CHAPTER 21
ELECTROMAGNET INDUCTION
FARADAY’S LAW
Units
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Induced EMF
Faraday’s Law of Induction; Lenz’s Law
EMF Induced in a Moving Conductor
Changing Magnetic Flux Produces an Electric Field
Electric Generators
Back EMF and Counter Torque; Eddy Currents
Transformers and Transmission of Power
Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI
Inductance
Energy Stored in a Magnetic Field
LR Circuit
AC Circuits and Reactance
LRC Series AC Circuit
Resonance in AC Circuits
Michael Faraday
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1791 – 1867
Great experimental scientist
Invented electric motor, generator and transformers
Discovered electromagnetic induction
Discovered laws of electrolysis
Induced EMF
Almost 200 years ago, Faraday looked for
evidence that a magnetic field would
induce an electric current with this
apparatus:
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Faraday concluded that although a constant magnetic field produces no current in a conductor, a
changing magnetic field can produce an electric current. Such a current is called an induced
current. This is also called electromagnetic induction.
Faraday’s Experiment – Set Up
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A current can be produced by a changing
magnetic field
– First shown in an experiment by Michael
Faraday
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A primary coil is connected to a battery
A secondary coil is connected to an
ammeter
The purpose of the secondary circuit is to detect
current that might be produced by the magnetic field
When the switch is closed, the ammeter reads a current and then returns to zero
When the switch is opened, the ammeter reads a current in the opposite direction and then
returns to zero
When there is a steady current in the primary circuit, the ammeter reads zero
Faraday’s Conclusions
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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
Induced EMF
He found no evidence when the current was steady, but did see a current induced when the switch
was turned on or off.
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Magnetic Flux
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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
You are given a loop of wire
The wire is in a uniform magnetic field
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
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
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Example 1: Calculate the magnetic flux
A conducting circular loop of radius 0.250 m is placed in the xy-plane in a uniform magnetic field or
0.60T that points in the positive z-direction, the same direction as the normal to the plane.
a) Calculate the magnetic flux through the loop.
A   r 2   (0.250m)2  0.196m2
 B  AB cos   (0.196m2 )(0.360T) cos(0o )  0.0706T  m2  0.0706Wb
Suppose the loop is rotated clockwise around the x-axis so the normal direction now points at a
45o angle with respect to the z-axis. Recalculate the magnetic flux through the loop.
 B  AB cos   (0.196m2 )(0.360T) cos(45.0o )  0.0499T  m2  0.0499Wb
c) What is the change in flux due to the rotation of the loop?
 B  AB cos   (0.196m2 )(0.360T) cos(45.0o )  0.0499T  m2  0.0499Wb
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Faraday’s Law of Induction; Lenz’s Law
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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)
Electromagnetic Induction – Results of the Experiment
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A current is set up in the circuit as long as there is relative motion between the magnet and
the loop
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The current is called an induced current because is it produced by an induced emf
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The same experimental results are found whether the loop moves or the magnet moves
Faraday’s Law and Electromagnetic Induction
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The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux
through the circuit
If a circuit contains N tightly wound loops and the flux changes by ΔΦB during a time
interval Δt, the average emf induced is given by Faraday’s Law:
Faraday’s law of induction:
[1 loop]
[N loops]
Faraday’s Law and Lenz’ Law
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The change in the flux, ΔΦB, can be produced by a change in B, A or θ
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Since ΦB = B A cos θ
The negative sign in Faraday’s Law is included to indicate the polarity of the induced emf,
which is found by Lenz’ Law
– The current caused by the induced emf travels in the direction that creates a
magnetic field with flux opposing the change in the original flux through the
circuit
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Magnetic flux will change if the area of the loop changes:
Magnetic flux will change if the angle between the loop and the field changes:
There are many devices that operate on the basis of Faraday’s law.
An electric guitar pickup:
Applications of Faraday’s Law – Electric
Guitar
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A vibrating string induces an emf in a coil
A permanent magnet inside the coil magnetizes a
portion of the string nearest the coil
As the string vibrates at some frequency, its magnetized
segment produces a changing flux through the pickup
coil
The changing flux produces an induced emf that is fed to
an amplifier
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Applications of Faraday’s Law – Apnea Monitor
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The coil of wire attached to the chest carries an
alternating current
An induced emf produced by the varying field
passes through a pick up coil
When breathing stops, the pattern of induced
voltages stabilizes and external monitors sound an
alert
Faraday’s Law of Induction
Tape recorder:
Faraday’s Law of Induction; Lenz’s Law
Problem Solving: Lenz’s Law
1. Determine whether the magnetic flux is increasing, decreasing, or unchanged.
2. The magnetic field due to the induced current points in the opposite direction to the
original field if the flux is increasing; in the same direction if it is decreasing; and is zero if
the flux is not changing.
3. Use the right-hand rule to determine the direction of the current.
4. Remember that the external field and the field due to the induced current are different.
Practice with Lenz’s law.
Initially, the magnetic field pointing out of the page passes
through the loop. If you pull the loop out of the field, magnetic
flux through the loop decreases; so the induced current will be
in a direction to maintain the decreasing flux through the loop:
the current will be counterclockwise to produce a magnetic field
outward.
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The external field is into the page. The coil area gets smaller, so
the flux will decrease; hence the induced current will be
clockwise, producing its own field into the page to make up for the
flux decrease.
Initially, the magnetic field pointing out of the page passes through the
loop. If you pull the loop out of the field, magnetic flux through the loop
decreases; so the induced current will be in a direction to maintain the
decreasing flux through the loop: the current will be counterclockwise
to produce a magnetic field outward.
Magnetic field lines point out from the N pole of a magnet, so as
the magnet moves toward the loop, the magnet’s field points
into the page and is getting stronger. The current in the loop will
be induced in the counterclockwise direction in order to produce
a field B out of the page.
Initially there is no flux through the loop. When you start to rotate the loop,
the external field through the loop begins increasing to the left. To
counteract this change in flux, the loop will have current induced in a
counterclockwise direction so as to produce its own field to the right.
Example 2: Pulling a coil from a magnetic field
A square coil of wire with side l  5.00cm contains 100
loops and is positioned perpendicular to a uniform 0.600T
magnetic field. It is quickly pulled from the field at constant
speed (moving perpendicular to B ) to a region where B
drops abruptly to zero. At t = 0, the right edge of the coil is
at the edge of the field.
It takes 0.100 s for the whole coil to reach the field-free
region. The coil’s total resistance is 100 . Find
a) The rate of change in flux through the coil.
A  l 2  (5.00 x102 m)2  2.5  x103 m2
 B  BA  (0.600T)(2.50x103 m2 )  1.50x103 Wb
 B 0  (1.50 x103 Wb)

 1.50 x102 Wb/s
t
0.100s
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b) the emf and current induced.
  N
I

R

 B
 (100)(1.50 x102 Wb/s)  1.50V
t
1.50V
 1.50 x102 A  15.0mA
100
By Lenz’s law, the current must be clockwise to produce more B into the page and thus oppose
the decreasing flux into the page.
c) How much energy is dissipated in the coil?
E  Pt  I 2 Rt  (1.50x102 A)2 (100)(0.100s)  2.25x103 J
d) What was the average force required?
F
W 2.25x103 J

 0.0450 N
d 5.00 x102 m
EMF Induced in a Moving Conductor
This image shows another way the magnetic flux can change:
The induced current is in a direction that tends to slow
the moving bar – it will take an external force to keep it
moving.
Assume that a uniform magnetic field B is perpendicular to the area bounded by the U-shaped
conductor and the movable rod resting on it. If the rod is made to move at a speed v, it travels a
distance x  vt in a time  t . Therefore, the area of the loop increases by an amount
A  l x  lvt in a time  t . By Faraday’s law there is an induced emf  whose magnitude is
given by
 B BA Blvt



 Blv
t
t
t
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Example 3: Does a moving airplane develop a large emf?
An airplane travels 1000km/h in a region where the Earth’s
magnetic field is 5.0 x105 T , and is nearly vertical. What is
the potential difference induced between the wing tips that
are 70m apart?
  Blv  (5.0 x105 T)(70m)(280m / s)  1.0V
Example 4: Electromagnetic blood-flow measurement.
The rate of blood flow in our body’s vessels can be
measured since blood contains charged ions. If a blood
vessel is 2.0 mm in diameter, the magnetic field is
0.080T and the measured emf is 0.10mV. What is the flow
velocity of the blood?
v

Bl

(1.0 x104 V )
 0.63m / s
(0.08T)(2.0 x103 m)
Mechanical Work and Electrical Energy
If the rod is to move at a constant speed, an external force must be exerted on it. This force should
have equal magnitude and opposite direction to the magnetic force:
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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
Lenz’s Law
This conducting rod completes the circuit. As it falls, the magnetic flux
decreases, and a current is induced.
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The force due to the induced current is upward, slowing the fall.
Mechanical Work and Electrical Energy
The mechanical power delivered by the external force is:
Compare this to the electrical power in the light bulb:
Therefore, mechanical power has been converted directly into electrical power.
Lenz’ Law – Moving Magnet Example
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A bar magnet is moved to the right toward a stationary loop of wire (a)
– As the magnet moves, the magnetic flux increases with time
The induced current produces a flux to the left, so the current is in the direction shown (b)
When applying Lenz’ Law, there are two magnetic fields to consider
– The external changing magnetic field that induces the current in the loop
– The magnetic field produced by the current in the loop
Application – Tape Recorder
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A magnetic tape moves past a recording and playback
head
– The tape is a plastic ribbon coated with iron oxide
or chromium oxide
To record, the sound is converted to an electrical signal
which passes to an electromagnet that magnetizes the
tape in a particular pattern
To playback, the magnetized pattern is converted back
into an induced current driving a speaker
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Generators
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Alternating Current (AC) generator
– Converts mechanical energy to electrical
energy
– Consists of a wire loop rotated by some
external 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
Basic operation of the generator
– As the loop rotates, the magnetic flux through it
changes with time
– This induces an emf and a current in the external circuit
– 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
Electric Generators
A dc generator is similar, except that it has a split-ring commutator
instead of slip rings.
AC Generators
• The emf generated by the rotating loop can be found by
ε =2 B ℓ v=2 B ℓ sin θ
If the loop rotates with a constant angular speed, ω, and N
turns
ε = N B A ω sin ω t
• ε = εmax when loop is parallel to the field
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ε = 0 when the loop is perpendicular to the field
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Joseph Henry
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1797 – 1878
First director of the Smithsonian
First president of the Academy of Natural Science
First to produce an electric current with a magnetic field
Improved the design of the electro-magnetic and
constructed a motor
Discovered self-inductance
Transformers and Transmission of Power
A transformer consists of two coils, either interwoven or linked
by an iron core. A changing emf in one induces an emf in the
other.
The ratio of the emfs is equal to the ratio of the number of turns
in each coil:
This is a step-up transformer – the emf in the secondary coil is
larger than the emf in the primary:
Transformers
A transformer is used to change voltage in an alternating
current from one value to another.
By applying Faraday’s law of induction to both coils, we
find:
Here, p stands for the primary coil and s the secondary.
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Example 5: Portable radio transformer.
A transformer for home use of a portable radio reduces 120-V ac to 9.0-V ac. (Such a device also
contains diodes to change the 9.0-V ac to dc, to be like its 9.0-V battery.) The secondary coil
contains 30 turns and the radio draws 400mA.
a) Calculate the number of turns in the primary coil.
V
(30)(120V )
NP  NS P 
400 turns
VS
9.0V )
b) Calculate the current in the primary.
N
 30 
I P  I S S  (0.40 A) 
  0.030 A
NP
 400 
c) Calculate the power transformed.
P  I SVS  (0.40 A)(9.0V )  3.6W
The power in the primary coil, P = (0.030A)(120V) =3.6W, is the same as the power in the
secondary coil.
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CHAPTER 21
ELECTROMAGNETIC INDUCTION
CONCEPTS
1. Units of magnetic flux consists of: T-m2, weber, and volt-second.
2. Magnetic flux density has the same units as magnetic field.
3. A wire moves across a magnetic field. The emf produced in the wire depends on: the
field’s magnetic flux, the length of the wire, and the orientation of the wire with respect to
the magnetic field vector.
4. Faraday’s law of induction states that the emf induced in a loop of wire is proportional to
the time variation of the magnetic flux.
5. Doubling the number of loops of wire in a coil produces a change on the induced emf that
is twice as much.
6. A coil lies flat on a table in a region where the magnetic field vector points straight up. The
magnetic field vanishes suddenly. When viewed from above, the induced current flows
counterclockwise.
7. A circular loop of wire is rotated at constant angular speed about an axis whose direction
can be varied. In a region where a uniform magnetic field points straight down, the
orientation of the loop’s axis of rotation must be vertical if the induced emf is to be zero.
8. A bar magnet is positioned inside a coil. In the following, “work”
refers to any work done as a consequence of the fact that the bar
is magnetic. If the bar is suddenly pulled out of the coil, more work
will be done if the switch is closed.
9. A generator coil rotates through 60 revolutions each second. The
frequency of the emf is 60 Hz.
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10. A lightbulb is plugged into a household outlet, and one of the wires leading to it is wound
into a coil. Now a slug of iron is slid into the coil. This will cause the lamp to dim because a
back emf will be produced by the coil, and this will reduce the current flowing in the coil
and the lamp.
11. A transformer is a device that operates only on AC.
12. The coil of a generator rotates through one complete turn in a uniform magnetic field 50
times every second. When this generator is connected to an external load, the current
through the load reverses directions 100 times every second.
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