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# Re: visualizing fields near charged objects

On Fri, 9 Feb 2001, John S. Denker wrote

At 08:12 AM 2/9/01 -0500, Ludwik Kowalski wrote:

1) Is it appropriate to perform simple averaging for usual
cylindrical cells (dr=const)?

No.

and went on to give the essential theory. To put it into practice
here's what I did (and, I assume, what others are doing as well.)

The key difference is the first derivative term in the cylindrical
Laplacian which involves r itself. This means that we need a
radial coordinate which can be simply generated as an index along
the side of the "universe."

27 ^ (+ r direction)
26 |
25 c
24 d a b -> (+z direction
23 e
22

The derivatives at the position of cell a are then numerically
implemented as follows:

Laplacian (V) = (c+e-2a) + (c-e)/(2*r_a) + (b+d-2a)

which corresponds respectively to the terms in John D's

... the Laplacian is ...
(d/dr)^2 + (1/r) (d/dr) + (d/dz)^2.

where r_a is the radial index for cell a. Setting the Laplacian =
0 and solving for a we get

a = (b+c+d+e)/4 + (c-e)/(2*r_a)

The only difference from the usual Cartesian implementation is the
last term which becomes progressively less importan (as one should
expect) as r gets bigger.

Once the calculation is finished, finding the Laplacian for those
cells whose value was specified as a part of the boundary
condition gives the charge density (times some appropriate
constant) in those cells.

John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm