Induced Voltages and Inductance

1
Chapter 20

• Induced Voltages and Inductance

2
Michael Faraday

• 1791 1867
• Great experimental scientist
• Invented electric motor, generator and
  transformers
• Discovered electromagnetic induction
• Discovered laws of electrolysis

3
Faradays Experiment Set Up

• A current can be produced by a changing magnetic
  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

4
Faradays Experiment

• 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

5
Faradays 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

6
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

7
Magnetic Flux, 2

• 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
• FB B?A B A cos ?
• ? is the angle between B and the normal to the
  plane

8
Magnetic Flux, 3

• When the field is perpendicular to the plane of
  the loop, as in a, ? 0 and FB FB, max BA
• When the field is parallel to the plane of the
  loop, as in b, ? 90 and FB 0
• The flux can be negative, for example if ? 180
• SI units of flux are T. m² Wb (Weber)

9
Magnetic Flux, final

• 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

10
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.
11
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.
12
Practice 1
Find the flux of the Earths 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.
13
Electromagnetic Induction An Experiment

• 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

14
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

15
Faradays Law and Electromagnetic Induction

• 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 ?FB during a time interval
  ?t, the average emf induced is given by Faradays
  Law

16
Faradays Law and Lenz Law

• The change in the flux, ?FB, can be produced by a
  change in B, A or ?
• Since FB B A cos ?
• The negative sign in Faradays 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

17
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.
18
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 O, 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?
19
Application of Faradays Law Motional emf

• A straight conductor of length l moves
  perpendicularly with constant velocity through a
  uniform field
• The electrons in the conductor experience a
  magnetic force
• F q v B
• The electrons tend to move to the lower end of
  the conductor

20
Motional emf

• 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

21
Motional emf, cont

• The potential difference between the ends of the
  conductor can be found by
• ?V 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
• If the motion is reversed, the polarity of the
  potential difference is also reversed

22
Motional emf in a Circuit

• 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

23
Motional emf in a Circuit, cont

• 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

24
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-O
resistor R? (b) What is the rate of energy
dissipation in the resistor? (c) What is the
mechanical power delivered by the constant force?
25
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 Earths magnetic field is 5.0 10-5
T, what is the emf induced between the blade tip
and the central hub?
26
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 Earths 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
Earths magnetic field.
27
Lenz Law Moving Magnet Example

• 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)

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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 loop
• The magnetic field produced by the current in the
  loop

29
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?
30
Generators

• 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

31
AC Generators, cont

• 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

32
AC Generators, final

• The emf generated by the rotating loop can be
  found by
• e 2 B l v?2 B l sin ?
• If the loop rotates with a constant angular
  speed, ?, and N turns
• e N B A ? sin ? t
• e emax when loop is parallel to the field
• e 0 when when the loop is perpendicular to the
  field

33
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

34
DC Generators, cont

• 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

35
Motors

• Motors are devices that convert electrical energy
  into mechanical energy
• 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

36
Motors and Back emf

• 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

37
Motors and Back emf, cont

• 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

38
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 Earths magnetic field at the
location of the loop is 2.0 10-5 T. Calculate
the maximum emf induced in the coil by the
Earths field.
39
Example 9
A motor has coils with a resistance of 30 O 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?
40
Self-inductance

• Self-inductance occurs when the changing flux
  through a circuit arises from the circuit itself
• 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

41
Self-inductance cont

• The self-induced emf must be proportional to the
  time rate of change of the current
• 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

42
Self-inductance, final

• The inductance of a coil depends on geometric
  factors
• The SI unit of self-inductance is the Henry
• 1 H 1 (V s) / A
• You can determine an expression for L

43
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.
44
Joseph Henry

• 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

45
Inductor in a Circuit

• Inductance can be interpreted as a measure of
  opposition to the rate of change in the current
• 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
• Therefore, the current doesnt change from 0 to
  its maximum instantaneously

46
RL Circuit

• 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

47
RL Circuit, cont

• The time constant depends on R and L
• The current at any time can be found by

48
Example 11
Consider the circuit shown in Figure P20.46. Take
e 6.00 V, L 8.00 mH, and R 4.00 O. (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?
49
Energy Stored in a Magnetic Field

• 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
• This work can be thought of as energy stored by
  the inductor in its magnetic field
• PEL ½ L I2

50
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.
51
Practice 3
How much energy is stored in a 70.0-mH inductor
at an instant when the current is 2.00 A?