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Re: capacitance of horned toroid



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

Tesla list wrote:
 >
 > Original poster: Ed Phillips <evp-at-pacbell-dot-net>

 >... If
 > I had access to a high performance computer and could
 > write code, I would calculated to ten significant
 > figures and end the matter in my mind tonight.
 >
 > Godfrey Loudner"
 >
 >         Good point.  Something doesn't have to have useful results in 
order to
 > be intellectually stimulating!  More decimal places please.
 >
 > Ed

I streamlined a bit my code for toroid capacitance and electric field,
removing some limitations. Now for the horned toroid, using the normal
series expressions for the "normal" toroid, I get the following
capacitances and maximum electric fields:
Using e0=8.85418781762038985E-0012
1x0.4999:       48.43687950554023140 pF, 2.65268461818073342 V/m/V
1x0.49999:      48.43849207477136340 pF, 2.65247794889824492 V/m/V
1x0.499999:     48.43865332230567610 pF, 2.65245728525155197 V/m/V
1x0.4999999:    48.43866944672453820 pF, 2.65245521891770083 V/m/V
1x0.49999999:   48.43867105943574900 pF, 2.65245501227832231 V/m/V
1x0.499999999:  48.43867121362083140 pF, 2.65245499159428584 V/m/V
1x0.4999999999: 48.43867118685169140 pF, 2.65245498946257243 V/m/V
Note the numerical error showing up in the last case. I stopped
adding terms to the series when the the terms were 1e14 times smaller
than
the sum (1e15 for the electric field). In the last case, 395704 terms
were used for the capacitance and 914724 for the electric field.
(For normal toroids at most 10-20 terms are needed.)
For comparison, a simulation with 200 rings gives:
C=48.4386640438 pF, Emax=2.6523918581 V/m/V

I would expect a simple expression for the capacitance of this toroid.
For the quite similar problem of two spheres in contact, the capacitance
is just 4*pi*e0*d*Ln(2)=4*pi*e0*d*0.69314718, where d is the diameter of
the spheres.
For a horned toroid with minor diameter d, it is something as:
C=4*pi*e0*d*0.87069013
87% of the capacitance of a sphere with the same major diameter.
Would this number be familiar to someone?

Antonio Carlos M. de Queiroz