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8/14/2019 PC Chapter 32
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Chapter 32
Inductance
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Some Terminology Use emfand currentwhen they are
caused by batteries or other sources
Use induced emfand induced currentwhen they are caused by changingmagnetic fields
When dealing with problems inelectromagnetism, it is important todistinguish between the two situations
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Self-Inductance When the switch is
closed, the current
does notimmediately reach
its maximum value
Faradays law can
be used to describethe effect
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Self-Inductance, 2 As the current increases with time, the
magnetic flux through the circuit loop
due to this current also increases withtime
This corresponding flux due to thiscurrent also increases
This increasing flux creates an inducedemf in the circuit
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Self-Inductance, 3 The direction of the induced emf is such that
it would cause an induced current in the loop
which would establish a magnetic fieldopposing 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
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Self-Inductance, 4 This effect is called self-inductance
Because the changing flux through the
circuit and the resultant induced emf arisefrom the circuit itself
The emfL is called a self-induced
emf
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Self-Inductance, Coil Example
A current in the coil produces a magnetic field
directed toward the left (a) If the current increases, the increasing flux creates an
induced emf of the polarity shown (b) The polarity of the induced emf reverses if the current
decreases (c)
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Self-Inductance, Equations A induced emf is always proportional to
the time rate of change of the current
L is a constant of proportionality called
the inductance of the coil and itdepends on the geometry of the coiland other physical characteristics
IL
d Ldt
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Inductance of a Coil A closely spaced coil ofNturns carrying
currentIhas an inductance of
The inductance is a measure of the
opposition to a change in current
I I
B LNLd dt
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Inductance Units The SI unit of
inductance is the
henry (H)
Named for JosephHenry (pictured
here)
A
sV1H1
=
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Inductance of a Solenoid Assume a uniformly wound solenoid
having Nturns and length Assume is much greater than the radius
of the solenoid
The interior magnetic field is
I Io oNB n l
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Inductance of a Solenoid, cont The magnetic flux through each turn is
Therefore, the inductance is
This shows that L depends on thegeometry of the object
IB o
NA
BA l
2
I
B oN N A
L
l
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RL Circuit, Introduction 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, even without a coil, a circuit will
have some self-inductance
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Effect of an Inductor in a
Circuit The inductance results in a back emf
Therefore, the inductor in a circuit
opposes changes in current in that
circuit
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RL Circuit, Analysis An RL circuit contains an
inductor and a resistor
When the switch is closed(at time t= 0), the current
begins to increase
At the same time, a back
emf is induced in theinductor that opposes the
original increasing current
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Active Figure 32.3
(SLIDESHOW MODE ONLY)
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RL Circuit, Analysis, cont. Applying Kirchhoffs loop rule to the
previous circuit gives
Looking at the current, we find
0I
Id
R Ldt
1I Rt L
eR
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RL Circuit, Analysis, Final The inductor affects the current
exponentially
The current does not instantly increaseto its final equilibrium value
If there is no inductor, the exponential
term goes to zero and the current wouldinstantaneously reach its maximumvalue as expected
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RL Circuit, Time Constant The expression for the current can also be
expressed in terms of the time constant, , of
the circuit
where = L / R
Physically, is the time required for the
current to reach 63.2% of its maximum value
1I t
eR
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RL Circuit, Current-Time
Graph, (1) The equilibrium value
of the current is /R
and is reached as tapproaches infinity
The current initiallyincreases veryrapidly
The current thengradually approaches
the equilibrium value
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RL Circuit, Current-Time
Graph, (2) The time rate of
change of the
current is amaximum at t= 0
It falls off
exponentially as t
approaches infinity In general,
I td edt L
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Active Figure 32.6
(SLIDESHOW MODE ONLY)
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Energy in a Magnetic Field In a circuit with an inductor, the battery
must supply more energy than in a
circuit without an inductor Part of the energy supplied by the
battery appears as internal energy inthe resistor
The remaining energy is stored in themagnetic field of the inductor
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Energy in a Magnetic Field,
cont. Looking at this energy (in terms of rate)
Iis the rate at which energy is beingsupplied by the battery
I2Ris the rate at which the energy is beingdelivered to the resistor
Therefore, LIdI/dtmust be the rate at whichthe energy is being stored in the magnetic
field
2 I I I I
d R L
dt
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Energy in a Magnetic Field,
final Let Udenote the energy stored in the
inductor at any time
The rate at which the energy is stored is
To find the total energy, integrate and
II
dU dL
dt dt
0
I
I IU L d
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Energy Density of a Magnetic
Field Given U = LI2
SinceAis the volume of the solenoid, themagnetic energy density, uB is
This applies to any region in which a magnetic
field exists (not just the solenoid)
22
21
2 2o
o o
B B
U n A A n
l l
2
2B
o
U Bu
A
l
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Example: The Coaxial Cable Calculate L for the cable
The total flux is
Therefore, L is
The total energy is
ln2 2
I Ib o oB
a bB dA dr
r a
ll
ln2I
B o bL a
l
2
21ln
2 4
I
Io b
U L a
l
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Mutual Inductance, 3 The mutual inductanceM12 of coil 2
with respect to coil 1 is
Mutual inductance depends on thegeometry of both circuits and on their
orientation with respect to each other
2 12
12
1I
NM
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Induced emf in Mutual
Inductance If currentI1 varies with time, the emf
induced by coil 1 in coil 2 is
If the current is in coil 2, there is a
mutual inductance M21 If current 2 varies with time, the emf
induced by coil 2 in coil 1 is
12 12 2 12
Id d N Mdt dt
2
1 21
Id M
dt
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Mutual Inductance, Final In mutual induction, the emf induced in one
coil is always proportional to the rate at which
the current in the other coil is changing The mutual inductance in one coil is equal to
the mutual inductance in the other coil M12 = M21 = M
The induced emfs can be expressed as
2 1
1 2and
I Id d M M
dt dt
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LCCircuits A capacitor is
connected to aninductor in an LCcircuit
Assume the capacitoris initially charged andthen the switch isclosed
Assume no resistanceand no energy lossesto radiation
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Oscillations in an LCCircuit Under the previous conditions, the current in
the circuit and the charge on the capacitoroscillate between maximum positive andnegative values
With zero resistance, no energy istransformed into internal energy
Ideally, the oscillations in the circuit persistindefinitely The idealizations are no resistance and no
radiation
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Oscillations in an LCCircuit, 2 The capacitor is fully charged
The energy Uin the circuit is stored in the
electric field of the capacitor The energy is equal to Q2max / 2C
The current in the circuit is zero
No energy is stored in the inductor The switch is closed
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Oscillations in an LCCircuit, 3 The current is equal to the rate at which
the charge changes on the capacitor
As the capacitor discharges, the energystored in the electric field decreases
Since there is now a current, some energyis stored in the magnetic field of the
inductor Energy is transferred from the electric field
to the magnetic field
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Oscillations in an LCCircuit, 4 The capacitor becomes fully discharged
It stores no energy
All of the energy is stored in the magneticfield of the inductor
The current reaches its maximum value
The current now decreases inmagnitude, recharging the capacitorwith its plates having opposite theirinitial polarity
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Oscillations in an LCCircuit,
final Eventually the capacitor becomes fully
charged and the cycle repeats
The energy continues to oscillatebetween the inductor and the capacitor
The total energy stored in the LC circuit
remains constant in time and equals2
21
2 2IC L
QU U U L
C
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LCCircuit Analogy to Spring-
Mass System, 1
The potential energy kx2 stored in the spring isanalogous to the electric potential energy C(V
max
)2
stored in the capacitor All the energy is stored in the capacitor at t= 0 This is analogous to the spring stretched to its
amplitude
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LCCircuit Analogy to Spring-
Mass System, 2
The kinetic energy ( mv2) of the spring is analogousto the magnetic energy ( LI2) stored in theinductor
At t= T, all the energy is stored as magneticenergy in the inductor
The maximum current occurs in the circuit This is analogous to the mass at equilibrium
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LCCircuit Analogy to Spring-
Mass System, 3
At t= T, the energy in the circuit is
completely stored in the capacitor The polarity of the capacitor is reversed
This is analogous to the spring stretched to -A
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LCCircuit Analogy to Spring-
Mass System, 4
At t = T, the energy is again stored in the
magnetic field of the inductor This is analogous to the mass again reaching
the equilibrium position
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Active Figure 32.17
(SLIDESHOW MODE ONLY)
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Time Functions of an LC
Circuit In an LC circuit, charge can be expressed as
a function of time
Q = Qmax cos (t+ ) This is for an ideal LCcircuit
The angular frequency, , of the circuit
depends on the inductance and the
capacitance It is the natural frequencyof oscillation of the
circuit
1LC
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Time Functions of an LC
Circuit, 2 The current can be expressed as a
function of time
The total energy can be expressed as a
function of time
max I sin( )dQ
Q t dt
2
2 2 21
2 2
maxmaxcos I sinC L
QU U Ut L t
c
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Charge and Current in an LC
Circuit The charge on the
capacitor oscillates
between Qmax
and
-Qmax The current in the
inductor oscillates
betweenImax and -Imax Q andIare 90o out of
phase with each other So when Q is a maximum, I
is zero, etc.
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Energy in an LCCircuit
Graphs The energy
continually oscillatesbetween the energystored in the electricand magnetic fields
When the totalenergy is stored inone field, the energystored in the otherfield is zero
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Notes About Real LCCircuits In actual circuits, there is always some
resistance
Therefore, there is some energy transformedto 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
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The RLCCircuit A circuit containing a
resistor, an inductor
and a capacitor iscalled an RLC
Circuit
Assume the resistor
represents the totalresistance of the
circuit
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Active Figure 32.21
(SLIDESHOW MODE ONLY)
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RLCCircuit, Analysis The total energy is not constant, since
there is a transformation to internal
energy in the resistor at the rate ofdU/dt= -I2R Radiation losses are still ignored
The circuits operation can beexpressed as
2
20
d Q dQ QL R
dt dt C
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RLCCircuit Compared to
Damped Oscillators The RLCcircuit is analogous to a
damped harmonic oscillator
When R= 0 The circuit reduces to an LCcircuit and is
equivalent to no damping in a mechanical
oscillator
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RLCCircuit Compared to
Damped Oscillators, cont. When Ris small:
The RLCcircuit is analogous to light
damping in a mechanical oscillator Q = Qmaxe
-Rt/2L cos dt
d is the angular frequency of oscillation
for the circuit and1
2 2
1
2d
R
LC L
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RLCCircuit Compared to
Damped Oscillators, final When Ris very large, the oscillations damp
out very rapidly There is a critical value ofRabove which no
oscillations occur
IfR= RC, the circuit is said to be critically
damped When R> RC, the circuit is said to be
overdamped
4 /CR L C
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Damped RLCCircuit, Graph The maximum value
ofQ decreases after
each oscillation R< RC
This is analogous to
the amplitude of a
damped spring-mass system
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Overdamped RLCCircuit,
Graph The oscillations
damp out very
rapidly Values ofRare
greater than RC
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