Electromagnetic Induction!
March 12, 2014
Chapter 29
1
Notes
!  Exam 4 next Tuesday
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Covers Chapters 27, 28, 29 in the book
Magnetism, Magnetic Fields, Electromagnetic Induction
Material from the week before spring break and this week
Homework sets 7 and 8
!  I will be out of town next week
•  Prof. Joey Huston will teach lectures
March 12, 2014
Chapter 29
2
Faraday’s Law of Induction
!  The magnitude of the potential difference, ΔVind,
induced in a conducting loop is equal to the time rate of
change of the magnetic flux through the loop.
ΔVind = −
dΦ B
dt
!  We can change the magnetic flux in several ways:
dB
A,θ constant: ΔVind = −Acosθ
dt
dA
B,θ constant: ΔVind = −Bcosθ
dt
A, B constant: ΔVind = ω ABsinθ
!  Where A is the area of the loop, B is the magnetic field strength,
and θ is the angle between the surface normal and the B field vector
March 12, 2014
Chapter 29
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Law of Induction
!  This potential difference means that the changing magnetic
field creates an electric field around the loop!
!  Thus, there are two ways of producing an electric field
!  If the electric field arises from a charge, the resulting electric
force on a test charge is conservative
!  Conservative forces do no work when they act on an object
whose path starts and ends at the same point in space
!  In contrast, electric fields generated by changing magnetic
fields give rise to electric forces that are not conservative
!  Thus, a test particle that moves around a circular loop once
will have work done on it by this electric field
!  In fact, the amount of the work done is the induced
potential difference times the charge of the test particle
March 12, 2014
Chapter 29
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Lenz’s Law
!  Lenz’s Law defines a rule for determining the direction of an
induced current in a loop
!  An induced current will have a direction such that the
magnetic field due to the induced current opposes the
change in the magnetic flux that induces the current
!  The direction of the induced current corresponds to the
direction of the induced emf
!  We can apply Lenz’s Law to the situations described in
yesterday’s lecture
March 12, 2014
Chapter 29
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Lenz’s Law
!  The physical situation shown here
involves moving a magnet toward a
loop with the north pole pointed
toward the loop
!  The magnetic field lines point
away from the north pole of the
magnet
!  As the magnet moves toward the loop,
the magnitude of the magnetic field within the loop, in the
direction pointing toward the loop, increases
!  Lenz’s law states that a current is induced in the loop that
tends to oppose the change in magnetic flux
!  This induced magnetic field then points in the opposite
direction from the field from the magnet
March 12, 2014
Chapter 29
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Lenz’s Law
!  The physical situation shown here
involves moving a magnet toward a
loop with the south pole pointed
toward the loop
!  The magnetic field lines point toward
the south pole of the magnet
!  As the magnet moves toward the
loop, the magnitude of the field
increases in the direction pointing toward the south pole
!  Lenz’s law states that a current is induced in the loop that
tends to oppose the change in magnetic flux
!  This induced magnetic field then points in the opposite
direction from the field from the magnet
March 12, 2014
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Lenz’s Law - Four Cases
!  (a) An increasing magnetic field pointing to the right
induces a current that creates a magnetic field to the left
!  (b) An increasing magnetic field pointing to the left induces
a current that creates a magnetic field to the right
!  (c) A decreasing magnetic field pointing to the right induces
a current that creates a magnetic field to the right
!  (d) A decreasing magnetic field pointing to the left induces a
current that creates a magnetic field to the left
March 12, 2014
Chapter 29
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Eddy Currents
!  Let’s consider two pendulums, each with a non-magnetic
conducting metal plate at the end that is designed to pass
through the gap of a strong permanent magnet
!  One metal plate is solid and the other has slots cut in it
!  We pull back both pendulums and release them
March 12, 2014
Chapter 29
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Eddy Currents
!  The pendulum with the solid metal sheet stops in the
gap of the magnet while the grooved sheet passes through the
magnetic field, only slowing slightly
!  This demonstrates the importance of induced eddy currents
!  As the pendulum with the solid plate enters the magnetic field,
Lenz’s law tells us that the changing magnetic flux will induce
currents that tend to oppose the change in flux
!  Just like eddy currents, swirls we see in turbulent water flow
!  These currents interact with the magnetic field to stop the
pendulum
•  Kinetic energy is converted into heat
!  Often, eddy currents have to be eliminated through segmenting
•  Pulsed magnets
•  Transformers
!  Induced eddy currents can also be employed in practical
situations such as braking railroad cars
March 12, 2014
Chapter 29
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Induced Potential Difference on a Moving Wire
in a Magnetic Field
!  Consider a conducting wire moving
with constant velocity perpendicular
to a constant magnetic field
!  The wire is oriented such that it is
perpendicular to the velocity and to
the magnetic field
!  The magnetic field exerts a force on
the conducting electrons in the wire,
causing them to move downward
!  This motion of the electrons produces a net negative charge
at the bottom of the wire and a net positive charge at the top
of the wire
!  This charge separation produces an electric field that exerts
a force on the conduction electrons that tends to cancel the
magnetic force
March 12, 2014
Chapter 29
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Induced Potential Difference on a
Moving Wire in a Magnetic Field
!  After some time the forces on the
electrons are in equilibrium and we
can write
FB = qvB = FE = qE
!  Thus we can express the induced
electric field as
E = vB
!  This electric field produces a potential
between the two ends of the wire given by
ΔVind
E=
= vB

!  We can then write the voltage between the ends of the wire
as ΔV = vB
ind
March 12, 2014
Chapter 29
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Satellite Tethered to a Space Shuttle
!  In 1996, the Space Shuttle Columbia
deployed a tethered satellite on a wire out to
a distance of 20 km
!  The wire was oriented perpendicular to the
Earth’s magnetic field at that point, and the
magnitude of the field was B = 5.1·10–5 T
!  Columbia was traveling at a speed of
7.6 km/s
PROBLEM
!  What was the potential difference induced
between the ends of the wire?
March 12, 2014
Chapter 29
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Satellite Tethered to a Space Shuttle
SOLUTION
!  The length of the wire is 20 km and the speed of the wire
through the magnetic field of the Earth is the same as the
speed of the Space Shuttle, which is 7.6 km/s
!  The induced potential difference is
ΔVind = vLB = ( 7.6⋅103 m/s )( 20⋅103 m )(5.1⋅10−5 T )
ΔVind = 7800 V
!  The astronauts on the Space Shuttle measured a current of
about 0.5 A at a voltage of 3500 V
!  The circuit consisted of the deployed wire and ionized
atoms in space as the return path for the current
!  The wire broke just as the deployment length reached 20 km
March 12, 2014
Chapter 29
17
Pulled Connecting Rod
!  A conducting rod is pulled
horizontally by a constant force of
F = 5.00 N along a set of
conducting rails separated by a
distance a = 0.500 m
!  The two rails are connected, and no friction occurs between
the rod and the rails
!  A uniform magnetic field with magnitude B = 0.500 T is
directed into the page
!  The rod moves at constant speed, v = 5.00 m/s
PROBLEM
!  What is the magnitude of the induced potential difference in
the loop created by the connected rails and the moving rod?
March 12, 2014
Chapter 29
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Pulled Connecting Rod
SOLUTION
!  The induced potential difference is
dA
ΔVind = −Bcosθ
dt
!  The area of the loop is increasing with time
A = A0 + a ( vt ) = A0 + vta
!  The change in the area as a function of time is
dA d
= ( A0 + vta ) = va
dt dt
!  The magnitude of the induced potential difference is
dA
ΔVind = −Bcosθ
= vab = (5.00 m/s )( 0.500 m )( 0.500 T ) = 1.25 V
dt
March 12, 2014
Chapter 29
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Generators and Motors
!  The third special case of our simple induction processes is
technologically by far the most interesting
!  It is the variation of the angle between the loop and the
magnetic field with time, while keeping the area of the loop
as well as the magnetic field strength constant in time
!  In this way, Faraday’s Law of Induction can be applied to
the generation and use of electric current
!  A device that produces electric current from mechanical
motion is called an electric generator
!  A device that produces mechanical motion from electric
current is called an electric motor
March 12, 2014
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Generators and Motors
!  A simple generator consists of a loop forced to rotate in a
fixed magnetic field
!  The driving force that causes the loop to rotate can be
supplied by hot steam running over a turbine
!  Or the loop can be made to rotate by water or wind in a
pollution-free way of generating electrical power
!  In a direct-current generator, the rotating loop is connected
to an external circuit through a split commutator ring
!  As the loop turns, the connection is
reversed twice per revolution, so the
induced potential difference always
has the same sign
March 12, 2014
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Generators and Motors
!  A similar arrangement is used to produce an alternating
current
!  An alternating current is a current that varies in time
between positive and negative values, with the variation
often showing a sinusoidal form
!  Each end of the loop is connected to the external circuit
through its own solid slip ring
!  Thus, this generator produces an
induced potential difference that
varies from positive to negative and back
!  A generator that produces alternating
voltages and the resulting alternating
current is also called an alternator
March 12, 2014
Chapter 29
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March 12, 2014
Chapter 29
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Induced Electric Field
!  Faraday’s Law of Induction states that a changing magnetic
flux produces an induced potential difference
!  Consider a test charge q moving in a circular path with
radius r in an electric field
!  The work done is equal to the integral of the scalar product
of the force and the displacement
!  For now we assume that the electric field is constant and
that is has field lines that are circular
!  Considering one revolution of the test charge we obtain the
work done on the test charge to be
 
 
W=
∫ Fids = ∫ qEids = (qE )( 2π r )
!  The work done by a constant electric field is qΔVind so
qΔVind = 2π rqE ⇒ ΔVind = 2π rE
March 12, 2014
Chapter 29
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Induced Electric Field
!  We can generalize this result by considering the work
on a test particle with charge q moving along an arbitrary
closed path as
 
 
W=
∫ Fids = q ∫ Eids
!  Again substituting qΔVind for the work we obtain
 
ΔVind = 
∫ Eids
!  Since the induced potential difference is the change in flux
 
dΦ B
∫ Eids = − dt
!  A changing magnetic field induces an electric field
!  This applies to any closed path drawn in a changing
magnetic field
March 12, 2014
Chapter 29
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Inductance of a Solenoid
!  Consider a long solenoid with N turns carrying a current i
!  This current creates a magnetic field in the center of the
solenoid resulting in a magnetic flux of ΦB
!  The same magnetic flux goes through each of the N
windings of the solenoid
!  It is customary to define the flux linkage as the product of
the number of windings and the magnetic flux, NΦB
!  Inside the solenoid, the magnetic flux is
 
ΦB = 
∫∫ B• dA = BA
!  Remembering that for a solenoid
N
B = µ0ni = µ0 i

March 12, 2014
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Inductance of a Solenoid
!  We can then write the flux linkage as
2
⎛
N
N
⎛
⎞
NΦ B = NBA = N ⎜ µ0 i ⎟ A = ⎜ µ0
⎝
 ⎠

⎝
⎞
A⎟ i = Li
⎠
!  The flux linkage is a constant times the current
!  We define this constant to be the inductance L
!  Thus the inductance is a measure of the flux linkage
produced by the solenoid per unit current
!  For an solenoid to behave in this manner, it must not have
any magnetic materials in its core
!  The unit of inductance is the henry (H), named after
American physicist Joseph Henry (1797 – 1878)
[Φ B ]
1 T m2
[L] =
⇒1 H =
[i]
1A
March 12, 2014
Chapter 29
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Inductance of a Solenoid
!  We can define the magnetic permeability of free space as
µ0 = 4π ⋅10−7 H/m
!  The inductance of a solenoid is then
N2
Lsolenoid = µ0
A

N
take n =

(n )2
Lsolenoid = µ0
A = µ0n 2 A

!  The inductance of a solenoid depends only on its geometry
!  When a solenoid is used in an electric circuit, it is called an
inductor
March 12, 2014
Chapter 29
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Self Inductance and Mutual
Induction
!  Consider the situation in which two coils, or inductors, are
close to each other
!  A current in the first coil produces magnetic flux in the
second coil
!  Changing the current in the first coil will induce a potential
difference in the second coil
!  However, the changing current in the first coil also induces
a potential difference in itself
!  This phenomenon is called self-induction
!  The resulting potential difference is termed the self-induced
potential difference
!  Changing the current in the first coil also induces a potential
difference in the second coil
!  This phenomenon is called mutual induction
March 12, 2014
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Self Induction
!  According to Faraday’s Law of Induction, the self-induced
potential difference for any inductor is given by
d ( NΦ B )
d ( Li )
di
ΔVind ,L = −
=−
= −L
dt
dt
dt
!  Thus, in any inductor, a self-induced potential difference
appears when the current changes with time
!  This self-induced potential difference depends on the time
rate of change of the current and the inductance of the
device
!  Lenz’s Law provides the direction of the self-induced
potential difference
!  The negative sign provides the clue that the induced
potential difference always opposes any change in current
March 12, 2014
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Self Induction
!  Consider a circuit with current flowing through
an inductor that increases with time
!  The self-induced potential difference will oppose
the increase in current
!  Consider a circuit with current flowing through
an inductor that is decreasing with time
!  A self-induced potential difference will oppose
the decrease in current
!  We have assumed that these inductors are ideal
inductors; that is, they have no resistance
!  Induced potential differences manifest
themselves across the connections of the inductor
March 12, 2014
Chapter 29
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