Induced Voltages and Inductance

1
Chapter 20

• Induced Voltages and Inductance

2
Faradays Experiment

• A primary coil is connected to a battery and a
  secondary coil is connected to an ammeter
• The purpose of the secondary circuit is to detect
  current that might be produced by a (changing)
  magnetic field
• When there is a steady current in the primary
  circuit, the ammeter reads zero

3
Faradays Experiment

• When the switch is opened, the ammeter reads a
  current and then returns to zero
• When the switch is closed, the ammeter reads a
  current in the opposite direction and then
  returns to zero
• An induced emf is produced in the secondary
  circuit by the changing magnetic field

4
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 (defined similar to that of
  electrical 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

• For a loop of wire with an area A in a uniform
  magnetic field, the flux is (? is the angle
  between B and the normal to the plane)
• FB B?A B A cos ?

5
Magnetic Flux

• When the field is perpendicular to the plane of
  the loop, ? 0 and FB FB, max BA
• When the field is parallel to the plane of the
  loop, ? 90 and FB 0
• The flux can be negative, for example if ? 180
• SI unit of flux Weber
• Wb T. m²

6
Magnetic Flux

• The value of the magnetic flux is proportional to
  the total number of magnetic field 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

7
Electromagnetic Induction

• When a magnet moves toward a loop of wire, the
  ammeter shows the presence of a current
• When the magnet moves away from the loop, the
  ammeter shows a current in the opposite direction
• When the magnet is held stationary, there is no
  current
• If the loop is moved instead of the magnet, a
  current is also detected

8
Electromagnetic Induction

• A current is set up in the circuit as long as
  there is relative motion between the magnet and
  the loop
• The current is called an induced current because
  is it produced by an induced emf

9
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

10
Faradays Law and Lenz Law

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

11
Faradays Law and Lenz Law

• Example
• The magnetic field, B, becomes smaller with time
  and this reduces the flux
• The induced current will produce an induced
  field, Bind, in the same direction as the
  original field

12
Chapter 20Problem 14

• A square, single-turn wire loop 1.00 cm on a side
  is placed inside a solenoid that has a circular
  cross section of radius 3.00 cm, as shown in
  Figure P20.14. The solenoid is 20.0 cm long and
  wound with 100 turns of wire. (a) If the current
  in the solenoid is 3.00 A, find the flux through
  the loop. (b) If the current in the solenoid is
  reduced to zero in 3.00 s, find the magnitude of
  the average induced emf in the loop.

13
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
• As the negative charges accumulate at the base, a
  net positive charge exists at the upper end of
  the conductor

14
Motional emf

• 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
• q E q v B E v B
• There is a potential difference between the upper
  and lower ends of the conductor

15
Motional emf

• The potential difference between the ends of the
  conductor (the upper end is at a higher potential
  than the lower end)
• ?V E l B l v
• 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

16
Motional emf in a Circuit

• As the bar (with zero resistance) is pulled to
  the right with a constant 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
• The changing magnetic flux through the loop and
  the corresponding induced emf in the bar result
  from the change in area of the loop

17
Motional emf in a Circuit

• The induced, motional emf, acts like a battery in
  the circuit
• 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 Law)

18
Motional emf in a Circuit

• The flux due to the external field is increasing
  into the page
• The flux due to the induced current must be out
  of the page
• Therefore the current must be counterclockwise
  when the bar moves to the right
• If the bar is moving toward the left, the
  magnetic flux through the loop is decreasing with
  time the induced current must be clockwise to
  produce its own flux into the page

19
Chapter 20Problem 57

• A conducting rod of length l moves on two
  horizontal frictionless rails. 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?

20
Lenz Law Moving Magnet Example

• As the bar magnet is moved to the right toward a
  stationary loop of wire, the magnetic flux
  increases with time
• The induced current produces a flux to the left,
  so the current is in the direction shown
• When applying Lenz Law, there are two magnetic
  fields to consider changing external and induced

21
AC Generators

• Alternating Current (AC) generators convert
  mechanical energy to electrical energy
• Consist of a wire loop rotated by some external
  means (falling water, heat by burning coal to
  produce steam, etc.)
• As the loop rotates, the magnetic flux through it
  changes with time inducing an emf and a current
  in the external circuit

22
AC Generators

• 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
• The emf generated by the rotating loop can be
  found by e 2 B l v? 2 B l v sin ?
• If the loop rotates with a constant angular
  speed, ?, and N turns e N B A ? sin ? t

23
AC Generators

• The magnetic force on the charges in the wires AB
  and CD is perpendicular to the length of the
  wires
• An emf is generated in wires BC and AD
• The emf produced in each of these wires is e B
  l v? B l v sin ?

24
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
• The output voltage always has the same polarity
• The current is a pulsing current

25
DC Generators

• 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

26
Motors

• Motors are devices that convert electrical energy
  into mechanical energy (generators run in
  reverse)
• A motor can perform useful mechanical work when a
  shaft connected to its rotating coil is attached
  to some external device
• As the coil begins to rotate, the induced back
  emf opposes the applied voltage and the current
  in the coil is reduced

27
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
  inducing 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

28
Self-inductance

• The self-induced emf must be proportional to the
  time rate of change of the current
• L inductance of a coil (depends on geometric
  factors)
• The negative sign indicates that a changing
  current induces an emf in opposition to that
  change
• The SI unit of self-inductance Henry
• 1 H 1 (V s) / A

29
Chapter 20Problem 40

• An emf of 24.0 mV is induced in a 500-turn coil
  when the current is changing at a rate of 10.0
  A/s. What is the magnetic flux through each turn
  of the coil at an instant when the current is
  4.00 A?

30
Inductor in a Circuit

• Inductance can be interpreted as a measure of
  opposition to the rate of change in the current
  (while resistance 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
• As a result, the current doesnt change from 0 to
  its maximum instantaneously

31
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
• The current can be found by

32
Chapter 20Problem 46

• Consider the circuit shown in the figure. 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?

33
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

34
Answers to Even Numbered Problems Chapter 20
Problem 10 34 mV
35
Answers to Even Numbered Problems Chapter 20
Problem 18 1.00 ms
36

• Answers to Even Numbered Problems
• Chapter 20
• Problem 28
• left to right
• no induced current
• right to left

37
Answers to Even Numbered Problems Chapter 20
Problem 30 13 mV
38

• Answers to Even Numbered Problems
• Chapter 20
• Problem 42
• 1.00 kO
• 3.00 ms