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Chapter 32
Inductance
Dr. Jie Zou
PHY 1361
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Outline
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Self-inductance (32.1)
Mutual induction (32.4)
RL circuits (32.2)
Energy in a magnetic field (32.3)
Oscillations in an LC circuit (32.5)
The RLC circuit (32.6, 33.5)
Dr. Jie Zou
PHY 1361
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Self inductance
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Due to self-induction, the
current in the circuit
does not jump from zero
to its maximum value
instantaneously when
the switch is thrown
closed. Dr. Jie Zou
Self induction: the changing flux
through the circuit and the resultant
induced emf arise from the circuit itself.
The emf L set up in this case is called a
self-induced emf.
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L = -L(dI/dt)
L = - L/(dI/dt): Inductance is a measure of
the opposition to a change in current.
Inductance of an N-turn coil: L = NB/I; SI
unit: henry (H).
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Mutual induction
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Dr. Jie Zou
Mutual induction: Very often,
the magnetic flux through the
area enclosed by a circuit
varies with time because of
time-varying currents in nearby
circuits. This condition induces
an emf through a process
known as mutual induction.
An application: An electric
toothbrush uses the mutual
induction of solenoids as part
of its battery-charging system.
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RL circuits
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An inductor: A circuit element that
has a large self-inductance is called
an inductor.
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A RL circuit:
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Dr. Jie Zou
An inductor in a circuit opposes
changes in the current in that circuit.
Kirchhoff’s rule:   IR  L
dI
0
dt
Solving for I: I = (/R)(1 – e-t/)
 = L/R: time constant of the RL circuit.
If L  0, i.e. removing the inductance
from the circuit, I reaches maximum
value (final equilibrium value) /R
instantaneously.
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Energy in a magnetic field
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Energy stored in an inductor:
U = (1/2)LI2.
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dI
  IR  L  0
dt
dI
2
I  I R  LI
dt
dU
dI
 LI
dt
dt

I
0
0
U   dU   LIdI  L  IdI 
Dr. Jie Zou
Magnetic energy density:
uB = B2/20

I
This expression represents the
energy stored in the magnetic
field of the inductor when the
current is I.
1 2
LI
2
The energy density is
proportional to the square of
the field magnitude.
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Oscillations in an LC circuit
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Total energy of the circuit: U = UC + UL =
Q2/2C + (1/2)LI2.
If the LC circuit is resistanceless and nonradiating, the total energy of the circuit must
remain constant in time: dU/dt = 0.
We obtain 2
d Q
1

Q
2
dt
LC
Solving for Q: Q = Qmaxcos(t + )
Solving for I: I = dQ/dt = - Qmaxsin(t + )
Natural frequency of oscillation of the LC
circuit:
1

Dr. Jie Zou
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Oscillations in an LC circuitfrom an energy point of view
Dr. Jie Zou
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The RLC circuit
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The rate of energy
transformation to internal
energy within a resistor: dU/dt
= - I 2R
d 2Q
dQ Q
 0
Equation for Q: L 2  R
dt
Dr. Jie Zou
C
Compare this with the equation
of motion for a damped blockspring system: d 2 x
dx
m

dt
dt
2
b
dt
 kx  0
Solving for Q:
Q = Qmaxe-Rt/2Lcos(dt)
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