Re: Capacitance of horned torus

["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
 >
 > 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.

The main problem with that series is how the function
Q is calculated. Using the same recursion for P accumulates
increasing error, and when Q gets very small the recursion may
generate negative values. After this, the recursion is totally
wrong.
To evaluate correctly that series with many terms, Q must
be calculated by some other method. A possibility is to start
the series from the last term considered, and use the recursion
to calculate the lower terms. Q can be calculated from
a mathematical nightmare called "hypergeometric functions".

But a better series for this case must be possible.

 > 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.

But a series with integrals? Something simpler must exist.

 > 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.

Verify if it's not again that problem with imprecise e0.
I am using the value derived from the speed of light
c=299792458.0, e0=8.85418781762038985E-12.

 > 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.

That paper is difficult, but has references.

 > 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.

Look at Maxwell's book. It has solutions for several problems with
two intersecting spheres, and they are simpler than the solutions
for nonintersecting spheres. This may happen in this case too.

Antonio Carlos M. de Queiroz