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>

A correction and some updates to my previous post:

 > 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}].

My code had an error. Now I am getting the right values all around the
surface of the toroid. Your expression is correct.
Your expression agrees with a formula that I have implemented in the
Inca
program, in the window "partial toroid capacitance". I had not noticed
that
it is valid at all the surface of the toroid (looked only at the maximum
and minimum radii) My expression calculates the
expression -dP/dB dB/dr, where P is the potential, B=(D-d)/d, and r is
the radial distance. It agrees with yours only at the surface.
I still find strange that my formula works...

Example: For a 0.9x0.3 m toroid:
       My formula         Your formula
t=0:  3.5342114604 V/m/V 3.5342112740 V/m/V
t=Pi: 0.3267284112 V/m/V 0.3267282829 V/m/V
top:  2.4412080294 V/m/V 2.4412081814 V/m/V

For the precision used, the results are the same.

It's good to eliminate the Q function from the calculations, since the
recursion for it becomes numerically unstable when the argument is >1,
as in this case.

Antonio Carlos M. de Queiroz