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:

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

Yes, you have to compute the charge distribution on the topload,
and it must be computed taking into account the presence and
geometry of nearby conductors and dielectrics.

Once you have the charge distribution, you immediately have
the surface field strength (directly proportional to surface
charge density) and by adding up the total charge you obtain
the capacitance.

Remember, the potential is the same everywhere on the surface,
and the charge density will vary as necessary to achieve this.

The surface field strength is just the surface charge density
divided by epsilon.

The tssp program tcap and the geotc library (as used in JavaTC)
perform these surface charge calculations using the boundary
element method.

In a nutshell the method proceeds as follows:

a) decompose the surface(s) under consideration into lots of
    small tiles, say N tiles, labelled 1..N

b) For each tile k = 1..N, compute the potential coefficient
    Pjk to tile j = 1..N;  The potential coefficient Pjk gives the
    potential at the location of j due to a charge of 1 Coulomb
    on tile k.  Each potential coefficient is just the Coulomb
    expression for the potential at a distance from a point charge.

c) Put all these coefficients into a square matrix [P], defined
    by
    [V] = [P][Q]
    where [V] is a column vector giving the potentials of the N
    tiles, and [Q] is a column vector giving the charge of each
    tile.

d) Set [V] to the known voltage distribution, ie for tiles which
    are part of the the topload it will be the topvolts value and
    for the groundplane tiles it will be zero.

e) We know [V] and [P], so solve the matrix equation for [Q].
    You then know the charge distribution [Q] which produces the
    specified potential distribution [V].

There's a little bit more to it than that, eg you have to allow
for the self-potential Pjj of each tile j, but essentially you
have N*N Coulomb coefficients followed by a matrix equation,
ie N simultaneous equations in N unknowns.

Once you have the charge distribution it is easy to compute
the surrounding E-field, if you need it.  For each point P in
the surroundings, simply sum up the Coulomb potentials due to
each of the N tiles.

By a little more work you can completely invert the potential
matrix [P] to produce the capacitance matrix [C], where Cjk gives
the charge induced on tile j when tile k is at a potential
of 1 volt.  You then have part of the information you need to
compute the actual voltage and current distributions.
--
Paul Nicholson
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