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Chapter 32
Inductance
Self-inductance
• Some terminology first:
• Use emf and current when they are caused by batteries
or other sources
• Use induced emf and induced current when they are
caused by changing magnetic fields
• It is important to distinguish between the two
situations
Self-inductance
• When the switch is closed, the current
does not immediately reach its maximum
value
• Faraday’s law can be used to describe the
effect
• As the current increases with time, the
magnetic flux through the circuit loop due
to this current also increases with time
• This increasing flux creates an induced
emf in the circuit
Self-inductance
• The direction of the induced emf is
such that it would cause an induced
current in the loop, which would establish
a magnetic field opposing the change in the
original magnetic field
• The direction of the induced emf is opposite the
direction of the emf of the battery
• This results in a gradual increase in the current to its
final equilibrium value
• This effect of self-inductance occurs when the changing
flux through the circuit and the resultant induced emf
arise from the circuit itself
Self-inductance
• The self-induced emf εL is always proportional to the time
rate of change of the current. (The emf is proportional to
the flux change, which is proportional to the field
change, which is proportional to the current change)
dI
εL  L
dt
• 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
Joseph Henry
1797 – 1878
Inductance of a Coil
• For a closely spaced coil of N turns carrying current I:
dI
d B
 L
 L  N
dt
dt
L
L
dI / dt
N
dI  d B
L
N
I  B
L
N B
L
I
• The inductance is a measure of the opposition to a
change in current
Chapter 32
Problem 4
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?
Inductance of a Solenoid
• Assume a uniformly wound solenoid having N turns and
length ℓ (ℓ is much greater than the radius of the
solenoid)
• The flux through each turn of area A is
N
 B  BA  0 nIA  0 IA
l
 N 
N   0 IA 
2
N B

N
A
l


0
L


I
I
l
• This shows that L depends on the
geometry of the object
L
0 N A
2
l
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 a back emf
• Thus the inductor in a circuit opposes changes in
current in that circuit and attempts to keep the current
the same way it was before the change
• As a result, inductor causes the circuit to be
“sluggish” as it reacts to changes in the voltage: the
current doesn’t change from 0 to its maximum
instantaneously
RL Circuit
• A circuit element that has a large self-inductance is
called an inductor
• The circuit symbol is
• We assume the self-inductance of the rest of the
circuit is negligible compared to the inductor
(However, in reality, even without a coil, a circuit will
have some self-inductance
• When switch is closed (at time t = 0),
the current begins to increase, and at
the same time, a back emf is
induced in the inductor that opposes
the original increasing current
I (t  0)  0
RL Circuit
• Applying Kirchhoff’s loop rule to the circuit in the
clockwise direction gives
   L  IR  0
dI
  L  IR  0
dt
Rdt
d (  IR )

L
  IR
e

Rt
L
  IR
dI
dt
dI


L
dt
L   IR
Rt

 ln(   IR )  ln( Const )
L
1
 Const (  IR)  (  IR )
e  Const  
1
Const 

0
I


1  e
R 

Rt
L




RL Circuit
• The inductor affects the current exponentially
• The current does not instantly increase to its final
equilibrium value
• If there is no inductor, the exponential term goes to
zero and the current would instantaneously reach its
maximum value as expected
• When the current reaches its maximum, the rate of
change and the back emf are zero
Rt




L
I  1  e 
R

RL Circuit
• The expression for the current can also be expressed
in terms of the time constant t, of the circuit:
• The time constant, t, for an RL circuit is the
time required for the current in the circuit
to reach 63.2% of its final value
t



I  1  e t
R




Rt




L
I  1  e 
R

L
t 
R
RL Circuit
• The current initially increases very rapidly and then
gradually approaches the equilibrium value
• The equilibrium value of the current is  /R and is
reached as t approaches infinity
Chapter 32
Problem 13
Consider the circuit shown in the figure. Take ε =
6.00 V, L = 8.00 mH, and R = 4.00 Ω. (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?
Energy Stored in a Magnetic Field
dI
dI
2
  L  IR  0
I  IL  I R
dt
dt
• In a circuit with an inductor, the battery must supply
more energy than in a circuit without an inductor
• I is the rate at which energy is being supplied by the
battery
• Part of the energy supplied by the battery appears as
internal energy in the resistor
• I2R is the rate at which the energy is being delivered to
the resistor
Energy Stored in a Magnetic Field
dI
dI
2
  L  IR  0
I  IL  I R
dt
dt
• The remaining energy is stored in the magnetic field of
the inductor
• Therefore, LI (dI/dt) must be the rate at which the
energy is being stored in the magnetic field dU/dt
dU L
dI
 IL
dt
dt
LI
UL 
2
2
Energy Storage Summary
• A resistor, inductor and capacitor all store energy
through different mechanisms
• Charged capacitor stores energy as electric potential
energy
• Inductor when it carries a current, stores energy as
magnetic potential energy
• Resistor energy delivered is transformed into internal
energy
Mutual Inductance
• The magnetic flux through the area enclosed by a
circuit often varies with time because of time-varying
currents in nearby circuits
• This process is known as mutual induction because it
depends on the interaction of two circuits
• The current in coil 1 sets up a
magnetic field
• Some of the magnetic field lines pass
through coil 2
• Coil 1 has a current I1 and N1 turns;
Coil 2 has N2 turns
Mutual Inductance
• The mutual inductance of coil 2 with respect to coil 1 is
N 2 12
N1 21
M 12 
 M 21 
I1
I2
M 12 I1
12 
N2
• Mutual inductance depends on the geometry of both
circuits and on their mutual orientation
• If current I1 varies with time, the
emf induced by coil 1 in coil 2 is
d M 12 I1
d12
 N2
 2  N2
dt N 2
dt
dI1
dI1
  M 12



M
2
12
dt
dt
LC Circuit
• A capacitor is connected to an inductor in an LC
circuit
• Assume the capacitor is initially charged and then the
switch is closed
• Assume no resistance and no energy losses to
radiation
• The current in the circuit and the
charge on the capacitor oscillate
between maximum positive and
negative values
LC Circuit
• With zero resistance, no energy is transformed into
internal energy
• Ideally, the oscillations in the circuit persist
indefinitely (assuming no resistance and no radiation)
• The capacitor is fully charged and the energy in the
circuit is stored in the electric field of the capacitor
Q2max / 2C
• No energy is stored in the inductor
• The current in the circuit is zero
LC Circuit
• The switch is then closed
• The current is equal to the rate at which the charge
changes on the capacitor
• As the capacitor discharges, the energy stored in the
electric field decreases
• Since there is now a current, some
energy is stored in the magnetic
field of the inductor
• Energy is transferred from the
electric field to the magnetic field
LC Circuit
• Eventually, the capacitor becomes fully discharged
and it stores no energy
• All of the energy is stored in the magnetic field of the
inductor and the current reaches its maximum value
• The current now decreases in magnitude, recharging
the capacitor with its plates having opposite their
initial polarity
• The capacitor becomes fully
charged and the cycle repeats
• The energy continues to oscillate
between the inductor and the capacitor
LC Circuit
• The total energy stored in the LC circuit remains
constant in time
2
2
Q
LI
U

2C
2
Q dQ
dI
 LI
0
C dt
dt
2
dU d Q
d LI


0
dt dt 2C dt 2
2
Q
d Q
L 2 0
C
dt
• Solution:
Q(t )  Qmax
2
 t

cos
 
 LC

2
d Q
Q

2
dt
LC
LC Circuit
• The angular frequency, ω, of the circuit depends on
the inductance and the capacitance ω  1
LC
• It is the natural frequency of oscillation of the circuit
• The current can be expressed as a function of time:
dQ d
I (t ) 
 Qmax cost     Qmax sin t   
dt dt
I (t )   I max sin t   
Q(t )  Qmax
 t

cos
 
 LC

Q(t )  Qmax cost   
LC Circuit
• Q and I are 90° out of phase with each other, so when
Q is a maximum, I is zero, etc.
I (t )   I max sin t   
Q(t )  Qmax cost   
Energy in LC Circuits
• The total energy can be expressed as a function of
time
2
2
2
2
Q
LI
Qmax
LI max
2
U


cos t    
sin 2 t   
2C
2
2C
2
• The energy continually oscillates
between the energy stored in the
electric and magnetic fields
• When the total energy is stored in
one field, the energy stored in the
other field is zero
Energy in LC Circuits
• In actual circuits, there is always some resistance
• Therefore, there is some energy transformed to
internal energy
• Radiation is also inevitable in this type of circuit
• The total energy in the circuit continuously decreases
as a result of these processes
Chapter 32
Problem 38
An LC circuit consists of a 20.0-mH inductor and a 0.500-μF capacitor.
If the maximum instantaneous current is 0.100 A, what is the greatest
potential difference across the capacitor?
RLC Circuit
• The total energy is not constant, since there is a
transformation to internal energy in the resistor at the
rate of dU/dt = – I2R
• Radiation losses are still ignored
• The circuit’s operation can be expressed as:
Q dQ
dI
2
 LI
 I R
C dt
dt
Q
dI
 L   IR
C
dt
2
Q
d Q dQ
L 2 
R0
C
dt
dt
RLC Circuit
• Solution:
d 
1
LC
Q(t )  Qmax e
 R 
 
 2L 

Rt
2L
cosd t   
2
• Analogous to a damped harmonic oscillator
• When R = 0, the circuit reduces to an LC circuit (no
damping in an oscillator)
2
Q
d Q dQ
L 2 
R0
C
dt
dt
Answers to Even Numbered Problems
Chapter 32:
Problem 2
1.36 µH