RE: Capacitance of horned torus

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

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

Hello Antonio

In the formula for a toroid with a hole in terms of
x = (D-d)/d, I took x = 1.0000000001 and the
Legendre type series came out to be 1.36744 for
400,000 terms and 1.36765 for 500,000 terms. Beyond
that my Mathematica stopped working and asked for
more memory. I have since added a lot more memory,
but have not tried to continue the calculation. The
sum of the series looked like it was still growing.
Whether the Mathematica calculation was reliable or
not, I have no idea.

Now using the exact formula for the capacitance with
no hole, Pi times the Bessel Type series becomes
1.34739 at 3000 terms with the thousandth place
value still growing. I've been sectioning the calculation
to 100 terms at a time. I just don't trust Mathematica
to do thousands of terms at one sweep. I stopped here,
hoping that those NASA people would do the calculation.
A senior scientist at NASA said that they would
review my request. But its looking like the value of
Pi times the Bessel type series is somewhat above
1.36744. A very accurate value would allow one to
test the accuracy of Inca's method of rings as the
hole shrinks to a point.

For your d = 0.4999, I get C = 48.42803 pF, using
the 1.36765 above. Well its somewhat close to your
value of 48.43867 pF. So 1.36765 must be getting
close to its true value.

I took a look at Hick's paper and I see that his
expression for the potential involves Bessel
functions, but I have not yet studied the pages.
But I am struck by his not appearing to use
tangent-sphere coordinates. Transforming Laplace's
equation to tangent-sphere coordinates was
indeed a horrible mess---my mind felt dulled after
doing it. Hick's appears to be using cylindrical
coordinates. While looking today at the book by Moon and
Spencer, I noticed the topic of
inversion coordinates, where it seemed like the
inversion of cylindrical coordinates was looking
like tangent-sphere coordinates (I'll have to check
that out in detail). Guess I'm going to study
that section of Hick's paper. Thinking wildly, the
method of images use an inverse distance. What if
the inverse distance thing takes a toroid with no
hole to a cylinder, then the capacitance problem
might also be solved by the method of images.
Maybe there is an inverse distance thing in
inversion coordinates. All this seems to close
to overlook. I got some studying to do.

I will try to solve the capacitance problem for
a spindle torus at http://mathworld.wolfram-dot-com/Torus.html.
If the two circles there are brought together into one
circle, then you have a sphere. Then you would have exact
formulas for a the entire transverse from torus ring,
horned torus, spindle torus, and sphere. These exact formulas
could be embedded in Inca, and future developments with your
ring method would be tested across the entire transverse.

Godfrey Loudner

 >So, 48.43867 pF is the best that I can calculate.

 >Best regards,
 >Antonio Carlos M. de Queiroz