Re: visualizing fields near charged objects

[John Denker <jsd@MONMOUTH.COM>]

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.

> When I have a converged result I can make another spreadsheet (not iterative)
> 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.  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
  http://www.monmouth.com/~jsd/physics/laplace.html
for more on this.