Re: Charge distribution on a Toroid (was spheres vs toroids)

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

Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk> 

Gerry wrote:

 > The voltage at point (p) would be:

 >     V(p) = sum{for all patches[i]; P[i] * q[i] * Q}

 > where  P[i] = 1/(4pi*epsilon*R[p,i])
 >        R[i] = distance between point (p) and the point charge of
 >               patch[i]
 >        q[i] = fraction of total charge Q represented by patch[i]

That's exactly right, with the proviso that if p coincides with
one of the patches, you replace that term in the sum with the
self potential of the offending patch.

 > I'm thinking that there is an added complication in that the
 > patches included in the summation are only those visible to
 > the point (p) implying determination of the shadowing affects
 > of the toroid.

No, all patches, all charges, are included, even those that are
'in the shadow' of some other conductor, and even those that are
totally enclosed by a conductor.  Coulomb's law applies throughout
all space, inside and through conductors, inside and through
dielectrics.

So how does the 'shielding effect' come about?  Charges in the
shield are redistributed by the Coulomb field from the source
charge, and the Coulomb field resulting from this new shield
charge distribution acts to produce the shielding you are thinking
of, by partially cancelling the source Coulomb potential.

So include all the Coulomb terms, and those that physically
will cancel one another out, will automatically do so.

 > The redistribution algorithm would also need to account for the
 > shadowing affects of the toroid shape.

Probably not.  As above, the shielding effect should emerge as
from your calculations as a 'phenomena predicted' and is not
something you have to put in - in a sense you're working at a level
deeper than the shielding effect.
--
Paul Nicholson
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