RE: capacitance of horned toroid

["Tesla list" <tesla-at-pupman-dot-com>]

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

Hello Dave

My comments are interspersed below.

 >It is curious that you use "(-1)^(n+1)", which will always be equal to -1.
 >Why is that?

Arranged in ascending order, the positive zeros a(n) of BesselJ[0,x] are
indexed by n. I'm summing over n = 1, 2, 3, 4, ... So (-1)^(n+1)
alternates in sign.

 >How did you come up with "(4n-1)^(-1/2)"?

I cannot show here all the details of the derivation of the asymptotic
expansion, but I can give some insight. For very large n, a(n) is
closely approximated by Pi(4n-1)/4. Upon substitution into the
asymptotic expansion, the (4n-1)^(-1/2) shows up. I use this
approximation for a(n) starting at n = 3001 to Infinity. I'll
scan and sent my derivation notes to your email after I get
them organized.

 >Shouldn't this be
 >C = 8(Pi)(permittivity)(1.3677)d?

I had Mathematica multiply Pi in during the calculations. Maybe I
should not do this. Three thousand multiplications by Pi might
be reducing accuracy during the Mathematica session. I really want
a more accurate value than 1.3677. Mathematica seems to have an
irresistible compulsion to fall back to a fixed machine accuracy.
I'll have a large table of very accurate numbers just calculated
and appearing on the screen. When I ask Mathematica to do something
with those numbers, all the numbers all get rounded off for the next
phase. I'm going to ask the Mathematica people for instructions on
how to get around this difficulty. Strangly enough, I never did
like computers before, but they have opened a new world to me.

 >Dave

Godfrey Loudner