RE: Spheres on toroids

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

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

Hello Antonio

I took a look at the electric field vector at the SURFACE of
a toroid. The magnitude of the electric field vector seems to
simplify to the expression below, where V is the assumed
potential on the surface, x = (D - d)/d > 1, and t is the theta
angle. The magnitude is given by the product of the two expressions below.

2[2^(1/2)]V[x - cos[t]]^(3/2)Pi^(-1)d^(-1)[x^2 -1]^(-1)

and the absolute value of

LegendreP[-1/2,x]^(-1)+ 2Sum[cos(nt)LegendreP[n-1/2,x]^(-1),{n,1,infinity}].

I fully believe the Legendre expression above is always positive, so the
absolute value command may not be required (I'm trying to rig a proof).

The expression for the magnitude might be in error. The simplification is
so messy, but I don't see an error yet. Maybe you have the means of
quickly testing it with numbers. If it works, at least you have only
one series to deal with numerically.

Godfrey Loudner

 >The exact solutions for potential and electric field for
 >a toroid are somewhat badly conditioned numerically. The precision that
 >can be obtained is not as good as with capacitances, specially in the
 >case of the surface field.

 >Antonio Carlos M. de Queiroz