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Inductance
©2011 by Bryan Pflueger
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Mutual Inductance
If two coils of wire are placed near each other and have a current
passing through them, they will each induce an emf on one another
because they each generate a magnetic flux through each of the coils
centers.
To show the proportionality of the induced magnetic field to the
current we take into account the number of turns as well as the
mutual inductance of the two coils, denoted M21.
Equation for Mutual
Inductance
Mutual Inductance
The emfs have a negative sign because according to Lenses Law
they have to oppose any change to the magnetic field.
Units of Mutual Inductance
The units of mutual inductance is the Henry, H, named in honor of
the American Physicist Joseph Henry.
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Self Inductance
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In mutual inductance we explained how two separate circuits effect
one another, but now we will discuss the case in which the circuit
induces an emf in itself by means of self inductance.
If a loop of wire with N number of coils is in a circuit and if the current
varies then so does the magnetic flux resulting in an induced emf.
Once again the induced emf will oppose any change to the current, so
it lessens the chance of any fluctuations in the magnitude of the
current.
Equation for
Self Inductance
Self Inductance
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We already know how to deal with a resistor, battery, and parallel plate
capacitors in circuits, but an inductor generates a non-conservative
electric field. In order to use them in a circuit we have to determine a
general principle which is analogous to Kirchoff's Loop rule.
For a circuit with a varying emf and an inductor there are two electric
fields. The first is a conservative electric field, Ec, which is produced by
the battery and the second is a non-conservative electric field, En,
produced by a varying magnetic flux within the inductor.
For now we will assume the inductor's coils have negligible resistance
and therefore only a relatively small electric field is required to move
charges throughout the circuit. Since the electric field in the circuit is
nonzero charge will begin to build up on the terminals of the inductors,
and the net electric field inside the inductor will be zero,Ec + En = 0.
Self Inductance
If we apply Faraday's Law to the previous scenario depicted here, we
can determine there is a true potential difference across the inductor
related to the conservative electric field, even though the inductor
produces a non-conservative electric field within its coils.
x
Varying
emf
Since the non-conservative electric field produced by the inductor is
nonzero everywhere except for inside the inductor we can change the
integration from the entire loop just to the segment containing the
inductor.
L
y
The sum of the electric fields within the inductor are
zero, Ec + En = 0, so we can rewrite the equation as:
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Self Inductance
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x
Varying
emf
L
The integration of this simply is the potential
difference at the points x and y along the circuit.
y
We can conclude that the inductor has a potential difference across its
terminals which are related to the conservative electric field and we
see that the inductor does not resist the current, rather it resists any
change (di/dt).
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1
An inductor with inductance L is placed in series with a
battery. The equation for the for the current in the
inductor is given by the I=be-2t. Which of the following
represents the emf induced in the inductor at t=1?
A -2Lb/e
B Lb/e2
C -2Lb/e2
D -Lb/3e3
E zero
Magnetic Field Energy
A battery has a potential difference of #, a resistor of resistance r, and
an inductor of self inductance L are all placed in a series circuit. The
current through the circuit will initially be zero, but it will eventually
reach its max value of #. This is because the inductor is resisting the
change in the current, but slowly the current will grow to its max
value as will the potential difference across the inductor.
In between these two points in time we can write several equations
which will allow us to calculate the power stored in the inductor after
a long time.
positive because the
current is increasing
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Magnetic Field Energy
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The equation we just found for the potential energy stored in an
inductor can be applied to any other shape, such as a toroidal
solenoid whose volume is equal to the circumfrence multiplied by its
area.
The value of its self inductance is:
The value of its Potential Energy is given by:
Magnetic Field Energy
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The magnetic energy density is given as the ratio of the potential
energy and the volume of the inductor. It is denote by u.
The magnetic energy density can also be represented in terms of the
magnetic field. The magnetic field inside the toroidal solenoid is:
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2
An inductor of Inductance .5H is placed in series with a
battery which supplies a steady current of 2A. After a
long time what is the energy stored within the inductor?
A 1J
B 2J
C 3J
D 4J
E 5J
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R-L Circuit
Initially when switch S1 is closed the current flowing through that
segment of the circuit is zero, but after a long time the current is at
its maximum of #/R; however, we want to discuss the case in
between these two points in time.
R
L
Initially the current is zero, so the rate
of change of current is:
When the current has reached its
maximum value, di/dt=0.
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R-L Circuit
Using Kirchoff's loop rule we know the net voltage drop in the loop
must be equal to zero, therefore:
Current for a R-L Circuit in
terms of time
R-L Circuit
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R-L Circuit
After the circuit has reached it maximum current, what happens if we
open S1 and close S2? We know that the inductor wants to prevent
any change in the current so initially it will remain at Imax, but
eventually it will dissipate and drop down to zero.
R
L
Current Decay with
Respect to time
R-L Circuit
Current Decay
R-L Circuit
Time Constant
The R-L Circuit is dependent on R/L. In one time constant the
currents value changes by I(1-1/e), when it is connected to the
battery and by 1/e, when it is decaying.
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3
An inductor of unknown inductance and a resistor of 12#
are placed in series with a battery that supplies a current
throughout the circuit. The current in the inductor is given
by the equation, I=Io-Ioe-3t. What is the value of the
inductance?
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A .167H
B .25H
C 4H
D 36H
E Not enough information provided
L-C Circuit
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If we place a completely charged capacitor whose potential difference
is V and has an initial charge of Q, Q=CV, in series with an inductor an
interesting case occurs. Since the capacitor stores energy in its
electric field and the inductor will store the energy in its magnetic field,
this will cause an oscillation of the energy back and forth from the
inductor to the capacitor.
As the Capacitor discharges the current will slowly reach its maximum
value, then it will be stored in the inductor's magnetic field. After the
current has reached its maximum value current will continue to flow
and start to recharge the capacitor, but with opposite polarity. This will
be a continuous process that will constantly reverse the polarity of the
capacitor.
L-C Circuit
Using Kirchoff's Loop Rule we set up an equation for
the net voltage drop in the circuit
Divide by -L and sub in for di/dt
This is the same equation you would derive for simple harmonic motion,
as well as an equation for its position and its angular frequency.
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L-C Circuit
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By confirming that the equations are analogous for the circuit and
the mass spring system, we can also setup one for the energy of the
system. Remember charge Q is the max amount of charge, and q
is just the charge sometime during the discharge and charging
states.
Energy in conserved in the circuit, and by solving for the current we
can find out how it is dependent on the charge on the capacitor.
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4
A fully charged capacitor and an inductor are placed in
series with each other and have a frequency of # . If
the inductance was quadrupled and the capacitance
was cut in half, which of the following represents the
new frequency?
A
B
C
D
E
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L-R-C Circuit
Before in the L-C Circuit the energy was conserved, but what if a
resistor were place in series with the other two components to form
an L-R-C circuit. In this case the energy would continue to dissipate
as it passes through the resistor each time.
When the value of r still allows for some oscillation of the energy it is
considered to be underdamped. If the circuit no longer oscillates
then it is considered critically damped. If there is an even greater
resistor in the circuit it will be overdamped and the capacitor will
approach zero even faster.
L-R-C Circuit
Charge with Respect to Time
Q
0
t
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