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Re: Flat conductors (was I need help).



I wrote:

Specifically, a disturbance in the charge pattern with
wavelength lambda will die out exponentially with a time
constant tau, where
tau = (eps0 lambda^2) / (k L)
where 1/k is the resistivity of the paper

kowalskil wrote:

Explain this LAMBDA, please.

What part of "wavelength" did you not understand?

I considered a disturbance of the form
psi(t, x) = exp(-t/tau) sin(2pi x/lambda)
verified that it satisfied the equation of motion,
(not too near the boundaries) and solved for tau
as a function of lambda.

When I plot the calculated
equipotential lines in the plane of symmetry ...
ignoring the paper size effect ...
The pattern for two point charges and the pattern for two
long cylinders are not identical.

So apply Gauss's law to each of the calculated
field-patterns and see which one is bogus.
I recommend a pancake-shaped pillbox, thin in the
Z direction and circular in the XY plane.

If you use a crummy voltmeter, you will get all sorts of
goofy answers, even with ordinary resistor paper, let
alone with any higher-resistivity substance. That's
especially true in regions far from the sources, i.e.
where the fields are relatively small.

Are you saying that static surface charges will be responsible
for the boundary conditions?

Yes. What else could it be? This isn't magic, it's
just Maxwell equations and a little bit of Ohm's law.
What else is there?

If so then blaming the "paper
boundary" (for the discrepancy we are addressing) and
blaming "surface charges" amounts to the same thing.

1) I'd rather call them "boundary" charges not "surface"
charges because in the ordinary meaning of the word,
the entire paper is a "surface".

2) I don't "blame" the boundary effects for anything.
My best guess is that the discrepancy is imaginary.
If I understand the reports, the cylinder calculation
agrees well-enough with observations. No problem.
The two-point-sources calculation doesn't agree, so
it seems a pretty safe bet that the calculation is
in error.