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Re: I need help.



1) Please, folks, let's use Subject: lines that describe the
topic. "I need help" is not descriptive.

2) People should have been assuming that the method of
images is theoretical and impractical. It's not!

Why not do the experiment? Take a piece of resistor paper and
divide it into four cells. Put the same pattern of electrodes
in each cell, with mirror symmetry:

+---------------------+---------------------+
| | |
| | |
| N | N |
| P | P |
| | |
| | |
+---------------------+---------------------+
| | |
| | |
| P | P |
| N | N |
| | |
| | |
+---------------------+---------------------+


where P denotes a positive electrode and N denotes a negative
electrode. Wire all the Ps together and all the Ns together so that
they are at the same potential. Make all the contacts the same size
and shape. The "dividing lines" in the diagram are purely imaginary
at this stage.

I predict that you will find that the potential in each cell in this
configuration is the same as you would get if you actually cut the
paper into four pieces. The method of images enforces the same
boundary conditions at the cell-boundary as the actual cut would.

Of course you can experiment with more than a 2x2 array of cells. Any
MxN will work.

Note that the correctness of all this depends on having a real
boundary at the edge of the 4-cell piece of paper. In other words, if
you put four cells (and nothing else) in the middle of a large sheet,
the potential would not quite agree with a single cut-out cell. If
you put lots and lots of cells in the middle of a large sheet, the
middle cells would almost agree with a single cut-out cell, but not
exactly.

On Mon, 25 Feb 2002, David Bowman wrote:

I believe the Ewald sum of potentials over the lattice of images
converges quite slowly with distance from the original unit cell of
the actual paper.

Right.

Then John Mallinckrodt wrote:

... I used the all purpose hedge word
"approximated." Nevertheless, the proof may be in the pudding.
Since the resulting potential visibly (almost) satisfies the
required boundary conditions, the uniqueness property
(approximately) guarantees the solution. How's that for more
hedging?

The right answer can be obtained easily. No hedging required. Just
impose the appropriate boundary condition at the edge of the universe.
If the universe contains an MxN array of cells, with both M and N
even, then you can use periodic boundary conditions. You can use
hall-of-mirrors boundary conditions for any M or N (even or odd).
Techniques for coding such boundary conditions were discussed
previously in connection with the spreadsheets for solving Laplace's
equation. Look at
http://www.monmouth.com/~jsd/physics/laplace.html
and search for "periodic boundary conditions" and "hall of mirrors".