Chapter 32 Inductance

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Chapter 32 Inductance

• PHYS 2326-21

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Concepts to Know

• Self Induction
• Mutual Inductance
• Inductors
• Magnetic Field Energy
• RL Circuit
• LC Circuit
• LRC Circuit

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Self Induction

• Must distinguish between emf from sources like a
  battery and induced emf from changing magnetic
  fields

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Mutual Inductance

• Magnetic flux variation in one circuit can cause
  a magnetic flux variation in another circuit.
  Note that this can be by intention or by
  accident.
• By intention, one can have a transformer
• By accident, one relay might cause another relay
  to close or open. or noise to be injected in one
  circuit by another

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Mutual Inductance

• emf2 -N2 dF12/dt -N2 d(M12 I1/N2)/dt
• -M12 dI1/dt , so emf1 -M21 dI2/dt
• Note that also M12 M21 M
• That is a current in coil 1 generates current in
  coil 2 and a current in coil 2 generates a
  current in coil 1

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Inductor

• An inductor is a circuit element that primarily
  has large inductance
• It can be a solenoid valve, or relay or even
  merely a component intended to provide inductance
  to a circuit for some purpose
• Inductors are subject to Lenzs law which states
  that with a changing magnetic field, there will
  be an action taken to oppose that change

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Magnetic Field Energy

• Given an applied emf across an inductor in series
  with a resistance,

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Transformer

• A transformer is two coils with total mutual
  induction Read Chapter 33.8

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Permeability

• Amperes law applied to a toroid
• Note that the magnetic field B depends upon µo
  the permeability of free space. This is for the
  area in the torus that the winding goes around
• What happens if its not a vacuum?
• Most transformers and solenoids have metal cores
  in the coils

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Permeability

• Note this doesnt seem to even be in this text
  book
• In the same concept as with dielectrics for
  capacitors there is a magnetic equivalent to the
  dielectric constant Km B/Bo relative
  permeability
• Note though that K for dielectrics ranged from 1
  for a vacuum upwards to ??
• Km is 1 for a vacuum, gt 1 for paramagnetic
  materials, slightly smaller than 1 for
  diagmagnetic materials and gtgt 1 for ferromagnetic
  materials

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Permeability

• µ µo Km is the permeability of a material

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Example 1

• Tesla coil has N1 turns along length l of a
  hollow tube and a second coil of N2 what is a)
  the mutual inductance b) if N110,000 and N2
  100, l 1m, radius 1cm what is the value of M?
  c) if a radio frequency of 1000 KHz is sent
  through coil 2 so that current oscillates with
  amplitude of 100ma what is the average magnetic
  flux through coil1 d)max current through coil1
  e)max induced emf in coil1 f)back emf in coil 1?

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Example 1

• M N2 FB2 /I1, FB2 FB1 B1A
• A pr2 3.14(0.01)2 0.000314 m2
• B1µo N1 I1/l substituting for B1
• M (µo N1 I1/l ) N2 A / I1 µo N2 N1 A I1 / l
• b) M 3.946 E -4 H
• c) max flux?
• I2 Iosin ?t , ? 2 pf (angular freq.)6.28E6
• rearranging M, FB2 FB1 M I2 / N2
• (3.946E-4)(0.1)/(100) 3.948E-7 Wb

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Example 1

• d) coil 1 current I1 I2 N2/N1
  (0.1)(100)/(10,000) 1.0 E-3A
• e) emf1 -M dI2 /dt , I2 Io sin ?t
• dI2 /dt Io ? cos ?t
• emf1 -M Io ? cos ?t
• emf1max M Io ? (3.948E-4)(0.1)(6.28E6) 248
  V
• f) emf2max emf1max N2/N1 (248)(100/10,000)
  2.48 V

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Example 2

• a) Inductance of a long solenoid length l and
  area A with N turns? b)if 2m long 2cm radius and
  2000 turns? c) if current decreased from 4A to 0
  in 2 microseconds what is magnitude and direction
  of the self induced emf ? d) what is the energy
  stored in the solenoid at the beginning of the 2
  microsecond interval? e) How much electrical
  power is dissipated during this time?

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Example 2

• inductance
• Substituting for B in L

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• b) Inductance value
• A 1.257E-3
• L (1.2566E-6)(2000)2(1.257E-3)/2.0
• 3.159E-3 H
• c) emf?
• emf (3.159E-3)(4-0)/(2.0E-6) 6318V
• in direction of current trying to stop field
  collapse by trying to maintain current

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• d) Energy?
• U1 (1/2) (3.158E-3)(4)2 2.52E-2 Joules
• e) Power
• P (2.52E-2)/(2E-6) 12,632 W
• why so high? Its also about timing too

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RL Circuits

• Review 32.2

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LC Circuits

• Study Chapter 32.5
• Given a capacitor and an inductor at time t0
  with the capacitor being connected in series to
  the inductor with Qmax charge on it there will be
  an oscillation. If no resistance is there to
  dissipate the energy, it will continue to
  oscillate

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RLC Circuit

• Study Chapter 32.6
• The difference between an LC and RLC circuit
  other than resistance exists in all circuits is
  that R is a dissipative element that absorbs
  energy as current flows