Transcript Chapter 20

Chapter 20
Induced Voltages and
Inductance
Michael Faraday
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1791 – 1867
Great experimental
scientist
Invented electric
motor, generator and
transformers
Discovered
electromagnetic
induction
Discovered laws of
electrolysis
Faraday’s Experiment –
Set Up
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A current can be produced by a
changing magnetic field
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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
Faraday’s Experiment
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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
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
Magnetic Flux, 2
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You are given a loop
of wire
The wire is in a
uniform magnetic
field B
The loop has an area
A
The flux is defined as
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ΦB = BA = B A cos θ
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θ is the angle
between B and the
normal to the plane
Magnetic Flux, 3
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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
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The flux can be negative, for example if θ = 180°
SI units of flux are T. m² = Wb (Weber)
Magnetic Flux, final
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The flux can be visualized with respect
to magnetic field lines
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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
Example 1
A square loop 2.00 m on a side is placed in a magnetic
field of magnitude 0.300 T. If the field makes an angle of
50.0° with the normal to the plane of the loop, find the
magnetic flux through the loop.
Example 2
A solenoid 4.00 cm in diameter and 20.0 cm long has 250
turns and carries a current of 15.0 A. Calculate the
magnetic flux through the circular cross-sectional area of
the solenoid.
Practice 1
Find the flux of the Earth’s magnetic field of magnitude 5.00 ×
10−5 T through a square loop of area 20.0 cm2 (a) when the
field is perpendicular to the plane of the loop, (b) when the field
makes a 40.0° angle with the normal to the plane of the loop,
and (c) when the field makes a 90.0° angle with the normal to
the plane.
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
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 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
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:
B
  N
t
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
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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
Example 3
A 300-turn solenoid with a length of 20 cm and a radius of
1.5 cm carries a current of 2.0 A. A second coil of four turns
is wrapped tightly about this solenoid so that it can be
considered to have the same radius as the solenoid. Find (a)
the change in the magnetic flux through the coil and (b) the
magnitude of the average induced emf in the coil when the
current in the solenoid increases to 5.0 A in a period of 0.90
s.
Example 4
A circular coil enclosing an area of 100 cm2 is made of 200
turns of copper wire. The wire making up the coil has
resistance of 5.0 Ω, and the ends of the wire are
connected to form a closed circuit. Initially, a 1.1-T uniform
magnetic field points perpendicularly upward through the
plane of the coil. The direction of the field then reverses so
that the final magnetic field has a magnitude of 1.1 T and
points downward through the coil. If the time required for
the field to reverse directions is 0.10 s, what average
current flows through the coil during that time?
Application of Faraday’s
Law – Motional emf
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A straight conductor of
length ℓ moves
perpendicularly with
constant velocity
through a uniform field
The electrons in the
conductor experience a
magnetic force
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F=qvB
The electrons tend to
move to the lower end
of the conductor
Motional emf
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As the negative charges accumulate at the
base, a net positive charge exists at the
upper end of the conductor
As a result of this charge separation, an
electric field is produced in the conductor
Charges build up at the ends of the
conductor until the downward magnetic
force is balanced by the upward electric
force
There is a potential difference between the
upper and lower ends of the conductor
Motional emf, cont
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The potential difference between the
ends of the conductor can be found by
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ΔV = B ℓ v
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
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If the motion is reversed, the polarity of the
potential difference is also reversed
Motional emf in a Circuit
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Assume the moving
bar has zero resistance
As the bar is pulled to
the right with a given
velocity under the
influence of an applied
force, the free charges
experience a magnetic
force along the length
of the bar
This force sets up an
induced current
because the charges
are free to move in the
closed path
Motional emf in a Circuit,
cont
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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
B v
  B v and I 
R
Example 5
A conducting rod of length ℓ moves on two horizontal
frictionless rails, as in Figure P20.18. A constant force of
magnitude 1.00 N moves the bar at a uniform speed of
2.00 m/s through a magnetic field that is directed into
the page. (a) What is the current in an 8.00-Ω resistor R?
(b) What is the rate of energy dissipation in the resistor?
(c) What is the mechanical power delivered by the
constant force?
Example 6
A helicopter has blades of length 3.0 m, rotating at 2.0
rev/s about a central hub. If the vertical component of
Earth’s magnetic field is 5.0 × 10−5 T, what is the emf
induced between the blade tip and the central hub?
Practice 2
A 12.0-m-long steel beam is accidentally dropped by a
construction crane from a height of 9.00 m. The horizontal
component of the Earth’s magnetic field over the region is
18.0 μT. What is the induced emf in the beam just before
impact with the Earth? Assume the long dimension of the
beam remains in a horizontal plane, oriented
perpendicular to the horizontal component of the Earth’s
magnetic field.
Lenz’ Law – Moving
Magnet Example
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A bar magnet is moved to the right toward a
stationary loop of wire (a)
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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)
Lenz’ Law, Final Note
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When applying Lenz’ Law, there
are two magnetic fields to consider
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The external changing magnetic field
that induces the current in the loop
The magnetic field produced by the
current in the loop
Example 7
A copper bar is moved to the right while its axis is
maintained in a direction perpendicular to a magnetic field,
as shown in Figure P20.27. If the top of the bar becomes
positive relative to the bottom, what is the direction of the
magnetic field?
Generators
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Alternating Current (AC) generator
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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
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These may include falling water, heat by burning
coal to produce steam
AC Generators, cont
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Basic operation of the
generator
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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
AC Generators, final
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The emf generated by
the rotating loop can be
found by
ε =2 B ℓ v=2 B ℓ sin θ
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If the loop rotates with a
constant angular speed,
ω, and N turns
ε = N B A ω sin ω t
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ε = εmax when loop is
parallel to the field
ε = 0 when when the
loop is perpendicular to
the field
DC Generators
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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, cont
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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
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The multiple outputs
are superimposed and
the output is almost
free of fluctuations
Motors
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Motors are devices that convert
electrical energy into mechanical
energy
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A motor is a generator run in reverse
A motor can perform useful
mechanical work when a shaft
connected to its rotating coil is
attached to some external device
Motors and Back emf
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The phrase back emf
is used for an emf
that tends to reduce
the applied current
When a motor is
turned on, there is
no back emf initially
The current is very
large because it is
limited only by the
resistance of the coil
Motors and Back emf, cont
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As the coil begins to rotate, the induced
back emf opposes the applied voltage
The current in the coil is reduced
The power requirements for starting a
motor and for running it under heavy
loads are greater than those for running
the motor under average loads
Example 8
A 100-turn square wire coil of area 0.040 m2 rotates about a
vertical axis at 1 500 rev/min, as indicated in Figure P20.30.
The horizontal component of the Earth’s magnetic field at
the location of the loop is 2.0 × 10−5 T. Calculate the
maximum emf induced in the coil by the Earth’s field.
Example 9
A motor has coils with a resistance of 30 Ω and operates
from a voltage of 240 V. When the motor is operating at its
maximum speed, the back emf is 145 V. Find the current in
the coils (a) when the motor is first turned on and (b)
when the motor has reached maximum speed. (c) If the
current in the motor is 6.0 A at some instant, what is the
back emf at that time?
Self-inductance
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Self-inductance occurs when the
changing flux through a circuit arises
from the circuit itself
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As the current increases, the magnetic flux
through a loop due to this current also increases
The increasing flux induces an emf that opposes
the change in magnetic flux
As the magnitude of the current increases, the
rate of increase lessens and the induced emf
decreases
This opposing emf results in a gradual increase
of the current
Self-inductance cont
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The self-induced emf must be
proportional to the time rate of change
of the current
I
  L
t
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L is a proportionality constant called the
inductance of the device
The negative sign indicates that a changing
current induces an emf in opposition to that
change
Self-inductance, final
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The inductance of a coil depends
on geometric factors
The SI unit of self-inductance is
the Henry
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1 H = 1 (V · s) / A
You can determine an expression
for L
 B N  B
LN

I
I
Example 10
A coiled telephone cord forms a spiral with 70.0 turns, a
diameter of 1.30 cm, and an unstretched length of 60.0 cm.
Determine the self-inductance of one conductor in the
unstretched cord.
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
Inductor in a Circuit
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Inductance can be interpreted as a
measure of opposition to the rate of
change in the current
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Remember resistance R is a measure of
opposition to the current
As a circuit is completed, the current
begins to increase, but the inductor
produces an emf that opposes the
increasing current
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Therefore, the current doesn’t change from
0 to its maximum instantaneously
RL Circuit
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When the current
reaches its
maximum, the rate
of change and the
back emf are zero
The time constant, ,
for an RL circuit is
the time required for
the current in the
circuit to reach
63.2% of its final
value
RL Circuit, cont
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The time constant depends on R
and L
L
 
R
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The current at any time can be
found by

I  1  et /  
R
Example 11
Consider the circuit shown in Figure P20.46. Take ε = 6.00
V, L = 8.00 mH, and R = 4.00 Ω. (a) What is the inductive
time constant of the circuit? (b) Calculate the current in the
circuit 250 μs after the switch is closed. (c) What is the
value of the final steady-state current? (d) How long does it
take the current to reach 80.0% of its maximum value?
Energy Stored in a
Magnetic Field
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The emf induced by an inductor
prevents a battery from establishing an
instantaneous current in a circuit
The battery has to do work to produce a
current
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This work can be thought of as energy
stored by the inductor in its magnetic field
PEL = ½ L I2
Example 12
A 300-turn solenoid has a radius of 5.00 cm and a length of
20.0 cm. Find (a) the inductance of the solenoid and (b) the
energy stored in the solenoid when the current in its
windings is 0.500 A.
Practice 3
How much energy is stored in a 70.0-mH inductor at an
instant when the current is 2.00 A?