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> 


Hi Paul,

 > Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk>
 >
 > I hope that Gerry is also following this thread, because I know he
 > is interested in getting to grips with some modelling too...

Yes, Im following the thread with much fascination.  As I think I mentioned
to Antonio (privately), Im taking a totally numerical approach to the
modeling and I can already see some simularities of what Im planning to do
with what you are talking about (using matricies, computing self potentials,
evaluating eliptical integrals, etc).  My sense of what you are doing is to
approximate the toroid with a composite of tubes, tapes, rings, etc. and
apply some exact math where ever it makes sense.

Im planning on decomposing the toroid into tiny surface patches described in
(R, phi) coordinates for major dimensions and (r, theta) coordinates for
minor dimensions, and assigning a weighted charge to each tiny patch so the
charge density approximation will initially be constant over the entire
surface and the total toroid charge will be Q.  Each patch will be
represented by a point charge at the centroid of the patch.  Im planning on
a two parts in the modeling.  First, I need to redistribute the charge in
theta (taking advantage of symmetry) until the summation of all forces on
each patch charge is zero tangental in minor curvature.  Second, I need to
compute the integral (from infinity to the toroid surface)(E.dl).  The
patches will be oriented so the integral will not terminate on a point
charge (to avoid singularities).  The intermediate results of the E field
calculations will be stored so the field gradient can later be determined.
The total voltage calculated from the integral will be proportional to the
charge Q initially assigned to the toroid.  The capacitance can then be
calculated as Q/V.  The integral will be evaluated only in the horizontal
plane which should coincide with the point of maxmum gradient for a toroid
in freespace with no other proximaty effects (such as with a secondary coil
present).

I hope this will be sufficient for my purposes in determining the "reach"
various toroid sizes have.  Accuracy will be mostly a function of patch
size, Im thinking, and it will be interesting to compare the results with
those from your more accurate models.  Hopefully, I can determine how small
a patch needs to be for sufficient accuracy.  Anyway, this will be my first
modeling attempt and will be a vehicle for learning some of the pitfalls.  I
will write the program in C and then figure out how to compile it (I don't
have a C compiler on my windows98 PC).  I may need to get my hands on a unix
machine for this purpose.

Gerry R
Ft. Collins, CO