Re: funny capacitor

[Ludwik Kowalski <KowalskiL@MAIL.MONTCLAIR.EDU>]

"Glenn A. Carlson" wrote:

> ...The idea that the solution to Laplace's equation represent
> states of  "minimal potential energy" is more intriguing.
> A variational calculation should show this. The potential
> energy density u is proportional to E^2 = (grad V)^2, and
> integrating u over all space gives the total potential energy U...

I implemented this suggestion and observed how the potential
energy of the field decreased monotonically as the function of
number of iterations. Here are the results:

After iteration #     Energy (arb. units)
    1                      2141
    2                      2079
    3                      1960
    4                      1908
    5                      1869

 After 90             1787.79 ~= const


It is a good pedagogical exercise to emphasize that the energy
resides only  in the cells of empty space. Here are some details:

a) For each of these cells I have only two candidates for the
gradient: dVx or dVy. I selected the one whose the absolute
was larger.

b) Then I calculated the (gradient V)^2, added the result to the
running sum and proceeded to the next empty space cell.

c) The running sum was displayed at the end of each iteration.

d) Then the running sum was reset to zero to start the new
iteration.

This method will not show that the equilibrated energy is the
lowest possible state, unless one starts changing cell contents
randomly to show that nothing less than 1787.79 can be found.
Ludwik Kowalski