Re: Spheres on toroids

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> 

Tesla list wrote:

 > Original poster: "Godfrey Loudner" <ggreen-at-gwtc-dot-net>

(returning to the subject after some time)

 > If your getting negative values, then that absolute value in the expression
 > for the magnitude of vector E at the surface will be necessary. Moon and
 > Spencer seems to treat the partial derivative of the potential with
 > respect to eta at the surface as always having non-negative values. Maybe
 > Moon and Spencer does not care about the sign---just something to fix up
 > on a physical intuition basis in an application. I was getting some
 > negative values at 100 terms, but using up to 200 terms of the series
 > made the result postive.

Your formula for the surface field, without the Q Legendre function, is
very  stable, and converges as well as the capacitance formula. The
formula that I had derived (that should simplify to yours at the
surface,
but I didn't yet try to figure out how) has problems because the
recursion for Q accumulates error quickly and eventually produces
numerical garbage. I am using your formula in the Inca program now, and
the maximum surface field can be calculated with full precision.

 > I have not looked much at Hick's paper, which is hard to follow. Some of
 > it does not make any sense to me.

The notation is irregular and difficult to follow, and there are some
strange things. The formula for the toroid capacitance, for example,
looks as the modern form with e0=1. There is a "c" in the formulas that
is cos v, although I don't see where this is said in the paper. The
notation for the P and Q functions are also different.

 > I think I'll do the simplification again to see if it comes out the same
 > as before. This time I'll use big sheets of art paper so I can write
 > big letters. Sorry if I wasted your time with something which might be
 > incorrect. But in the final part of the simplification, a mass of terms
 > cancel all at once. This simply does not happen unless your on the
 > right track.

I will see if I can simplify my formula, eliminating Q.

A new version of the Inca program is available at:
http://www.coe.ufrj.br/~acmq/programs
The electric field formulas were updated, and some minor changes were
made.

Antonio Carlos M. de Queiroz