Electromagnetic
Induction
emf – electromotive force
When you studied electric circuits, you
learned that a source of electrical energy,
such as a battery is needed to produce a
continuous current.
 The potential difference, or voltage given to
the charges by the battery is called the
electromotive force, or emf.
 Emf is not actually a force, it is a potential
difference measured in volts.

Faraday’s Experiment Induction
• From Ch. 24, a B field is
produced by a moving
charge. Well it works
the other way too.

A current can be produced by a changing
magnetic field (not a constant B field!)

First shown in an experiment by Michael Faraday
 A primary coil is connected to a battery
 A secondary coil is connected to an ammeter
Faraday’s
Experiment
The secondary circuit has a
current that is induced by the
magnetic field that is created by
current through the primary coil.
 When the switch is closed, the ammeter deflects in one
direction and then returns to zero. It responds to the
changing magnetic field as the primary current ramps up.
 When there is a steady current in the primary circuit, the
ammeter reads zero.
 When the switch is opened, the ammeter deflects in the
opposite direction and then returns to zero (as the current
ramps down.)
•
Faraday’s Conclusions
An electrical current is produced by a
changing magnetic field
 The secondary circuit acts as if a source
of emf were connected to it for a short
time
 It is customary to say that an induced

emf is produced in the secondary circuit
by the changing magnetic field
Magnetic Flux
The emf is actually induced by a change in the
quantity called the magnetic flux rather than
simply by a change in the magnetic field.
 Magnetic flux is defined in a manner similar to
that of electrical flux.
 Magnetic flux is proportional to both the
strength of the magnetic field passing through
the plane of a loop of wire and the area of the
loop.

Magnetic Flux, 2

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The box is a loop of wire.
The wire is in a uniform
magnetic field B.
The loop has an area A.
The flux is defined as
ΦB = BA = B A cos θ


θ is the angle between B
and the normal to the plane
Flux ONLY counts the
component of the B field that
is perpendicular to the loop
Magnetic Flux, 3
ΦB = BA = B A cos θ
θ is the angle between B and
the normal to the plane of the
loop (one loop).




When the field is perpendicular to the plane of the
loop, as in (a), θ = 0 and ΦB = ΦB, max = BA
When the field is parallel to the plane of the loop, as
in b, θ = 90° and ΦB = 0
 The flux can be negative, for example if θ = 180°
SI units of flux are T m² = Wb (Weber)
(so if you see the unit of Weber/m2, that is a Tesla)
Magnetic Flux

The flux can be visualized with respect to
magnetic field lines

The value of the magnetic flux is proportional to
the total number of lines passing through the loop
When the area is perpendicular to the lines,
the maximum number of lines pass through
the area and the flux is a maximum
 When the area is parallel to the lines, no lines
pass through the area and the flux is 0

emf and magnetic flux

Faraday’s law of induction states
that the instantaneous emf induced in a
circuit equals the rate of change of
magnetic flux through the circuit:
𝒅𝝋
∆(𝑵𝑩𝑨𝒄𝒐𝒔𝜽)
 𝜺 = −𝑵
or 𝜺 = −
𝒅𝒕
∆𝒕
 N = number of loops of wire
 emf = ε = induced voltage
 You will need to judge the – sign
Electromagnetic Induction –
An Experiment
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When a magnet moves toward a
loop of wire, the ammeter shows
the presence of a current (a)
When the magnet is held
stationary, there is no current (b)
When the magnet moves away
from the loop, the ammeter
shows a current in the opposite
direction (c)
If the loop is moved instead of
the magnet, a current is also
detected
Induction
Note the direction of
the induced current.
 Top picture, N end magnet
moves towards loop.
 Use RHR B on the current
in the loop. Your thumb
will point in opposite
direction of motion of
magnet.
 Induced current always
opposes change in B field
(that’s why there’s a
negative sign in the eq.)

Electromagnetic Induction –
Results of the Experiment

A current is set up in the circuit as long
as there is relative motion between the
magnet and the loop


The same experimental results are found
whether the loop moves or the magnet
moves
The current is called an induced current
because is it produced by an induced
emf
PhET simulations on induction

faraday_en.jar
 Go to “pickup coil” tab
 Note that these are electrons moving
in the coil, so use your left hand.
Applications of Faraday’s Law
– Ground Fault Interrupters

The ground fault interrupter
(GFI) is a safety device that
protects against electrical
shock

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Wire 1 leads from the wall outlet
to the appliance
Wire 2 leads from the appliance
back to the wall outlet
The iron ring confines the
magnetic field, which is
generally 0 (B fields from
currents in opposite directions
cancel out)
If a leakage occurs, the field is
no longer 0 and the induced
voltage triggers a circuit breaker
shutting off the current
Application of Faraday’s Law –
Motional emf
A straight conductor of length ℓ
moves perpendicularly with
constant velocity through a
uniform field
 RHR says + charges go up
 LHR says - charges go down
 The electrons in the conductor
experience a magnetic force



l
F=qvB
The electrons move to the lower
end of the conductor (LHR)
B field goes into page
Motional emf
E
As the - charges accumulate at the base, that is
like + charge moving upwards.
 As a result of this charge separation, an electric
field is produced in the conductor (see above)
 Charges build up at the ends of the conductor
until the downward magnetic force is balanced by
the upward electric force F=qE = qvB so E = vB
 There is a potential difference between the upper
and lower ends of the conductor V=E*l (recall
that E = Volts/meter, so V = E*length)

Motional emf

Therefore the potential difference between the
ends of the conductor can be found by
 emf
= ΔV = E l = B l v
(ΔV = voltage or emf, L= length of wire, v=velocity)
 The upper end is at a higher potential than the
lower end

A potential difference is maintained across the
conductor as long as there is motion through
the field

If the motion is reversed, the polarity of the
potential difference is also reversed
A wire of length L, moving through a magnetic field, B,
induces an electromotive force (EMF=Voltage).
• If the wire is part of a circuit, then there will be a current in
the direction shown (+ charge moves downwards by RHR)
• This current will interact with the magnetic field and
produce a force, F = BIL.
• Notice that the resulting force F= BIL, opposes the
original motion, v, of the wire.
•
Motional emf in a closed loop
circuit
Assume all of the circuit except R
has zero resistance. As the bar is
pulled to the right with velocity v
under the influence of an applied
EMF
force, Fapp, the resulting emf is:
∆𝜑𝐵
∆𝑥
‫= ׀𝑓𝑚𝑒׀‬
= 𝐵𝑙
= 𝐵𝑙𝑣
∆𝑡
∆𝑡
along the length of the bar
 This force sets up an induced current (see direction of I)
because the charges are free to move in the closed path

Motional emf in a closed loop
circuit


The changing magnetic flux
through the loop and the
corresponding induced emf in
the bar result from the change
in area of the loop
The induced, motional emf,
acts like a battery in the circuit
𝑬𝑴𝑭 = 𝑩𝒍𝒗 𝒂𝒏𝒅 𝑽 = 𝑰𝑹
SO
𝑰=
𝑩𝒍𝒗
𝑹
‫׀‬ε‫ = ׀‬EMF = B l v
Lenz’ Law Revisitedopposing the motion
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As the bar moves to the right,
the magnetic flux through the
circuit increases with time
because the area of the loop
increases
The induced current must be
in a direction such that it
opposes the change in the
external magnetic flux
This is why there is a
negative sign in Faraday’s law
Original: Use RHR A: fingers
into, thumb right, palm up,
therefore current flows up.
Induced: Use RHR A:
fingers in, thumb up, palm
left, induced force goes left.
Lenz’ Law, Opposing the
Motion
If the magnetic flux due
to the external field is
increasing into the page,
then the flux due to the
induced current must be
out of the page
 Therefore the current
must be counterclockwise (RHR B: Put thumb in
direction of current,
when the bar moves to
curl fingers, they point
the right

out of page!)
Lenz’ Law, Opposing the
Motion
Now imagine the bar
is moving toward the
left (less area)
 The magnetic flux
through the loop is
decreasing with time
 The induced current
must be clockwise to
to produce its own
flux into the page

RHR B: Thumb in direction of
I, curl fingers, and they go
into page (within the loop)
adding to flux.
Lenz’ Law, Moving Magnet
Example
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A bar magnet is moved to the right toward a
stationary loop of wire (a) (the diagram on the left)
 As the magnet gets closer, magnetic flux increases
The induced current (note direction of I) produces a
flux to the left, so the current is in the direction
shown for loop (b)
Note B field for (b) opposes change in B field for (a)
Eddy Current Videos
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2’ Neodymium magnet in a copper pipe
http://www.youtube.com/watch?v=E97CYWlALEs
4’ Eddy current demo w/ Alum pendulum
http://www.youtube.com/watch?v=OJvEOXsSuaQ
2’ Russian guy neodymium magnet/Eddy current
http://www.youtube.com/watch?v=tDBigHmn-FQ
Lenz’ Law, Final Note

When applying Lenz’ Law, there are two
magnetic fields to consider
The external changing magnetic field that
induces the current in the first loop
 The magnetic field produced by the current
in the second loop

Generators

Alternating Current (AC) generator
Converts mechanical energy to electrical
energy
 Consists of a wire loop rotated by some
external mechanical means
 There are a variety of sources that can
supply the energy to rotate the loop
 These may include falling water, heat by
burning coal to produce steam, nuclear
power, wind power, etc.

AC Generators
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As the loop rotates, the
magnetic flux through it changes
with time (max when it’s
perpendicular to the B field)
This induces an emf, which
creates a current in the external
circuit that runs the toaster
The ends of the loop are
connected to slip rings that
rotate with the loop
Connections to the external
circuit are made by stationary
brushes in contact with the slip
rings
Generators

The emf generated by the rotating
loop can be found by

𝜺 = 𝟐 𝑩 𝒍 𝒗 = 𝟐𝑩 𝒍 𝒗 𝐬𝐢𝐧 𝜽
The reason for the 2 above is that
there is an emf coming from
segment BC and one from AD
(both with the same
sense/direction of rotation).
 Since the axis of rotation r= a/2
(see graphic >) and v=rω and
θ=ωt, [ω = 2 f, f= freq. in Hz)


𝜀 = 2𝐵𝑙
𝑎
2
𝜔𝑠𝑖𝑛𝜔𝑡 = 𝐵𝑙𝑎𝜔𝑠𝑖𝑛𝜔𝑡
Generators (cont.)
𝑎
2

𝜀 = 2𝐵𝑙

If the coil has N turns, the emf
is N times as large because each
loop has the same emf in it.
Also using the fact that A=la
(see diagram), the total emf is
𝜀 = 𝑁𝐵𝐴 𝜔𝑠𝑖𝑛𝜔𝑡
And since the sin function is a
maximum at 90 or 270 degrees,
at those angles,
𝜀𝑚𝑎𝑥 = 𝑁𝐵𝐴𝜔
When loop is parallel to B field
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𝜔𝑠𝑖𝑛𝜔𝑡 = 𝐵𝑙𝑎𝜔𝑠𝑖𝑛𝜔𝑡
AC Generators
– Summary
If the loop rotates with a constant
angular speed, ω, and has N turns

ε = N B A ω sin ω t
A=area of loop, N = no. of loops
 ε = εmax when loop is parallel to B because the
component of velocity perpendicular to the B field is
the greatest at that point.
 ε = 0 when when the loop is perpendicular to B
 ω = 2𝜋𝑓 where f is the frequency in Hertz (60 Hz in
USA) so ω is in radians/sec and =angular velocity.

DC Generators
Components are
essentially the same as
that of an AC
generator
 The major difference is
the contacts to the
rotating loop are made
by a split ring or
commutator.

DC Generators
The output voltage
always has the same
polarity.
 The current is a pulsing
current
 To produce a steady current, many loops and
commutators around the axis of rotation are
used
 The multiple outputs are superimposed and the
output is almost free of fluctuations (not like the
graph above)

Motor example / commutator
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http://www.youtube.com/watch?v=qqkUeQ0nsF8
About 46 minutes into video
MIT 8.02 Lecture 11
RMS voltage and current
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For the sine wave signal from an AC
generator, the effective voltage (also known
as root-mean-square voltage) is calculated
by
Vrms = 0.707 Vmax (.707 =
and likewise for current
 Irms = 0.707 I max
2
2
)
QUICK QUIZ 1
The figure below is a graph of magnitude B versus time t for a magnetic
field that passes through a fixed loop and is oriented perpendicular to the
plane of the loop. Rank the magnitudes of the emf generated in the loop at
the three instants indicated (a, b, c), from largest to smallest.
QUICK QUIZ 1 ANSWER
(b), (c), (a). At each instant, the magnitude
of the induced emf is proportional to the
rate of change of the magnetic field
(hence, proportional to the slope of the
curve shown on the graph).