chapter 30 inductance
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Fall 2008Physics 231 Lecture 10-1
Chapter 30Inductance
Fall 2008Physics 231 Lecture 10-2
Magnetic Effects
As we have seen previously, changes in the magnetic flux due to one circuit can effect what goes on in other circuits
The changing magnetic flux induces an emf in the second circuit
Fall 2008Physics 231 Lecture 10-3
Suppose that we have two coils,Coil 1 with N1 turns and Coil 2 with N2 turns
dt
dN B2
22
Coil 1 has a current i1 which produces a magnetic flux, , going through one turn of Coil 2If i1 changes, then the flux changes and an emf is induced in Coil 2 which is given by
Mutual Inductance
Fall 2008Physics 231 Lecture 10-4
Mutual InductanceThe flux through the second coil is proportional to the current in the first coil
12122 iMN B
where M21 is called the mutual inductance
dt
diM
dt
dN B 1
212
2 or
dt
diM 1
212
Taking the time derivative of this we get
Fall 2008Physics 231 Lecture 10-5
If we were to start with the second coil having a varying current, we would end up with a similar equation with an M12
We would find that MMM 1221
The two mutual inductances are the same because the mutual inductance is a geometrical property of the arrangement of the two coils
To measure the value of the mutual inductance you can use either
dt
dIM 1
2 ordt
dIM 2
1
Mutual Inductance
Fall 2008Physics 231 Lecture 10-6
Units of Inductance
2Amp
J1
Amp
secV1 Henry 1
Fall 2008Physics 231 Lecture 10-7
Self InductanceSuppose that we have a coil having N turns carrying a current I
That means that there is a magnetic flux through the coil
This flux can also be written as being proportional to the current
ILN B
with L being the self inductance having the same units as the mutual inductance
Fall 2008Physics 231 Lecture 10-8
If the current changes, then the magnetic flux through the coil will also change, giving rise to an induced emf in the coil
This induced emf will be such as to oppose the change in the current with its value given by
dt
dIL
If the current I is increasing, then
If the current I is decreasing, then
Self Inductance
Fall 2008Physics 231 Lecture 10-9
There are circuit elements that behave in this manner and they are called inductors and they are used to oppose any change in the current in the circuit
As to how they actually affect a circuit’s behavior will be discussed shortly
Self Inductance
Fall 2008Physics 231 Lecture 10-10
What Haven’t We Talked AboutThere is one topic that we have not mentioned with
respect to magnetic fields
Just as with the electric field, the magnetic field has energy stored in it
We will derive the general relation from a special case
Fall 2008Physics 231 Lecture 10-11
Magnetic Field EnergyWhen a current is being established in a circuit, work has to be done
If the current is i at a given instant and its rate of change is given by di/dt then the power being supplied by the external source is given by
dt
diiLiVP L
The energy supplied is given by PdtdU
The total energy stored in the inductor is then
2
02
1ILdiiLU
I
Fall 2008Physics 231 Lecture 10-12
This energy that is stored in the magnetic field is available to act as source of emf in case the current starts to decrease
We will just present the result for the energy density of the magnetic field
0
2
2
1
B
uB
This can then be compared to the energy density of an electric field
202
1EuE
Magnetic Field Energy
Fall 2008Physics 231 Lecture 10-13
R-L CircuitWe are given the following circuit
and we then close S1 andleave S2 open
It will take some finite amount of time for the circuit to reach its maximum current which is given by RI
Kirchoff’s Law for potential drops still holds
Fall 2008Physics 231 Lecture 10-14
Suppose that at some time t the current is i
The voltage drop across the resistor is given by
RiVab The magnitude of the voltage drop across the inductor is given by
dtdiLVbc
The sense of this voltage drop is that point b is at a higher potential than point c so that it adds in as a negative quantity
0dt
diLRi
R-L Circuit
Fall 2008Physics 231 Lecture 10-15
We take this last equation and solve for di/dt
iL
R
Ldt
di
Notice that at t = 0 when I = 0 we have that Ldt
di
initial
Also that when the current is no longer changing, di/dt = 0, that the current is given by R
I
as expected
But what about the behavior between t = 0 and t =
R-L Circuit
Fall 2008Physics 231 Lecture 10-16
R-L CircuitWe rearrange the original equation and then integrate
dtL
R
Ri
di
/
ti
dtL
R
Ri
di
00
'/'
'
The solution for this is tLReR
i /1
Which looks like
Fall 2008Physics 231 Lecture 10-17
As we had with the R-C Circuit, there is a time constant associated with R-L Circuits
R
L
Initially the power supplied by the emf goes into dissipative heating in the resistor and energy stored in the magnetic field
dt
diiLRii 2
After a long time has elapsed, the energy supplied by the emf goes strictly into dissipative heating in the resistor
R-L Circuit
Fall 2008Physics 231 Lecture 10-18
We now quickly open S1 and close S2
The current does not immediatelygo to zero
The inductor will try to keep the current, in the same direction, at its initial value to maintain the magnetic flux through it
R-L Circuit
Fall 2008Physics 231 Lecture 10-19
R-L CircuitApplying Kirchoff’s Law to the bottom loop we get
where I0 is the current at t = 0
0dt
diLiR
Rearranging this we have dtL
R
i
di
and then integrating this
t
I
dtL
R
i
di
0
0
''
'
0
tLReII /0
Fall 2008Physics 231 Lecture 10-20
This is a decaying exponentialwhich looks like
The energy that was stored in the inductor will be dissipated in the resistor
R-L Circuit
Fall 2008Physics 231 Lecture 10-21
L-C CircuitSuppose that we are now given a fully charged capacitor and an inductor that are hooked together in a circuit
Since the capacitor is fully charged there is a potential difference across it given by Vc = Q / C
The capacitor will begin to discharge as soon as the switch is closed
Fall 2008Physics 231 Lecture 10-22
We apply Kirchoff’s Law to this circuit
0C
q
dt
diL
Remembering thatdt
dqi
We then have that 2
2
dt
qd
dt
di
The circuit equation then becomes 01
2
2
qLCdt
qd
L-C Circuit
Fall 2008Physics 231 Lecture 10-23
This equation is the same as that for the Simple Harmonic Oscillator and the solution will be similar
)cos(0 tQq
The system oscillates with angular frequency
The current is given by )sin(0 tQdt
dqi
LC
1
is a phase angle determined from initial conditions
L-C Circuit
Fall 2008Physics 231 Lecture 10-24
Both the charge on the capacitor and the current in the circuit are oscillatory
For an ideal situation, this circuit will oscillate forever
The maximum charge and the maximum current occur seconds apart
L-C Circuit
Fall 2008Physics 231 Lecture 10-25
L-C Circuit
Fall 2008Physics 231 Lecture 10-26
Just as both the charge on the capacitor and the current through the inductor oscillate with time, so does the energy that is contained in the electric field of the capacitor and the magnetic field of the inductor
Even though the energy content of the electric and magnetic fields are varying with time, the sum of the two at any given time is a constant
BETotal UUU
L-C Circuit
Fall 2008Physics 231 Lecture 10-27
Instead of just having an L-C circuit with no resistance, what happens when there is a resistance R in the circuit
Again let us start with the capacitor fully charged with a charge Q0 on it
The switch is now closed
L-R-C Circuit
Fall 2008Physics 231 Lecture 10-28
L-R-C CircuitThe circuit now looks like
The capacitor will start to discharge and a current will start to flow
We apply Kirchoff’s Law to this circuit and get
0C
q
dt
diLiR
And remembering thatdt
dqi we get
01
2
2
qLCdt
dq
L
R
dt
qd
Fall 2008Physics 231 Lecture 10-29
The solution to this second order differential equation is similar to that of the damped harmonic oscillator
The are three different solutions
Underdamped
Critically Damped
Overdamped
Which solution we have is dependent upon the relative values of R2 and 4L/C
L-R-C Circuit
Fall 2008Physics 231 Lecture 10-30
L-R-C CircuitUnderdamped:
C
LR
42
The solution to the second differential equation is then
tL
R
LCeQq
tL
R
2
22
04
1cos
This solution looks like The system still oscillates but with decreasing amplitude, which is represented by the decaying exponential
This decaying amplitude is often referred to as the envelope
Fall 2008Physics 231 Lecture 10-31
L-R-C CircuitCritically Damped:
C
LR
42
Here the solution is given by
tL
R
etL
RQq
2
0 21
This solution looks like
This is the situation when the system most quickly reachesq = 0
Fall 2008Physics 231 Lecture 10-32
L-R-C Circuit
Overdamped:C
LR
42
Here the solution has the form
ttt
LR
eLR
eLR
eQ
q ''20
'21
'21
2
withLCL
R 1
4'
2
2
This solution looks like
Fall 2008Physics 231 Lecture 10-33
L-R-C Circuit
The solutions that have been developed for this L-R-C circuit are only good for the initial conditions at t = 0 that q = Q0 and that i = 0
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