Re: Spheres vs Toroids

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Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> 

Hi:

I have just finished a routine that calculates exact capacitances
of toroids in free space. The routine uses that series with
Legendre functions, that is not after all so difficult to evaluate
with precision. The values that I am obtaining, however, are a bit
different from the ones that I see in the document prepared by
Gary L. Johnson.
The formula looks the same, but I am evaluating the functions in a
different way, using elliptic integrals, that should be almost
exact.

 >From the paper:
Major diameter  Minor diameter  C         My C
90 cm           30 cm          40.46 pF   40.5648 pF
60 cm           20 cm          26.97 pF   27.0432 pF
30 cm           10 cm          13.49 pF   13.5216 pF
56 cm           16 cm          24.55 pF   24.6005 pF
52 cm           12 cm          22.02 pF   22.0615 pF
48 cm            8 cm          19.28 pF   19.3223 pF
44 cm            4 cm          16.00 pF   16.0522 pF
42 cm            2 cm          13.72 pF   13.7969 pF
41 cm            1 cm          12.00 pF   12.1214 pF
40.5 cm          0.5 cm        10.62 pF   10.8164 pF
40.2 cm          0.2 cm         9.09 pF    9.47161 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.

 >From the example in the Fantc program:
Major diameter  Minor diameter    C            My C
50 cm           10 cm             20.73804 pF  20.7282 pF

Of course, most of the difference is insignificant for practical
purposes, but I want the calculations correct. Maybe the difference
is just e0, that I used as 8.85e-12. Only 3 digits are significant
in the calculations.

The calculation was implemented in the Inca program:
http://www.coe.ufrj.br/~acmq/programs

Antonio Carlos M. de Queiroz