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
 >
 > A horned torus is a toroid with no hole. If one
 > tries to get the capacitance of a horned toroid
 > with toroidal coordinates, the essential calculations
 > collapse to 0 = 0. After looking around the internet,
 > I came across a list of strange coordinate system.
 > Tangent-sphere coordinates looked very interesting.
 > After trying the new coordinates, the solution to the
 > boundary value problem just did not look right.
 > After a long time, I finally realized that I was
 > missing a differential boundary condition (good
 > at math, but low on physical sense). With the
 > additional boundary condition, the rest of the
 > calculation was walk in the park. With d for the
 > inside diameter of a horned torus, the exact
 > expression for the capacitance is below.
 >
 > Let a(n) be the ascending sequence of the positive zeros
 > of the zero order Bessel function J[0,x]. Then C =
 >
 > 8Pi(permittivity)d times the below
 >
 > Sum[BesselJ[1,a(n)]^(-1)Integrate[E^(-a(n)Sinh[t]),{t,0,Infinity}],{n,1,Infi
 > nity}].
 >
 > BesselJ[0,x] and BesselJ[1,x] are the zero and first order
 > Bessel function respectively.
 >
 > The problem is computing the infinite series, but its a one
 > time calculation. My Mathematica can calculate the zeros of
 > Bessel J[0,x] and the integrals to 30 significant figures, but
 > I'm still having trouble getting the value. I have contacted
 > NASA, and they might run the calculation as a courtesy to my
 > school. I don't know yet if they will do the arithmetic. I've
 > asked them to calculate Pi times the infinite series to 20
 > significant figures.

There is an integral in the capacitance? Maybe you can expand the
integrand
in a series and integrate the series.

In Hick's difficult paper there is a section about a torus without a
hole.
He lists a solution for the potential as a series in Bessel functions.

The formulas for a toroid with hole get lost in this case, but still can
calculate the values, specially the capacitance, with great accuracy.
The best value that I can get with less than 1000 terms in the series
for a
1x0.5 toroid is:
1x0.49986 -> 48.43616275361470810 pF (990 terms)
But the value is not more precise than the smaller diameter:
1x0.49985 -> 48.43598356039409650 pF (957 terms)
1x0.49984 -> 48.43580436507744290 pF (928 terms)
I should check if the recursion for Q is not producing garbage with so
many terms.
The best value appears to be for 1x0.4999, ading 1001 terms:
48.43687950554023140 pF
After this the terms don't decrease, but increase due to the failure
of the recursion for Q.
Using decomposition in rings:
20 rings:  48.4312295893 pF 1x0.49986 ->48.4287234030 pF
50 rings:  48.4382016634 pF 1x0.49986 ->48.4356933175 pF
100 rings: 48.4386131163 pF 1x0.49986 ->48.4361046481 pF
200 rings: 48.4386640438 pF 1x0.49986 ->48.4361555608 pF

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

Best regards,

Antonio Carlos M. de Queiroz