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*From*: Ludwik Kowalski <KowalskiL@MAIL.MONTCLAIR.EDU>*Date*: Tue, 6 Feb 2001 23:01:46 -0500

John's odd object (28 cells) looks like a dog on my screen.

The problem you solved, if I infer correctly, is only the

first, most demanding, step. Let me guess what you would

do to find the S(r) over the surface of Leigh's disk.

The cylindrical disk, at 100 V, would be represented by a

flat rectangle (the cross section of the disk) and you would

produce the distribution of potential in space surrounding

it, as you did for the "dog". The next step would be what?

Finding gradients along the top or bottom boundary,

identifying them with E, and calculating sigma=eps_o*E

at various distances from the center. Right?

Now that you have a working program why don't you use

it to produce the S(r) distribution? You are so close from it.

How strongly "anizotropic" (ANIZ== Smax/Smin) is the

S(r) distribution? Does the anisotropy depend on the disk

radius? Does it depend on the disk thickness? I suppose

many people would be interested in shapes of S(r).

The fact that you could do this in Excel is remarkable. By the

way, I do not have Office 2000 but my older version of Excel

showed your impressive display (and the graph below it)

without creating any problem.

Ludwik Kowalski

"John S. Denker" wrote:

At 06:57 PM 2/5/01 -0500, Ludwik Kowalski wrote:

I could not resist to write a little program .... Monte Carlo approach.

I also couldn't resist writing a little program.

Rather than take the Monte Carlo approach, I took the Finite Element

Modeling approach using a spreadsheet.

http://www.monmouth.com/~jsd/physics/laplace.xls

Overview: I set up a system to solve Laplace's equation, with more-or-less

any boundary conditions you want. My demo has an odd-shaped object in the

middle, with potential 100 volts, inside a moderately large box at 0 volts.

You can easily change any of these conditions. The machine calculates and

displays the potential at each point.

Just to the right of the "potential" grid there is another grid that

calculates and displays the magnitude of the electric field at each point.

Principle of operation: Consider a cross-shaped group of 5 elements

somewhere on the spreadsheet, and label them as

a

b w c

d

Now the discrete approximation to the second derivative in the horizontal

direction is b+c-2w, and in the vertical direction it is a+d-2w. The D=2

Laplacian vanishes if w=(a+b+c+d)/4, i.e. if the central element is equal

to the average of its four neighbors. This leads to an algorithm that says

that for all elements (except for ones that are constrained because they

represent the boundary condition), just set its value to the average of the

four neighbors. After doing this, we have to recalculate the neighbors,

and iterate. For 841 elements (a 29x29 grid) it converges in a few

seconds. That's fast enough that it's not boring, but slow enough that you

can observe the propagation of changes if you fiddle with the boundary

conditions. (Cell V2 at the top of the graph changes the voltage on the

object in my demo.)

Below the potential grid is a graph with many traces; each trace shows the

potential as a function of x; different traces show different y values

(rows). Clicking on one of the traces highlights the corresponding row.

This may help you locate extremal values.

Below the field grid is a similar graph.

Edge effects are nontrivial; if the box were any smaller it would be a

problem.

You can make the box bigger by adding more rows and colums if you like;

use the "fill across" and "fill down" features to propagate the vacuum

formula into the new cells. Make sure you fill FROM a vacuum cell that is

NOT ADJACENT to the newly-added cells or the results will be incorrect.

This is for D=2. You could extend it to D=3 in the obvious way: make 29

different 29x29 grids and put the appropriately-generalized formula in them.

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

Additional details.....

The following assumes Excel version 9 (the one that comes with office 2000).

Suggestion: If you are going to seriously play with this, you will want to

play with the spreadsheet's iteration controls. I've got iteration turned

on, and presumably you want to leave it on; hint:

Tools -> Options -> Calculation -> Iteration.

You might want to delay recalculation if you are making numerous changes to

the grid:

Tools -> Options -> Calculation -> Manual.

which is the opposite of:

Tools -> Options -> Calculation -> Automatic.

And you can invoke the manual "recalculate now" function with the F9 key.

Another small feature uses "Conditional Formatting" to highlight the

contours of potential and field-strength.

Another small feature: You can double-click on a cell, and it will

highlight which cells contribute to the value of that cell. This works

fine for the |field| grid; on the potential grid it is somewhat confusing

because some of the contributor-cells are hidden under the displayed

formula. Clicking puts you into "edit mode" for editing that cell, which is

probably not what you want. Hit "escape" to get out of this mode. It is

less confusing if you format the potential grid as flush-left or

flush-right (not centered) but if you're not using this clicking feature

then centering looks better.

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