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CPHY 122
Class Notes 14
Instructor: H. L. Neal
1 Self Inductance
R
I
E
Created usingUNREGISTEREDTop Draw 3.10 Nov 16,'106 4:06:20 PM
Consider a single circuit around which a current I will áow after the switch is closed. This current generates a magnetic Öeld B which gives rise to a magnetic áux B through the circuit. We expect the áux B to be directly proportional to the current I, given the linear nature of the laws of magnetostatics and the deÖnition of magnetic áux. Thus, we can write
B = LI;
where the constant of proportionality L is called the self inductance of the circuit. If the current áowing around the circuit changes with time, then according to Faradayís law, an emf
EL =
dB dt = L
dI dt Thus, the emf generated around the circuit due to its own current is directly proportional to the rate at which the current changes. Lenzís law, and common sense, demand that if the current is increasing then the emf should always act to reduce the current, and vice versa. This is easily appreciated, since if the emf acted to increase the current when the current
was increasing then we would clearly get an unphysical positive feedback e§ect in which the current continued to increase without limit. The self inductance L of a circuit is necessarily a positive number.
2 Example 1
Here we will analyze carefully the ináuence of the self inductance L in the circuit above after the switch is closed. Applying Kircho§ís rules gives
E + EL = IR:
or
E = IR + L
dI dt
The solution to this equation is found in two steps: First we not that the steady state solution is
Isteady =
E
R
Second, we obtain the homogeneous solution by solving
IR + L
dI dt
Note that we may write dI I
L
R
dt:
Integrating both sides gives
ln I =
L
R
t + C;
so that
Ihomo = C exp
L
R
t
Then
I (t) = Isteady + Ihomo
=
E
R
L
R
t
At t = 0
I (0) =
E
R
+ C = 0;
so that
C =
E
R
Therefore
I (t) =
E
R
1 e ^
L R t
the same current áows around circuit 2: this is true irrespective of the size, number of turns, relative position, and relative orientation of the two circuits. Because of this, we can write
M 12 = M 21 = M;
where M is termed the mutual inductance of the two circuits. Note that M is a purely geometric quantity, depending only on the size, number of turns, relative position, and relative orientation of the two circuits. The SI units of mutual inductance are called Henries (H). One henry is equivalent to a volt-second per ampere:
1H 1V s A ^1 :
It turns out that a henry is a rather unwieldy unit. The mutual inductances of the circuits typically encountered in laboratory experiments are measured in milli-henries. Suppose that the current áowing around circuit 1 changes with time. Then the áux linking circuit 2 changes with time. According to Faradayís law, an emf
E 2 =
dB 2 dt
is generated around the second circuit due to the changing magnetic áux linking that circuit. This emf can also be written
E 2 = M
dI 1 dt
Thus, the emf generated around the second circuit due to the current áowing around the Örst circuit is directly proportional to the rate at which that current changes. Likewise, the emf generated around the Örst circuit is
E 1 = M
dI 2 dt
Note that there is no direct physical coupling between the two circuits: the coupling is due entirely to the magnetic Öeld generated by the currents áowing around the circuits.
