Transcript Chapter 31

Chapter 23
Faraday’s Law
and
Induction
Michael Faraday



1791 – 1867
Great experimental
physicist
Contributions to early
electricity include

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Invention of motor,
generator, and
transformer
Electromagnetic
induction
Laws of electrolysis
Induction
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An induced current is produced by a
changing magnetic field
There is an induced emf associated with
the induced current
A current can be produced without a
battery present in the circuit
Faraday’s Law of Induction describes
the induced emf
EMF Produced by a
Changing Magnetic Field, 1
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A loop of wire is connected to a sensitive ammeter
When a magnet is moved toward the loop, the
ammeter deflects
The deflection indicates a current induced in the wire
EMF Produced by a
Changing Magnetic Field, 2
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When the magnet is held stationary, there is
no deflection of the ammeter
Therefore, there is no induced current
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Even though the magnet is inside the loop
EMF Produced by a
Changing Magnetic Field, 3
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The magnet is moved away from the loop
The ammeter deflects in the opposite
direction
EMF Produced by a Changing
Magnetic Field, Summary
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The ammeter deflects when the magnet is
moving toward or away from the loop
The ammeter also deflects when the loop is
moved toward or away from the magnet
An electric current is set up in the coil as long
as relative motion occurs between the
magnet and the coil
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This is the induced current that is produced by an
induced emf
Faraday’s Experiment –
Set Up
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A primary coil is connected to
a switch and a battery
The wire is wrapped around
an iron ring
A secondary coil is also
wrapped around the iron ring
There is no battery present in
the secondary coil
The secondary coil is not
electrically connected to the
primary coil
Faraday’s Experiment –
Findings
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At the instant the switch is closed, the
galvanometer (ammeter) needle deflects in
one direction and then returns to zero
When the switch is opened, the galvanometer
needle deflects in the opposite direction and
then returns to zero
The galvanometer reads zero when there is a
steady current or when there is no current in
the primary circuit
Faraday’s Experiment –
Conclusions
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An electric current can be produced by a timevarying magnetic field
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This would be the current in the secondary circuit of
this experimental set-up
The induced current exists only for a short time
while the magnetic field is changing
This is generally expressed as: an induced
emf is produced in the secondary circuit by
the changing magnetic field
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The actual existence of the magnetic field is not
sufficient to produce the induced emf, the field must
be changing
Magnetic Flux
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To express
Faraday’s finding
mathematically, the
magnetic flux is
used
The flux depends on
the magnetic field
and the area:
 B   B  dA
Flux and Induced emf
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An emf is induced in a circuit when the
magnetic flux through the surface
bounded by the circuit changes with
time
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This summarizes Faraday’s experimental
results
Faraday’s Law – Statements
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Faraday’s Law of Induction states that
the emf induced in a circuit is equal
to the time rate of change of the
magnetic flux through the circuit
Mathematically,
d B
 
dt
Faraday’s Law –
Statements, cont
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If the circuit consists of N identical and
concentric loops, and if the field lines
pass through all loops, the induced emf
becomes
d B
  N
dt
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The loops are in series, so the emfs in the
individual loops add to give the total emf
Faraday’s Law – Example
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Assume a loop
enclosing an area A
lies in a uniform
magnetic field
The magnetic flux
through the loop is
B = B A cos q
The induced emf is
d
    BA cosq 
dt
Ways of Inducing an emf
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The magnitude of the field can change
with time
The area enclosed by the loop can
change with time
The angle q between the field and the
normal to the loop can change with time
Any combination of the above can occur
Applications of
Faraday’s Law – Pickup Coil
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The pickup coil of an electric
guitar uses Faraday’s Law
The coil is placed near the
vibrating string and causes a
portion of the string to
become magnetized
When the string vibrates at
the some frequency, the
magnetized segment
produces a changing flux
through the coil
The induced emf is fed to an
amplifier
Motional emf
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A motional emf is one
induced in a conductor
moving through a
magnetic field
The electrons in the
conductor experience
a force that is directed
along l
FB  qv  B
Motional emf, cont
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Under the influence of the force, the electrons
move to the lower end of the conductor and
accumulate there
As a result of the charge separation, an
electric field is produced inside the conductor
The charges accumulate at both ends of the
conductor until they are in equilibrium with
regard to the electric and magnetic forces
Motional emf, final
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For equilibrium, q E = q v B or E = v B
A potential difference is maintained
between the ends of the conductor as
long as the conductor continues to
move through the uniform magnetic field
If the direction of the motion is reversed,
the polarity of the potential difference is
also reversed
Sliding Conducting Bar
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A bar moving through a uniform field and the
equivalent circuit diagram
Assume the bar has zero resistance
The work done by the applied force appears as
internal energy in the resistor R
Sliding Conducting Bar, cont
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The induced emf is
d B
dx
 
 B
 B v
dt
dt
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Since the resistance in the circuit is R,
the current is
 B v
I 
R
R
Sliding Conducting Bar,
Energy Considerations
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The applied force does work on the
conducting bar
This moves the charges through a magnetic
field
The change in energy of the system during
some time interval must be equal to the
transfer of energy into the system by work
The power input is equal to the rate at which
energy is delivered to the resistor
2
  Fappv  I B  v 
R
AC Generators
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Electric generators take
in energy by work and
transfer it out by
electrical transmission
The AC generator
consists of a loop of
wire rotated by some
external means in a
magnetic field
Induced emf
In an AC Generator
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The induced emf in
the loops is
d B
  N
dt
 NABw sinw t
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This is sinusoidal,
with max = N A B w
Lenz’ Law
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Faraday’s Law indicates the induced
emf and the change in flux have
opposite algebraic signs
This has a physical interpretation that
has come to be known as Lenz’ Law
It was developed by a German
physicist, Heinrich Lenz
Lenz’ Law, cont
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Lenz’ Law states the polarity of the
induced emf in a loop is such that it
produces a current whose magnetic
field opposes the change in magnetic
flux through the loop
The induced current tends to keep the
original magnetic flux through the circuit
from changing
Lenz’ Law – Example 1
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When the magnet is moved toward the stationary
loop, a current is induced as shown in a
This induced current produces its own magnetic field
that is directed as shown in b to counteract the
increasing external flux
Lenz’ Law – Example 2
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When the magnet is moved away the stationary loop,
a current is induced as shown in c
This induced current produces its own magnetic field
that is directed as shown in d to counteract the
decreasing external flux
Induced emf
and Electric Fields
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An electric field is created in the conductor
as a result of the changing magnetic flux
Even in the absence of a conducting loop,
a changing magnetic field will generate an
electric field in empty space
This induced electric field has different
properties than a field produced by
stationary charges
Induced emf
and Electric Fields, cont
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The emf for any closed path can be
expressed as the line integral of E  ds
over the path
Faraday’s Law can be written in a
general form
d B
   E  ds  
dt
Induced emf
and Electric Fields, final
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The induced electric field is a
nonconservative field that is generated
by a changing magnetic field
The field cannot be an electrostatic field
because if the field were electrostatic,
and hence conservative, the line
integral would be zero and it isn’t
Self-Induction
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When the switch is
closed, the current
does not
immediately reach
its maximum value
Faraday’s Law can
be used to describe
the effect
Self-Induction, 2
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As the current increases with time, the
magnetic flux through the circuit loop
due to this current also increases with
time
The corresponding flux due to this
current also increases
This increasing flux creates an induced
emf in the circuit
Self-Inductance, 3
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The direction of the induced emf is such that
it would cause an induced current in the loop
which would establish a magnetic field
opposing the change in the original magnetic
field
The direction of the induced emf is opposite
the direction of the emf of the battery
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Sometimes called a back emf
This results in a gradual increase in the
current to its final equilibrium value
Self-Induction, 4
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This effect is called self-inductance
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Because the changing flux through the
circuit and the resultant induced emf arise
from the circuit itself
The emf L is called a self-induced
emf
Self-Inductance, Equations
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An induced emf is always proportional
to the time rate of change of the current
dI
 L  L
dt
L is a constant of proportionality called
the inductance of the coil
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It depends on the geometry of the coil and
other physical characteristics
Inductance of a Coil
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A closely spaced coil of N turns carrying
current I has an inductance of
N B

L

I
dI dt
The inductance is a measure of the
opposition to a change in current
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Compared to resistance which was
opposition to the current
Inductance Units
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The SI unit of inductance is a Henry (H)
V s
1H  1
A
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Named for Joseph Henry
Joseph Henry
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1797 – 1878
Improved the design of
the electromagnet
Constructed one of the
first motors
Discovered the
phenomena of selfinductance
Inductance of a Solenoid
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Assume a uniformly wound solenoid
having N turns and length l
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Assume l is much greater than the radius
of the solenoid
The interior magnetic field is
N
B  o nI  o I
Inductance of a Solenoid, cont
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The magnetic flux through each turn is
 B  BA  o
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I
Therefore, the inductance is
2
N  B o N A
L
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NA
I

This shows that L depends on the
geometry of the object
RL Circuit, Introduction
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A circuit element that has a large selfinductance is called an inductor
The circuit symbol is
We assume the self-inductance of the
rest of the circuit is negligible compared
to the inductor
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However, even without a coil, a circuit will
have some self-inductance
RL Circuit, Analysis
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An RL circuit contains
an inductor and a
resistor
When the switch is
closed (at time t=0), the
current begins to
increase
At the same time, a
back emf is induced in
the inductor that
opposes the original
increasing current
RL Circuit, Analysis, cont
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Applying Kirchhoff’s Loop Rule to the
previous circuit gives
dI
  IR  L  0
dt
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Looking at the current, we find

Rt L
I
1 e

R

RL Circuit, Analysis, Final
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The inductor affects the current
exponentially
The current does not instantly increase
to its final equilibrium value
If there is no inductor, the exponential
term goes to zero and the current would
instantaneously reach its maximum
value as expected
RL Circuit, Time Constant
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The expression for the current can also be
expressed in terms of the time constant, t, of
the circuit
I t  
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
1 e 

R
t t
where t = L / R
Physically, t is the time required for the
current to reach 63.2% of its maximum value
RL Circuit,
Current-Time Graph, 1
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The equilibrium value
of the current is /R
and is reached as t
approaches infinity
The current initially
increases very rapidly
The current then
gradually approaches
the equilibrium value
RL Circuit,
Current-Time Graph, 2
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The time rate of
change of the
current is a
maximum at t = 0
It falls off
exponentially as t
approaches infinity
In general,
dI  t t
 e
dt L
Energy in a Magnetic Field
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In a circuit with an inductor, the battery
must supply more energy than in a
circuit without an inductor
Part of the energy supplied by the
battery appears as internal energy in
the resistor
The remaining energy is stored in the
magnetic field of the inductor
Energy in a
Magnetic Field, cont
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Looking at this energy (in terms of rate)
dI
2
I  I R  LI
dt
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I is the rate at which energy is being supplied by
the battery
I2R is the rate at which the energy is being
delivered to the resistor
Therefore, LI dI/dt must be the rate at which the
energy is being delivered to the inductor
Energy in a
Magnetic Field, final
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Let U denote the energy stored in the
inductor at any time
The rate at which the energy is stored is
dUB
dI
 LI
dt
dt
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To find the total energy, integrate and
UB = ½ L I 2
Energy Density
of a Magnetic Field
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Given U = ½ L I2,
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Since Al is the volume of the solenoid, the
magnetic energy density, uB is
U B2
uB  
V 2o
This applies to any region in which a
magnetic field exists
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2
2


1
B
B
U  o n 2 A 
A
 
2
2 o
 o n 

not just the solenoid
Inductance Example –
Coaxial Cable
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Calculate L and
energy for the cable
The total flux is
B   BdA  
b
a

Therefore, L is
L

o I
 I b
dr  o ln  
2 r
2  a 
 B o
b

ln  
I
2  a 
The total energy is
1 2 o I 2  b 
U  LI 
ln  
2
4
a
Magnetic Levitation –
Repulsive Model
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A second major model for magnetic
levitation is the EDS (electrodynamic
system) model
The system uses superconducting
magnets
This results in improved energy
effieciency
Magnetic Levitation –
Repulsive Model, 2
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The vehicle carries a magnet
As the magnet passes over a metal plate that
runs along the center of the track, currents
are induced in the plate
The result is a repulsive force
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This force tends to lift the vehicle
There is a large amount of metal required

Makes it very expensive
Japan’s Maglev Vehicle

The current is
induced by magnets
passing by coils
located on the side
of the railway
chamber
EDS Advantages
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Includes a natural stabilizing feature
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If the vehicle drops, the repulsion becomes
stronger, pushing the vehicle back up
If the vehicle rises, the force decreases
and it drops back down
Larger separation than EMS
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About 10 cm compared to 10 mm
EDS Disadvantages
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Levitation only exists while the train is in
motion
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Depends on a change in the magnetic flux
Must include landing wheels for stopping and
starting
The induced currents produce a drag force as
well as a lift force
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High speeds minimize the drag
Significant drag at low speeds must be overcome
every time the vehicle starts up