Re: 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> 


 > Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk>
 >
 > Gerry wrote:
 >  > I assumed a charge and need to determine the voltage the charge
 >  > corresponds to in order to scale.
 >
 >  > The E field will need to be computed from Coulombs Law for each
 >  > point on the path of E.dl,
 >
 > Ok, so the steps are:
 > a) Assume an arbitrary total charge Q on the topload, with an
 >     arbitrary initial distribution.
 > b) By some process, redistribute the charge so that the potential is
 >     the same at all points on the surface.
 > c) Compute the surrounding E-field field, as a function of Q.
 > d) Integrate E along a path from infinity to the topload to get
 >     the topvolts as a function of Q, and thus the capacitance.
 >
 > I think that steps (c) and (d) are the reverse of one another, and
 > that you can expect the topvolts potential to pop out at step (b).
 >
 > Let's say that you have arrived in step (b) at a charge distribution
 > that gives a uniform potential to all the tiles of the topload.
 >
 > So you now have a list of fractional tile charges, ie against each
 > tile 'j' you know the what fraction q[j] (range 0 to 1) of the total
 > charge it holds.
 >
 > In other words, you have the charge distribution as a function of the
 > total charge.  Now just consider any one of the tile charges and
 > note that the potential at its location is the sum of the Coulomb
 > potentials due to all the other tile charges, plus its own self-
 > potential.  This tile is at the same potential as the rest of the top-
 > load, so we can go on to say, where 'i' labels any 'test' charge:
 >
 >   V_topload = self_potential_coefficient[i] * q[i] * Q +
 >               sum{ for all j != i; q[j] * coulomb_coefficient[i,j]} * Q
 >
 > Thus, since you know all the coefficients and all the q[i],
 > you can immediately calculate the constant P in the equation
 >
 >     V_topload = P * Q
 >
 > and so the total topload capacitance is then just 1/P.
 >
 > Thus, really, all the work will be done by your charge re-distribution
 > algorithm in step (b).   Then you'll run step (c), not to determine
 > topvolts, but maybe to assess the likely streamer paths.
 > --
 > Paul Nicholson

OK, I think I figured out what you're saying, similar to an application of
the potential function summed for each of the surface point charges, each
evaluated at the point that the voltage is being evaluated.  Agreed much
easier than what I was thinking.  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]

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.  The redistribution
algorithm would also need to account for the shadowing affects of the toroid
shape.  Would this be correct??

Gerry R
Ft Collins, CO