RE: capacitance of horned toroid

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

Original poster: "David Thomson" <dave-at-volantis-dot-org> 

Hi Godfrey,

This post became interesting to me when you gave your approximation:

 > With the above, the capacitance can be written as C =
 >
 > 8(Pi)(permittivity)d times the below
 >
 > Sum[a(n)^(-1)BesselJ[1,a(n)]^(-2)(-1+Integrate[BesselJ[0,t],{t,0,a
 > (n)}]),{n,1,Infinity}].
 >
 > Borrowing an asymptotic expansion for the Struve function H(z)
 > and some properties of Bessel functions, I have derived
 > an asymptotic expansion for the terms of the series above
 > where n is large. I'll send the details to your University
 > address after I get them written up in logical order. The
 > expansion is a finite sum of p terms. By increasing p,
 > the accuracy of the expansion increases. But p = 1 gives the
 > dominate term. For p > 1, the terms are very small and would
 > only become valuable if one were going for many significant
 > figures. Using the dominate term only,
 >
 > BesselJ[1,a(n)}^(-1)Integrate[E^(-a(n)Sinh[t]),{t,0,Infinity}]
 >
 > is approximated by (-1)^(n+1) 2^(1/2) (4n-1)^(-1/2).

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

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

 > So the best so far in the MKS system of units is
 >
 > C = 8(permittivity)(1.3677)d.

Shouldn't this be

C = 8(Pi)(permittivity)(1.3677)d?

Dave