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Re: funny capacitor (URL)



At 06:30 PM 3/17/01 -0500, Bob Sciamanda wrote:
I defined the potentials only on the conductors and then applied
the averaging algorithm to all other cells - including the boundaries of
the region of calculation.
...
Just try removing your container and see what happens.

This is a fine thing to do. I haven't seen Bob's code, or even a
description thereof, so I don't know exactly how he handled the boundary
conditions. The code Ludwik posted has no provision for dealing with the
boundary conditions, so "just" removing the container is likely to lead to
loads of confusion.

In general, you have to be careful with the boundary conditions. At the
boundary, it is never 100% obvious what "averaging" is supposed to mean or
what "Laplacian" is supposed to mean. If you're not careful, you could
stumble across a formula that violates gauge invariance and/or violates
local charge conservation.

I know of a couple of nice techniques of handling the boundary
conditions: Born-von-Karman periodic boundary conditions, and
hall-of-mirrors periodic boundary conditions. These are discussed at
http://www.monmouth.com/~jsd/physics/laplace.html#gauge
which contains pointers to downloadable spreadsheets that demonstrate each
technique.

Can anybody suggest additional gauge-invariant charge-conserving boundary
techniques?