Chapter 27 – Magnetic Induction
Motional EMF
Consider a conductor in a B-field moving to the right.
In which direction will an electron
in the bar experience a magnetic
force?
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V
FB = q(v × B)
The electrons in the bar
will move toward the
bottom of the bar.
This charge separation creates an electric field in the bar and
results in a potential difference between the top and bottom of
the bar. What is the electric field? E = vB
The motional EMF is
ε = vBL
What if the bar were placed across conducting rails (in red) so
that there is a closed loop for the electrons to follow?
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V
In this circuit, what direction is the current?
a) clockwise b) counterclockwise
L
Now the rod has a current through it. What is the direction of the
magnetic force on the rod due to the external magnetic field?
F = I (L × B)
Using the right hand rule, the force on the bar is directed to the
left.
The magnitude of the magnetic force on the rod is:
vBL
vB 2 L2
F = ILB sin 90° = ILB =
LB =
R
R
To maintain a constant EMF, the rod must be towed to the right
with constant speed. An external agent must do work on the bar.
Power:
Where does the energy go?
Ohmic dissipation:
Induction
•  Motion of wire in B field induces an
emf (thus a current if circuit is
closed). This is motional emf, and
the emf is due to the magnetic field.
•  Are there other ways? Can an emf
be produced (induced) if the circuit
is stationary?
Magnetic flux
ΦB =
�
� · dA
�
B
Faraday’s Law of Induction
•  The induced emf in a closed loop equals the negative of the rate of
change of the magnetic flux through the loop:
Vemf
dΦB
=−
dt
•  Valid regardless of the reason for the change in magnetic
flux (could be a motional emf, a changing magnetic field,
changing circuit geometry, etc)
•  Note: the fact this equation is valid for either motional emf
or for a changing magnetic field is what led Einstein to
come up with the theory of relativity.
•  Technically, Faraday’s law of induction refers to situations
where the B field is changing (but the equation also works
for motional emf.)
CT 33.3
A loop of wire is moving rapidly through a
uniform magnetic field as shown. Is a nonzero EMF induced in the loop?
A: Yes, there is
B: No, there is not
Lenz’s Law
Vemf
dΦB
=−
dt
•  What is with the minus sign in Faraday’s Law?
•  If current in loop is solely due to induction, direction of current is
such that the magnetic field associated with this induced current
causes a reduction in the rate of change of the net magnetic flux.
•  Consistent with conservation of energy; prevents situations of
perpetual motion.
Recall the sliding bar: Induced emf drives a current that produces a
magnetic field directed out of page through the circuit. This reduces
the net rate of change of the magnetic flux.
Note: Induced current “tries” to cause magnetic flux to not change. It
always fails.
Lenz’s Law
CT 33.6
A current-carrying wire is pulled away from a
conducting loop. As the wire moves, is there a
current induced around the loop?
A: Yes, CW
B: Yes, CCW
C: No
CT 33.6c
A loop of wire is near a long straight wire that is
carrying a large current I, which is decreasing
with time.
The loop and wire are in the same plane. The
current induced in the loop is
I to the right, but decreasing
loop
A: CW
B: CCW
C: No current
Example
A long straight wire has a current I(t)
changing in time by the equation I(t) = d t.
A current loop of length l and width w is
situated such that the nearest end is a
distance a from the straight wire.
What is the induced current if the resistance
of the loop is R?
Applications of Induction
Magnetic recording and playback:
•  hard drives, tape, credit cards, answering machines,
etc
Recording:
varying current in solenoid produced varying magnetic
field, which aligns magnetic dipoles in material.
Playback:
Varying B in time induces emf in
solenoid which produces varying
current and voltage.
Applications Cont’d
•  Spark Plugs (sudden change in current produces huge EMF)
•  Dynamic Microphones (sound waves move diaphragm and
attached coil in magnetic field, inducing current)
•  Transformers and inductors (next chapter)
•  Metal Detectors (discussed with Eddy currents)
•  Electric guitar pickup (strings are magnetized by permanent
magnet)
Applications: Electric Generators
A coil of wire is spun in a magnetic field. This produces an
EMF and also a current; both vary with time. (AC-alternating
current)
•  Power plant
•  Alternator
•  Portable home generator
Real generators use more than one coil!
http://www.sciencejoywagon.com/physicszone/otherpub/wfendt/
generatorengl.htm
CT 33.8b
What can you say about the current generated
by the loop at this moment shown?
A) Maximum
B) Zero
C) Nonzero, and
changing
The EMF produced by an AC generator is:
dΦB
d
�=−
= − (BA cos(ωt)) = ωBA sin(ωt) = �0 sin(ωt)
dt
dt
In the United States and Canada Vemf = 170 Volts and f = ω/2π
= 60 Hz (for home outlets). (Vrms = 120 Volts)
An energy source is needed to turn the wire coil. Examples
include burning coal or natural gas to produce steam (which
drives pistons); falling water; engine in hybrid car
Example
An electric generator consists of a 100-turn circular coil 50 cm in
diameter. Its rotated at f=60 Hz inside a solenoid of radius 75 cm
and winding density n = 50 cm-1. What DC current in the solenoid is
needed for the the maximum emf of the generator to be 170 V?
Eddy Currents
Eddy currents are induced currents in a large (2D or 3D) chunk of
a conductor (moving through a B field or in a changing B field)
Applications: Roller coaster car breaking,
induction stoves, magnetic levitation (trains),
metal detectors, etc
•  The windings of a long solenoid carrying a current I
•  Where does the emf in the wire loop come from? What actually
drives the current? How does the wire know that the B field
changed inside the solenoid?
Induced Electric Field
An electric field is generated (induced) by the changing B field
(regardless of whether a wire is there or not).
This induced electric field is what produces the EMF.
�
Alternate expression of Faraday’s
dΦ
B
� =−
� · dr
Vemf = E
Law of Induction (associated with
dt
a changing B field)
What is the electric field?
�
dΦB
�
E · d�r = −
dt
Outside solenoid:
dB
2πrE =
πR2
dt
R2 dB
(r > R)
E(r, t) =
2r dt
�
dΦB
�
E · d�r = −
dt
Inside solenoid:
dB 2
2πrE =
πr
dt
dB r
E=
(r < R)
dt 2
Mutual Inductance
•  Consider two stationary circuits in the vicinity of each other.
Current in one circuit produces a magnetic field, and thus a magnetic
flux through the other circuit.
•  Mutual inductance is the constant of proportion between I1 and the
magnetic flux in circuit #2
ΦB2 = M21 I1
•  If I1 is changing, an emf is induced in circuit #2
dΦB2
dI1
�2 = −
= −M21
dt
dt
• Mutual inductance is the basis of transformers (covered next chapter)
Self-Inductance and Inductors
• Self inductance determines the magnetic flux in a single circuit due
to the circuit’s own current.
ΦB = LI
•  Every circuit has some inductance!
•  Inductor – a circuit device designed to have a particularly high
inductance (typically a solenoid)
dI
•  Back emf: emf due to Faraday’s Law: �L = −L
dt
CT 33.23
Two long solenoids, each of inductance L, are
connected together to form a single very long
solenoid of inductance Ltotal. What is Ltotal?
L(total)=?
+
A: L
B: 2L
C: 4L
D: 8L
E: other
LL
=
L
L
Ltotal = ?
Clicker Question
CT 33.28
What is the current through the resistor
immediately after the switch is closed ?
R = 20!
V = 10V
A) 0 A
B) 0.5 A/s
C) 1 A/s
D) 10 A/s
E) other
L = 10H
CT 33.29
The switch is closed at t=0.
What is the current through the resistor
after a very long time?
R = 20!
V = 10V
A) 0 A
B) 0.5 A
C) 1 A
D) 10 A
E) other
L = 10H
What is voltage at top and bo1om of inductor right a7er switch is closed (I=0)? 1.  Vtop = ξ0, Vbo-om= 0 2.  Vtop = 0, Vbo-om = 0 3.  Vtop = ξ0, Vbo-om=ξ0 4.  WTF (huh?)?! This seems wrong. Seems to depend how I look at it… LR circuit with constant voltage source:
•  Faraday’s Law of induction (a
law):
�revision of Kirchoff’s loop
�
dI
� = −L
� · dl
∆Vi
E
=−
dt
i
always integrate in
direction of current
dI
−L
= IR − ξ0
dt
E to the right in Resistor
E downward in battery
E=0 in wire
of inductor
Note: not proper to think of a
voltage difference across inductor
(as many books do)
Thinking of circuits solely in terms of voltages is problematic sinc
the electric field is no longer conservative (it has curl)
�
dI
�
E · d� = IR − ξ0 = −L
dt
�
0
I
dI
−1
=
IR − ξ0
L
�
t
dt
0
�
ξ0 �
I(t) =
1 − e−Rt/L
R
timescale to reach steady state: τ = L/R
|L dI/dt|
Back emf in motor circuit
Notice that an electric motor can act as a generator (this is the
principle behind hybrid cars)
So, for an electric motor circuit, the rotating coil will generate
a back emf (this emf causes the current to be smaller than
what it would be without a back emf)