
loop is equal to the rate at which the magnetic flux
M through the loop changes.
dA
field through an open surface A
as:
Φ B = BA cosθ
B
B
ε=−
ΔΦ B
Δt
Unit: 1 Weber=1Wb=1 Tm2
The minus sign indicates opposition
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2
EMF for a Coil of N turns
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How to Change
Magnetic Flux in
a Coil
9
Lenz’ Law
Demo: E&M Cannon
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.
Opposition to Flux:
B
v
B
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N
S
N
S

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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v
11
•  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.
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10
v
B
~
side view
12
3
Faraday’s Law in terms of E Field
Summary
•  Faraday’s Law (Lenz’ Law)
x x xEx x x x x x x
E
xxxxxxxxxx
r
xxxxxxxxxx
B
xxxxxxxxxx
E
x x x x x x x xEx x
–  a changing magnetic flux through a loop
induces a current in that loop
ε=−
dΦ M
dt
negative sign indicates that
the induced EMF opposes
the change in flux
•  Faraday’s Law in terms of Electric Fields
 
dΦ
E
∫ • dl = − dtM
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