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Re: charge is conserved



At 02:45 PM 2/9/01 -0800, Leigh Palmer wrote:

I have no charge grid; it is unnecessary to produce one, so I
don't.

If you want to see HOW and WHERE the charge is conserved, you need to
produce one.

I infer surface charge density by calculating the
normal component of the potential gradient at the surface. A
surface charge is calculated only after the relaxation has
converged sufficiently to make that a reasonable calculation.

This incorporates the completely unfounded assumption that all charge
resides on surfaces. The real physics of the situation includes the
possibility of space charge. The model must describe this. The model does
describe this.

You choose an object. The object has a certain capacitance. You choose a
potential. That implies a certain amount of charge. That charge has to
get from the boundary to the object. The charge must flow through the
intervening space. The model must describe this. The model does describe
this.

This initial condition does not satisfy Laplace's equation.

That is irrelevant. There are many potentials, some of which do or do not
satisfy Laplace's equation. Yet Gauss's theorem applies to them all.

> Your contention that charge must be conserved is obscure to me.
Conservation of charge is, of course, a requirement in any
physical process, but a relaxation calculation does not in any
way represent a physical process; it is purely a mathematical
exercise

It should have been clear from my previous note that the conservation of
charge has nothing to do with the relaxation process. An !!arbitrary!!
potential, whether relaxed or unrelaxed, has zero net charge (assuming any
halfway reasonable boundary conditions). This zero is conserved.

The relaxation algorithm, or any other algorithm, cannot change this zero.

I suggest that it is not a good test for
soundness of procedure and I offer my sound procedure as a
Gegenbeispiel.

Calling it a sound procedure doesn't make it a sound procedure. Saying
that charge is not conserved, when most of the region is not examined for
the possible presence of charge, is highly unsound.

It seems to me that, whatever your procedure is,
you build in charge conservation.

I am not assuming it. There are simple mathematical theorems that
guarantee it, whether I like it or not.

===============================

I can even state more general boundary conditions than given in my previous
note. An esthetically appealing example is Born-von-Karman periodic
boundary conditions: Imagine an NxN grid, extended such that column N+1 is
the same as column 1, and row N+1 is the same as row 1, and so on _ad
infinitum_ in all directions. It is easy to show that the net charge in
this system is zero.

(I am taking the charge operator to be a+b+c+d-4w as discussed in my
previous note.)

In addition it is easy to show that this system has a gauge
invariance. Adding a uniform constant to all cells leaves the charge
distribution unchanged.

After proving the foregoing results, take the limit as N becomes
exceedingly large.