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Re: visualizing fields near charged objects

At 8:52 AM -0500 2/8/01, John Denker wrote:
At 04:33 PM 2/7/01 -0800, Leigh Palmer wrote in part:
I have set up a relaxation calculation in Excel based on the Laplace
equation in cylindrical coordinates.
The charge density on the surface of the disc must now be inferred by
looking at the difference between numbers at the surface. The electric
field intensity is proportional to the surface charge density, and the
gradient of the potential field is the relevant parameter here.

One could argue that the Laplacian is even more relevant than the
gradient. It's also a whole lot easier to calculate on a spreadsheet,
since it's a scalar.

I calculated the potential itself, as you did. The Laplacian is
identically zero everywhere in free space. The surface charge
density itself is proportional to the magnitude of the gradient
of the potential just outside the surface

>When I have a converged result I can make another spreadsheet (not
which will show the charge density on the surface. It won't be great
right on the sharp edges

The charge calculation should be just as accurate as the potential
calculation for any shape you can represent.

That is correct for smooth surfaces, but when one gets to sharp
edges (corners in two dimensions) it breaks down (unconscious

The formula for the Laplacian
that gives the charge density in this second step is essentially the same
formula used in the relaxation algorithm in the previous step, so at the
very least there should be excellent consistency. Checking to see that
overall charge is conserved is a good diagnostic. See
for more on this.

Since I set potential boundary conditions in my technique charge
is not conserved. I don't see how you accommodated to azimuthal
symmetry in your relaxation calculation, John. It looks like you
have used the rectangular form of the Laplacian everywhere, even
on and near the axis. That will surely lead to funniness. Why do
you not use the cylindrical Laplacian? The calculation is still
two dimensional when you do so.

I see your comment on "conditional formatting", a term I'd never
seen before. I'm really creaky on Excel, having progressed from
v.1.0 -> v.3.0 -> Excel 98 without having really ever read a
manual*. I will backslide and learn about conditional formatting.
Yesterday I found a zoom function in one of the menus. I can
zoom out to 25% and see the some of the equipotentials quite
easily, directly in the page as displayed on my 17" monitor.
Printing the page makes them even clearer.


* In our religion (orthodox Macish) we are taught that reading
manuals is sinful, thus most Mac users are illiterate, preferring
the Columbus approach (called by us physicists "suck it and see")
to using a new application.