W9_parallel_resonance.eps

Week 8: Faraday’s Law and Induction 275

Of course we’ve formulated this result in a completely general way, but forarbitraryconducting
pathwaysMijhides a whole lot of integration evil that we just won’t be able to manage. In simple
cases, however, wecanevaluate it analytically (and we will, in examples and for homework), andin
others we can evaluate it numerically, and when both of these fail wecan at the very leastmeasure
it in a lab, so this is a useful decomposition. We call theMijthemutual inductanceof theith and
jth circuit and give it a set of SI units all its, own,Henries. We will specify Henries more precisely
shortly, as they are still obscure.

Note that there is no real reason fori 6 =jin this expression. There is a magnetic field through
the loopCidue to the currentIiinCi; this current creates a flux through the loop due to its own
current:

φii =

∫

Si/Ci

B~i·ˆnidAi

=

μ 0

4 π

∫

Si/Ci

(∫

Ci

μ 0 Iid~li′×(~ri−~ri′)

|~ri−~ri′|^3

)

·nˆidAi

=

μ 0

4 π

∮

C 1

∮

C 1

d~li·d~li′

|~ri−~ri′|

Ii

= MiiIi

= LiIi (594)

where we define theself-inductanceof theith loop to be the symbolLi. Note that I had to add
primes to the “j” coordinates in the previous expression to differentiate between the integral over
the current loop and the integral over the area.

In practical terms, the self-inductance will be very important to us as design elements in elec-
tronic circuits designed to process information and as an importantaspect of any piece of electrical
equipment based on coils of wire with many turns, e.g. electrical motors and generators.

Inductance is the magnetic equivalent of capacitance. Inductances can (as we will see) store
energy, generate voltages, and do many useful things for us. Before we move on to see how by
actually computing inductances and the potentials they can generate, we should complete the formal
work we have begun by introducing theLiandMijsymbols. In terms of these, we can now write
the total magnetic flux through theith circuit loop due to the currents inallof the loops:

φi=LiIi+

∑

j 6 =i

MijIj (595)

If we then differentiate this with respect to time and use Faraday’s Law, we get the following
expression for the induced voltage in theith loop:

Vi=−LidIi

dt

+

∑

j 6 =i

MijddtIj (596)

Finally, in many, if not most, cases of interest, we can neglect mutual inductance because the
magnetic field dies off rapidly with distance. For that reason we will often speak of the self-inductance
onlyof specific circuit elements, especially “inductors”, the magnetic equivalent of capacitors in a
circuit, labelled with a plainLwith or without an index. The key equation for a single self-inductance
will be:

VL=−L

dI

dt

(597)

whereVLis the voltage drop or rise across the inductor andIis the current through the inductor.
This expression finally gives us a good way of specifying the SI units for inductance. One Henry
is a Volt-Second/Ampere, or a Volt-Second^2 /Coulomb, or (since a Volt is a Joule/Coulomb) a
Joule/Ampere^2.