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

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

Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> 

Tesla list wrote:

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

 > How about looking at mutual capacitance between two objects?  You'll
 > have to tag the rings to remember which electrode they belong to, then
 > sum the charges separately for each object.
 >
 > An easy one is two concentric spheres, say radius 0.5m and 0.3m,
 > C = 4 * pi * epsilon Ra * Rb / (Rb - Ra) = 83.448756 pF

I have just completed a program (named "cacal") that computes the
capacitance matrix of a set of objects composed of lines and curves,
with cylindrical symmetry. The algorithm uses decomposition in thin
rings. Still needs some polishing, but appears to work well. For two
concentrical spheres with radii 0.3 and 0.5 m:

  rings      tssp        cacal
    10      80.341 pF    83.3801277244 pF
    20      81.887 pF    83.4395579512 pF
    40      82.643 pF    83.4475644027 pF
    80      83.045 pF    83.4486029642 pF
   200      83.289 pF    83.4487445528 pF

 > In these figures, each object is given the specified number of rings.
 >
 > If neither object encloses the other, we must be specific about
 > which capacitance we are measuring...
 >
 > For two discs, 1 metre diameter, spaced 10cm apart, I get

  rings   tssp total    tssp mutual  cacal total      cacal mutual
    10      94.596 pF     75.263 pF  98.1496072537 pF 78.6827468834 pF
    20      97.845 pF     78.245 pF  99.8002154701 pF 80.1207334151 pF
    40      99.633 pF     79.896 pF 100.6788391600 pF 80.8969570508 pF
    80     100.604 pF     80.796 pF 101.1321593111 pF 81.3000670740 pF
   200     101.572 pF     81.699 pF 101.4087787914 pF 81.5468815768 pF

 > The capacitances are obtained by putting 1 volt on one of the objects,
 > with the other(s) fixed at zero volts.

The diagonal terms of the capacitance matrix.

 > The 'mutual' capacitance is
 > obtained by determining the charge induced on the zero volt object(s),
 > and the 'total' capacitance of the 1 volt object is obtained by
 > looking at the charge on the 1 volt object.

The terms out of the diagonal of the capacitance matrix (in absolute
value, since they are negative).

 > In this example the
 > two objects are the same size, but if you modelled two different
 > sized objects, you would have to present four capacitances.

Or just three. The mutual capacitances are always identical.

Antonio Carlos M. de Queiroz