RE: Spheres on toroids

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

Hello Antonio

Yes, t is the theta of toroidal coordinates as in Moon and Spencer.
Toroidal coordinates are (eta, theta, psi). Now fix eta, then theta
and psi will graph a toroid. Let C = (x, y, z) be a point on the toroid,
a be as in Moon and Spencer, and k = (a^2 + (d/2)^2)^(1/2). Now we
make a list of points.

A = (aCos[psi], aSin[psi], 0],

B = (-aCos[psi], aSin[psi], 0),

D = (kCos[psi], kSin[psi], 0],

E = (x, y, 0).

Theta is the angle between the lines CA and CB. This angle is not so
natural to use, but it makes the equations nice. Its more natural to
use the angle alpha between the lines DC and DE. Now Cos[alpha] is
equal to the product of the two expressions below:

sign{d-(D-d)Cos[theta]} and

{d-(D-d)Cos[theta]}/{D-d-dCos[theta]}.

The sign function above is 1 or -1.

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.

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

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.

Godfrey Loudner





 >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