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>

 > It's hard to get any work done with windows - ditch it for a proper
 > OS!  For C compiling, install gcc.

Delphi (Pascal) is good. I am doing now everything with a free Delphi 3.
And it's easy (apparently) to port to Linux using Kylix.

 > For a sphere with 200 rings, the value is 31.83135%.
 >
 > Ah, 1/pi then?

Indeed! I found it by trying. This number looked familiar...

 > Wonder how
 > you get on with objects that have edges?

I can try an open hemisphere, that has also a simple expression for
capacitance (now using 1/pi of the maximum radius as tube radius):
Hemisphere with 1 meter of diameter:
Exact:     45.5246270366 pF
10 rings:  44.9288700205 pF
20 rings:  45.2259355350 pF
40 rings:  45.3749442201 pF
80 rings:  45.4496825615 pF
200 rings: 45.4946218478 pF

 > I suspect that the ideal
 > spacing ratio depends on the curvature of the modelled surface
 > at that 'latitude', and would therefore vary for each pair of
 > adjacent rings:  1/pi +/- some curvature-related term?

Possibly. I will try to prove that 1/pi. Maybe it's the correct
value for a flat surface, producing the same far field.

 >  > Gauss' law. Really, I don't have to calculate the electric field.
 >
 > Yup, easy isn't it!

For a closed surface, yes.

 >  > and the electric field is E=p/(2*e0).  [hmm]

I looked again at that book. This division by 2 is exactly at the center
of the charged layer. For breakdown voltage over a closed surface, the
value to use is really E=p/e0.

 > If the charge density is p, then the total flux from an infinitesimal
 > area dA is p*dA/e0.  If this flux were to come out equally on both
 > faces of the area then the flux out of each face is p*dA/(2*e0) and
 > the flux density on each face is p/(2*e0).  But here we have a closed
 > surface, so we are entitled to say that all the flux comes off the
 > outer surface, thus the surface field strength is actually E=p/e0.

My error was trivial. I was considering the diameter of a sphere as
its radius, what produces 2 V/m/V for a "1 m sphere" (1m of diameter).

 > Note that our use of single layer potentials doesn't tell us how the
 > flux is shared between the two faces, only the total.  For closed
 > objects we already know the answer, but for partially open objects,
 > eg the tube of the secondary we would have to calculate some E values
 > for each side of the surface and apportion the charge pro-rata.
 > (Or, model two surfaces for each conductor, and put E=0 inside the
 >   conductor to complete the system of equations).

The last approach is realistic. I was imagining a kind of "image method"
where I calculate the charges in a set of rings inside the object,
considering the voltages at points of the surface. The electric field
could then be calculated directly at the surface.

 > Sorry, I got my units mixed up!  It should be 3.53 V/m/V, I forgot
 > my program reported in V/cm/V.

I get a maximum of 3.5341105020 V/m/V with 200 rings at the largest
diameter. Breakdown voltage=848.86989 kV

 > Yes, tssp gives 1.01436 V/m/V for the 1m sphere in 135 rings. That's
 > the highest field that it finds on the surface, so shows, if you like,
 > the max positive error.  C = 111.185pF (tssp), 111.265pF (4*pi*e0)

I get  C=111.2650023882 pF with 200 rings. (Cexact=111.2650056054 pF)

 > Something wrong there.  Your're ok for the sphere but not the toroid.

Removing the /2 everything seems to work. Reviewing the toroids:
(Rmax/pi radius, 200 rings, Maximum electric field = 3 MV/m)

Size      Bela    Tssp (C  Max field   Breakout) Inca (C   Breakout)
12 x 3:   302 kV  14.01pF, 9.87 V/m/V,  304kV    13.11 pF  250.6 kV
16 x 4:   386 kV  18.68pF, 7.40 V/m/V,  405kV    17.48 pF  334.2 kV
20 x 5:   458 kV  23.35pF, 5.92 V/m/V,  507kV    21.84 pF  417.7 kV
26 x 6:   532 kV  30.48pF, 4.50 V/m/V,  667kV    28.03 pF  522.0 kV
34 x 8.5: 795 kV  39.69pF, 3.48 V/m/V,  862kV    37.14 pF  710.1 kV
48 x 12: 1353 kV  56.03pF, 2.47 V/m/V, 1215kV    52.43 pF  1003. kV
12  sph:  850 kV  33.89pF  3.33 V/m/V,  901kV    33.91 pF  914.4 kV

 > The tssp figures are for the toroid in free space, not in a room.
 > I expect the agreement would be better if we put in your room sizes,
 > since tssp diverges from Bela when the toroid size increases.
 > There's something a bit funny about the 48x12 model.

I will redo the simulation. The toroid capacitances calculated by
Bela seem to be always wrong.

Antonio Carlos M. de Queiroz