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>
 >
 > 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.

I have just tested your expression. It gives the correct value at the
major diameter, with t=0. Simpler than my formula.
The formula looks as the formula shown in Hicks' paper when he
calculated
the ratio between the minimum and the maximum fields.
But there is something wrong at other angles.
Or maybe I am not using the right t. Is t the angle of the toroidal
coordinates? With t=Pi, that I think that corresponds to the minimum
radius of the toroid, the formula gives a larger value than at the
maximum radius, and close to that angle I am getting negative values.

Antonio Carlos M. de Queiroz