Re: Spherical vs. toroidal top loads on tube coils

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

Original poster: "Antonio Carlos M. de Queiroz by way of Terry Fritz <twftesla-at-qwest-dot-net>" <acmq-at-compuland-dot-com.br>

Tesla list wrote:

> Original poster: "Ed Phillips by way of Terry Fritz <twftesla-at-qwest-dot-net>"
<evp-at-pacbell-dot-net>

>         Here are some capacitance and inductance data which I have found
quite
> useful; the capacitance of the sphere is exact from first principles,
> the others are approximations..  Note that the capacitance (of a toroid
> is somewhere between that of a disk and that of a sphere of the same
> diameter. 

A very useful post. Another thing that is interesting is that the 
capacitance of a toroid made with thin tube is about 70% of the 
capacitance of a sphere with the same diameter. A ring of thick wire has 
70% of the capacitance of a sphere of the same diameter, and is much 
easier to make. This finds applications in antenna design, and
explains the function of the rings, known as "Winter rings" that were
used with old electrostatic friction machines for the production
of long sparks.
A pair of crossed rings is almost equivalent to a sphere:
http://www.coe.ufrj.br/~acmq/winter.jpg

Of course, a sphere results in maximum breakout voltage. Do someone
know a formula (or a table) showing how the breakout voltage of a
toroid compares with the one of a sphere? 
Maybe something as: 

Vtoroid=Vsphere*1.4*d1/d2

This assumes (d1=small diameter) < (d2=large diameter)/2.
I come to this as follows:
Surface charge density in a sphere: 
ps=Csphere*Vsphere/(pi*d2^2)
Maximum surface charge density in a toroid:
pt=Ctoroid*Vtoroid/(pi*d1*pi*d2)*k
where it is assumed d1<<d2, and k accounts for an effective fraction
of the toroid surface, where an uniform charge distribution would
produce the correct maximum surface charge density.
Making ps=pt, and assuming for convenience k~=1/pi (1/3.14 of the area):
Vtoroid=Vsphere*Csphere/Ctoroid*d1/d2=Vsphere*1.4*d1/d2.
 
Antonio Carlos M. de Queiroz