RE: Q about E-tesla-6

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

Original poster: "Peter Lawrence by way of Terry Fritz <twftesla-at-qwest-dot-net>" <Peter.Lawrence-at-Sun-dot-com>

Geofrey,
        thanks for taking the time
        
>Because a charge placed on a toroid will not uniformly distribute itself
>over the surface, I have no idea of how to compute the charge density.
>Without an expression for the charge density, the calculation of the
>potential cannot proceed. Anyway I thought it was an interesting question
>and I'll think about it some more.

OK, glad that at least one person besides myself thinks the charge might
not distribute itself uniformly over the toroid.

One possible analogy is that the electrons are a "gas" that are in an enclosed
volume (the sheet metal aluminum of the toroid is the "volume", not the whole
solid toriodal volume), in which case the "pressure" would of course force
a uniform distribution. But the electrons are not a "gas" they exert a pressure
not only to their neighbors but also the inverse-square force across free space
to all other electrons in other relatively remote regions of the toroid.

Clearly the electrons will distribute themselves in such a way as to minimize
the total potential energy of the total repulsive forces between them all,
but whether that results in a uniform or non-uniform distribution is not yet
obvious to me.

Do you understand E-tesla-6 enough to know whether it is the charge
distribution on the TC surfaces that is being calculated when Terry talks of
iteratively converging to the solution, or is it the E-field around the
surfaces and up to the chosen gaussian surface that is being computed,
or both?


Peter Lawrence.



-------------------------------------------------------------------------------
>
>Hi Peter
>
>I am responding to your question regarding an exact formula for the self
>capacitance of a toroid. One would have to center a toroid inside a sphere
>of radius r, and compute an expression for the potential difference between
>the toroid and the sphere. Once such an expression is in hand, one can give
>an expression for the capacitance of the arrangement of the toroid and the
>sphere. Then the self capacitance of a toroid can be obtained by finding the
>limit of the capacitance expression as r approaches positive infinity. 
>
>Using toroidal coordinates, the surface integral to compute the potential
>can be set up in terms of the charge density on a toroid. Once an expression
>for the charge density is in hand, I am betting that the resulting integrals
>will be so horrific that they cannot be evaluated in terms of familiar
>special functions.
>
>Because a charge placed on a toroid will not uniformly distribute itself
>over the surface, I have no idea of how to compute the charge density.
>Without an expression for the charge density, the calculation of the
>potential cannot proceed. Anyway I thought it was an interesting question
>and I'll think about it some more.


>
>Godfrey Loudner  
>
>> -----Original Message-----
>> From:	Tesla list [SMTP:tesla-at-pupman-dot-com]
>> Sent:	Monday, September 10, 2001 8:02 PM
>> To:	tesla-at-pupman-dot-com
>> Subject:	Q about E-tesla-6
>> 
>> Original poster: "Peter Lawrence by way of Terry Fritz
>> <twftesla-at-qwest-dot-net>" <Peter.Lawrence-at-Sun-dot-com>
>> 
>> Terry,
>>       I've started using E-Tesla-6, and have read your short overview
>> about
>> it but I have some questions about its design.
>> 
>> I think I understand how the electric field is computed at any point (the
>> sum
>> of the electric fields from the charge at many individual points for all
>> the
>> TC "parts"), but how is the distribution of charge on a toroid computed -
>> do
>> you assume it is evenly distributed over the entire surface of the toroid
>> plus
>> the center plate, or is there an iterative method to determine the charge 
>> distribution, or is there a known formula?
>> 
>> I am wondering both to understand E-Tesla-6, and because I would be
>> interested
>> in knowing how an "exact" formula for the free-space capacitace of a
>> toroid
>> could be determined, I've seen the derivation in physics books for a
>> sphere,
>> but never anything more complicated.
>> 
>> thanks,
>> Peter Lawrence.
>> 
>> 
>
>
>