Charge distribution on a Toroid (was spheres vs toroids)

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

Original poster: "Gerry Reynolds" <gerryreynolds-at-earthlink-dot-net> 

To all,

I posted a question about whether or not it makes any difference on what 
the charge distribution on a toroid is for purposes of calculating its 
capacitance using a common method of putting a charge (Q) on the toroid and 
calculating the voltage that it would result in.

C = Q / V

V = -integral (infinity, the toroid surface) (E.dl)

{the work per unit charge to move the charge from infinity to the charged 
toroid surface}

I almost talked myself into thinking the charge distribution doesn't 
matter.  The more I think about it, the more I believe it does 
matter.  This is my rational.  If the charge was concentrated at a point on 
the surface,  it would have the greatest repelling force on the unit charge 
that was being brought from infinity to the toroid surface.  If the charge 
were to spread out, the repelling force would not be as great and it would 
take less energy to bring this unit charge to the surface.

Also, if didn't make any difference, we could move the charge distribution 
at will and not effect the surface voltage.  If we were to do this, 
however, the toroid surface would not be equipotential since it would take 
work to redistribute the charge from its natural seeking distribution.

I'm now thinking that one needs to compute an accurate charge distribution 
in order to calculate the capacitance of the toroid.

My purpose ultimately is NOT to calculate the toroid capacitance but to 
calculate the field gradient as a function of: horizontal distance from, 
and the size parameters of the toroid since the method involves calculating 
the field strength as a function of distance from the toroid and would a 
common requirement for calculating capacitance.

Paul Nicholson is trying to calculate the toroid capacitance precisely for 
purposes of establishing a reference for evaluating finite element 
approximations of the capacitance.  I believe he is also wanting to find 
the field gradient for toroids of different sizes and ultimately evaluate 
other shapes for toploads.

I would like to try a finite element approximation of the field gradient 
for purposes of determining an optimum size for streamer growth.  Precision 
would not necessary be one of my top objectives, since I would probably be 
modeling the toroid without any proximity affects from the secondary.  The 
results of the calculation would mostly be used for comparative purposes in 
choosing a toroid size.

I would appreciate any comments or feedback on this proposed approach.

Thanks in advance for any input,

Gerry R.
Ft Collins, CO