W9_parallel_resonance.eps

168 Week 5: Resistance

something – it is very similar to the pictures we used to talk aboutelectric fluxin the context of
Gauss’s Law!

The problem we face is that there aremanysurfaces that pass through any given point, so
talking about how much charge passes a point on the wire isn’t very well defined. We would do
better talking about how much charge passes through a closed curve drawn around the wire (or
other, arbitrarily shaped) conductor, but even so, there are aninfinite number of surfaces bounded
by any closed curve. We need the electric current through such a loop not to depend on the surface
chosen, at least in the (quasi) stead-state dynamical equilibrium currents we are talking about here.

We can achieve this by recapitulating the reasoning of electric flux for (again) a single, simple
cylindrical wire where we can count on help from geometry. We want the current through our surface
Aperpendicular to the direction of motion of the charge to be the same as the current through a
second surfaceA′that is cut through the wire at more or less the same place but is tipped at an angle
θrelative to the direction of the current. As before, the tipped surface area areaA′=A/cos(θ) is
larger thanA. In order to get thesamecurrentIfrom these two surfaces, we need to compensate
for the cosine on the bottom with one on the top:

I=nqAvd=nq A

cos(θ)

cos(θ)vd=nqA′vdcos(θ) (293)

We can get the cosine out of a dot product between thelocal directionof thevectordrift velocity
~vd(assumed to be parallel to the actual current at any point in the wire) andˆn, thedirected normal
unit vectorto the surfaceAorA′:

I=nqA~vd·ˆn=nqA′~vd·nˆ′ (294)

We have a single choice to make in this expression – there are two possible directions perpendicular
to the surface and we have to choose (for example) either left to right or right to left as being positive
nˆ.

Again as before in our discussions of electric flux, we can take an arbitrarycurvedsurface and
break it up into tiny differential chunksdA, each with its own normal vectornˆselected with the
same left-to-right or vice-versa sense. The chunks are small enough that we can treat all the charges
that pass through them aslocallyall going in the same, unambiguous direction~vd. For each of
these, the differential current through the chunk is:

dI=nq~vd·nˆdA (295)

and we can now unambiguously sum up all of the current through an arbitrarycurvedsurface or
through plane surfaces where the flow of charge isnotall parallel and perpendicular to the surface.

If we chose as our surfaceany open surfaceSthat cuts completely across a branch of our
conductor, we will find that it is always bounded by a closed curveCon the surface of the branch.
We can then write the following, completely general and correct definition for the “current in the
branch” in the steady state:

IC=

∫

S/C

nq~vd·nˆdA=

∫

S/C

J~·nˆdA (296)

whereS/Cis read “through the surfaceSbounded by the closed curveCand where:

J~=nq~vd=ρfree~vd (297)

is called thecurrent density. In other words,the current through an open surfaceSbounded
by a closed curveCis the flux of the current density through that surface. Note well
that this is still justI=nqvdA=ρfreevdA=JAfor the simple cylindrical wire and perpendicular
surfaceAwe began with, but it can now handle far more general flows of current.