W9_parallel_resonance.eps

348 Week 10: Maxwell’s Equations and Light

through a surface bounded by the curveC) is notinvariantwhen we vary the surfaceS

in perfectly reasonable ways.

One way we can try to deal with this is to insist that we have to use “nice” curvesC(ones

in a plane, for example) and “nice” surfacesS(ones in that same plane, for example)

but that isn’t very satisfactory – it seems like just a way of saying that Ampere’s Law is

really just Ampere’s Sort of OK Rule That Sometimes Works, Sometimes, If We Cheat.

We want a natural law toalwayswork – it has to be “unbreakable”, especially by as

simple a thing as bendingCinto twisted loop (like a crumpled coat hanger) or choosing

“the wrong”S(by what standard? how can we decide it is wrong without knowing the

answer some other way?).

Physicists get very anal about this sort of thing. If they don’t, the bugaboo of all human

efforts to reason,inconsistency, creeps into our set of beliefs, and mathematicians all well

know that you can proveanythingfrom a contradiction (and hence knownothingon the

basis of your proofs)^96.

Our job, it appears, is to try to make the current in Ampere’s Law invariant so that

it gives us the exact same current for any surfaceS/Cwe mighthappento choose to

solve a problem. That way we’ll all get the same answer forBφ, and if we choose the

rightinvariant current (there may be more than one) that answer will even agree with

experiment!

J

S 2

ρ

n = n’

n = n’

n = n’

S 1

n’

n

I out

I out

I in

C

Figure 139: A very general current density flows through space. Some current flows in from the
left and exits on the right, but some builds up in the current densityρin the volume between the
two surfacesS 1 andS 2. The point is that thedifferencebetween the flux (current) in throughS 1
and out throughS 2 must be equal to the rate that charge builds up in between, because charge is
conserved.

The picture that will best help us find the invariant current is drawn in139. We are

going to take this picture and think about it in the light of another physical law that

we really believe in, theLaw of Charge Conservation. We will discover that Ampere’s

Law fails to account for charge conservationandGauss’s Law for Electricity consistently.

Even better, thecorrectinvariant current will more or less fall out of our analysis at our

feet, ready to be plugged into Ampere’s Law to make it correct.

(^96) In fact, by insisting that Maxwell’s Equationsasnatural laws ought to be invariant under changes of inertial
reference frame, Einstein threw out more or lessall of classical non-relatistic physics– and was backed up by numerous
experients that showed that he wasrightto do so! Kind of scary, that...