RE: Spheres vs Toroids

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

Hello Antonio

The messy part of the exact formula is
f(x) = Sum[Q(n-1/2,x)/P(n-1/2,x), {n, 1, infinity}]
where x = (D-d)/d.
Set f(x,k) = Sum[Q(n-1/2,x)/P(n-1/2,x), {n, 1, k}], i.e.,
the sum of the first k terms. By playing around with integral
inequalities, one can show the below.

absolutevalue[f(x)-f(x,k)] < e whenever x > 1 and

k > ln[Pi(2x)^(1/2)/e(x-1)(x^2-1)^(1/4)]/ln[x].

This can be slightly improved if one admits having
Q(-1/2,x) into the estimate. The problem with my
estimates is that they may over shoot the number of
terms necessary to achieve a given accuracy. I don't
know how optimal the estimates are.

Let's take your D = 90cm and d = 30cm, then x = 2.
Now with e = 1/10^m with accuracy to m places in
mind.

place value accuracy     number of terms
          1                 6 or more
          2                 9 or more
          3                13 or more
          4                16 or more
          5                19 or more
          6                23 or more

 >From my notes, Mathematica gives 40.56477807 pf

However with x very close to 1 from above, my estimate
indicates that an astronomical number of terms might be
required. If D = 2d, we have a torus with no hole and x = 1.
I have tried to compute the limit of the formula as
x approaches 1 from above, but without success so far.

Godfrey Loudner







Example:
Major diameter: 90 cm
Minor diameter: 30 cm
I get: Exact capacitance: 40.5648 pF (all the digits correct)
The document lists:       40.46   pF

There is also a table in the documentation of the program Mandk,
but that table appears to use an approximate formula, the same
listed in Gary's paper.

Antonio Carlos M. de Queiroz