# Re: Sphere/Toroid Comparison Chart

```Original poster: "Jim Lux by way of Terry Fritz <twftesla-at-uswest-dot-net>" <jimlux-at-earthlink-dot-net>

The charge distribution should be independent of voltage (excepting effects
like space charge or corona).  They are all linear equations so
superposition should hold.

Where it all breaks down (pun intended, or not), is when the field gets high
enough to ionize (break down) the air, and that is, most definitely, a
nonlinear process.

The electrostatic equations are actually quite simple. It's the analytic
solutions for an arbitrary geometry that are complex (cf. analytic
expression for E-field between two spheres).  The whole beauty of Finite
Element Methods (FEM) is that if you break your problem up into sufficiently
small parts, you can essentially do numerical integration and get an
approximate answer that is so close to the real thing that it's not worth

The nifty thing these days, with piles of CPU, is that you can run the
numerical integration multiple times, get a whole raft of data points, and
then come up with a relatively simple empirically derived equation.  Back in
the 20's, Medhurst, et al, had to build a pile of capacitors, measure them
with excruciating precision, etc. Now, you can build those caps (or
inductors) on a computer, FEM them once, and be done with it.

----- Original Message -----
From: "Tesla list" <tesla-at-pupman-dot-com>
To: <tesla-at-pupman-dot-com>
Sent: Wednesday, May 09, 2001 7:51 PM
Subject: Re: Sphere/Toroid Comparison Chart

> Original poster: "Luc by way of Terry Fritz <twftesla-at-uswest-dot-net>"
<ludev-at-videotron.ca>
>
> Hi Matt, Kurt, all
>
> Tx guy, both of your answer are logical, I know that
> electrostatic equations are not particularly easy .... To have a
> final word on that we probably need measurement and if Matt
> theory is good; measurement at high voltage ( not sure about that
> but I think the repulsive force could be higher at higher voltage
> pushing the charge farther on the exterior ???).
>
> Cheers,
>
> Luc Benard
>
> P.S. I need to try to understand the electrostatic equations ;-)
>
> Tesla list wrote:

```