Induced Voltages and Inductance

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Chapter 20

• Induced Voltages and Inductance

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20.1 Induced emf

• A current can be produced by a changing magnetic
  field Bf (t), i.e., B varies over time
• First shown in an experiment by Michael Faraday
• A primary coil is connected to a battery
• A secondary coil is connected to an ammeter

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Michael Faraday

• Faraday is often regarded as the greatest
  experimental scientist of the 1800s. His
  contributions to the study of electricity include
  the invention of the electric motor, generator,
  and transformer.

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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 deflects
  in one direction and then returns to zero
• When the switch is opened, the ammeter deflects
  in the opposite direction and then returns to
  zero
• When there is a steady current in the primary
  circuit, the ammeter reads zero

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Faradays Conclusions

• An electrical current is produced by a changing
  magnetic field
• It is customary to say that an induced emf is
  produced in the secondary circuit by the changing
  magnetic field

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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 wire loop wire and the area of the
  loop

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Magnetic Flux, 2

• You are given a loop of wire
• The wire is in an uniform magnetic field B
• 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

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Magnetic Flux, 3

• (a) When the field is perpendicular to the plane
  of the loop, ? 0 and FB FB, max BA
• (b) 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 units of flux are T m² Wb (Weber)

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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

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20.2 Electromagnetic Induction

• 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

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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
  it is produced by an induced emf

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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 through each loop changes by ?F during
  an interval ?t, the average emf induced is given
  by Faradays Law

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Faradays Law and Lenz Law

• The minus sign is included because of the
  polarity of the emf. The induced emf in the coil
  gives rise to a current whose magnetic field
  OPPOSES (? Lenzs law) the change in magnetic
  flux that produced it

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There are three possibilities to produce an emf

• 1) Time-varying magnetic field
• e-N(A cos?)(DB/Dt)
• 2) Time-varying loop area  
• (B cos ?)(DA/Dt) 
• 3) Turning of the loop (generator) 
• BA(Dcos?/Dt)

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Applications of Faradays Law Ground Fault
Interrupters

• The ground fault interrupter (GFI) is a safety
  device that protects against electrical shock
• Wire 1 leads from the wall outlet to the
  appliance
• Wire 2 leads from the appliance back to the wall
  outlet
• The iron ring confines the magnetic field, which
  is generally 0
• If a leakage occurs, the field is no longer 0 and
  the induced voltage triggers a circuit breaker
  shutting off the current

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Applications of Faradays Law Electric Guitar

• A vibrating string induces an emf in a coil
• A permanent magnet inside the coil magnetizes a
  portion of the string nearest the coil
• As the string vibrates at some frequency, its
  magnetized segment produces a changing flux
  through the pickup coil
• The changing flux produces an induced emf that is
  fed to an amplifier

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Applications of Faradays Law Apnea Monitor

• The coil of wire attached to the chest carries an
  alternating current
• An induced emf produced by the varying field
  passes through a pick up coil
• When breathing stops, the pattern of induced
  voltages stabilizes and external monitors sound
  an alert

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20.3 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

l
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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

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Motional emf, cont.

• V El
• FqvB  
• VBlv, voltage across the conductor  
• If the motion is reversed, the polarity of the
  potential difference is also reversed

FqEq (V/l ) qvB
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Magnitude of the Motional emf
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Motional emf in a Circuit

• A conducting bar sliding with v along two
  conducting rails under the action of an applied
  force Fapp. The magnetic force Fm opposes the
  motion, and a counterclockwise current is
  induced.

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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

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Example Operating a light bulb
Rod and rail have negligible resistance but the
bulb has a resistance of 96 W, B0.80 T, v5.0
m/s and l 1.6 m. Calculate (a) emf in the rod,
(b) induced current (c) power delivered to the
bulb and (d) the energy used by the bulb in 60
s. (a) evBl e (5.0 m/s)(0.80 T)(1.6 m)6.4
V (b) Ie/R I(6.4V)/(96 W)0.067 A (c)
PeI PeI(6.4 V)(0.067 A)0.43 W (d)
EPt E(0.43 W)(60 s)26 J (26 Ws)
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20.4 Lenz Law Revisited Moving Bar Example

• 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

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Lenz Law, Bar Example, cont

• 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

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Lenz Law, Bar Example, final

• The bar is moving toward the left
• The magnetic flux through the loop is decreasing
  with time
• The induced current must be clockwise to to
  produce its own flux into the page

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Lenz Law Revisited, Conservation of Energy

• Assume the bar is moving to the right
• Assume the induced current is clockwise
• The magnetic force on the bar would be to the
  right
• The force would cause an acceleration and the
  velocity would increase
• This would cause the flux to increase and the
  current to increase and the velocity to increase
• This would violate Conservation of Energy and so
  therefore, the current must be counterclockwise

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Lenz Law, Moving Magnet Example

• (a) A bar magnet is moved to the right toward a
  stationary loop of wire. As the magnet moves, the
  magnetic flux increases with time
• (b) The induced current produces a flux to the
  left to counteract the increasing external flux
  to the right

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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

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Application Tape Recorder

• A magnetic tape moves past a recording and
  playback head
• The tape is a plastic ribbon coated with iron
  oxide or chromium oxide

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Application Tape Recorder, cont.

• To record, the sound is converted to an
  electrical signal which passes to an
  electromagnet that magnetizes the tape in a
  particular pattern
• To playback, the magnetized pattern is converted
  back into an induced current driving a speaker