W9_parallel_resonance.eps

274 Week 8: Faraday’s Law and Induction

nearby conducting loops to change in time. This, according to Faraday’s Law, will induce a voltage
around those loops and, assuming they have some resistance, cause current to flow in the direction
predicted by Lenz’s Law.

Forloops of fixed size and orientation, the field produced by them at any given point in space
isdirectly proportional to the current they carry (from the Biot-Savart Law, which contains the
current in the wire on top and constant so it can be pulled out of the integral over the geometry of
the wire). The magnetic flux both through the loop itself and through allotherloops that its field
passes through is thusalsoproportional to the current.

This general state of affairs is pictured in figure 100. In this figure,loop 1 (we suppose) carries a
currentI 1. At the instant shown, this current produces a magnetic that swirls up through loop 1 in
field line loops that go around the current in the right-handed direction. These field lines pass both
through any surfaceS 1 we might draw that is bounded by the curveC 1 and through the surfaces
Si 6 =1bounded by the other curvesCi. These fields createmagnetic fluxthat is proportional toI 1
in all of the loops.

We can write this in an algebraic form. The flux through theith loop caused by the current in
thejth loop is:

φij =

∫

Si/Ci

B~j·nˆidAi

=

μ 0

4 π

∫

Si/Ci

(∫

Cj

Ijd~lj×(~ri−~rj)

|~ri−~rj|^3

)

·ˆnidAi

=

μ 0

4 π

(∫

Si/Ci

∫

Cj

d~lj×(~ri−~rj)

|~ri−~rj|^3 ·nˆidAi

)

Ij

= MijIj (591)

where I’ve take some pains to label the coordinates with the object:nˆinormal to the surfaceSi
bounded by the curveCi, wheredAiis the area element of this surface and~rithe vector coordinate
of a point on its surface; coordinatesd~ljand~rjon the curveCj.

There are a few very interesting things to observe about this pair of integrals. One is that
the integral over the surfaceSicannot depend on the particular surface chosen out of theinfinite
number of surfacesSibounded by any particular curveCi. Understanding how integrals like this
can be invariant as one selects different surfaces will be a key aspect of our addition of the Maxwell
Displacement Current in two more weeks, so consider this a hint.

Ultimately, it can therefore only depend onCiitself, so both integralscanbe represented as
integrals around the closed loopsCiandCjusing theorems from multivariate calculus that you do
probably do not yet know^73. The result is (eventually):

Mij =

μ 0

4 π

(∫

Si/Ci

∫

Cj

d~lj×(~ri−~rj)

|~ri−~rj|^3

·nˆidAi

)

=

μ 0

4 π

∮

C 1

∮

C 2

d~li·d~lj

|~ri−~rj|

(592)

which is obviously symmetric under interchange ofiandj:

Mij=Mji (593)

for any two loopsCiandCjcarrying currentsIiandIjrespectively.

(^73) Wikipedia: http://www.wikipedia.org/wiki/derivation of self inductance. It uses Stoke’s Theorem and the defi-
nition of the magnetic field in terms of the vector potential,both things that are beyond the scope of this course, but
it actually isn’t terribly difficult. I link the wikipedia page so that interested students (or students in a more advanced
course trying to connect back to simpler concepts by readingthis book) can take a look.