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>

 > Here are the corrected figures...
 >
 > Size    Bela  Tssp________________   Inca_________
 >          kV    C pf   V/m/V    kV      C pF     kV
 > 12x3    302   13.10  11.97   250.6   13.11   250.6
 > 16x4    386   17.47   8.97   334.4   17.48   334.2
 > 20x5    458   21.84   7.18   417.8   21.84   417.7
 > 26x6    532   28.02   5.75   521.7   28.03   522.0
 > 34x8.5  795   37.12   4.22   710.9   37.14   710.1
 > 48x12  1353   52.41   2.99  1003     52.43  1003.
 > 12 sph  850   33.89   3.33   901     33.91   914.4
 >
 > Now we're back on track. I think this demonstrates that for toroids
 > and spheres the decomposition into tubular rings with 1/pi spacing
 > ratio is working at least as well as decomposition into flat tape
 > rings, and offers the added advantage that the potential coefficients
 > are easier and quicker to calculate.

Yes. For closed surfaces this rule works well. It remains to be seen
why..

 > Antonio, could you summarise for us now the formulas you're using for
 > the Pij and the Pii coefficients?

I made a document showing the algorithms:
http://www.coe.ufrj.br/~acmq/tesla/capcalc.pdf

 > The Bela breakout voltages now seem to be on the high side.

The capacitances too.

 > I don't have an open hemisphere in tcap, I'd better put one in...
 > ...ok 'tis done, here's what we get:
 >
 >  > Hemisphere with 1 meter of diameter:
 >  > Exact:     45.5246270366 pF
 >              Inca               Tssp (tcap)
 > 10 rings:  44.9288700205 pF    44.53 pF
 > 20 rings:  45.2259355350 pF    45.00 pF
 > 40 rings:  45.3749442201 pF    45.26 pF
 > 80 rings:  45.4496825615 pF    45.39 pF
 > 200 rings: 45.4946218478 pF    45.48 pF
 >
 > The tubular rings seem to be converging better than the flat tapes do.

The edge causes difficulties to both algorithms.

 > Can we have a look at a thin circular disk?
 > Say 1 metre diam, C = 4 * e0 * diam = 35.41675 pF

Ok. I added this to the program (allowed a hole in the center too).
I even extended the calculation to a general truncated cone, that
can be a disk, a cylinder, or anything between them. I didn't test
the general case yet. Do you have some cylinders or cones calculated?
I apparently canīt calculate the electric field by Gauss' law for
these figures, except for a disk (even with a hole), where it's just
a question of dividing by 2 the normal result. The values at the edges
diverge to infinity as they should as the number of rings is increased.

rings      tssp          Inca
   10       34.297 pF   34.648 pF
   20       34.833 pF   35.034 pF
   40       35.114 pF   35.225 pF
   80       35.263 pF   35.321 pF
  200       35.405 pF   35.379 pF

There is a sign of numerical error for 200 rings. The edge causes
difficulties with a disk too, that is probably a worst case. I kept
the maximum radius/Pi idea. The rings are centered at their portions
of the disk.

 > These are GNU 'gzip' compressed postscript files.  Your web browser
 > should know how to uncompress gz files on-the-fly.  Can 'ms-word'
 > read postscript - I'm not sure?  Maybe you'll have to install
 > a tool such as ghostview.   Postscript is a very good, very well
 > established standard that's been around for decades. Everything
 > speaks postscript - except in windows land!

The site www.ps2pdf-dot-com can convert ps to pdf. Has all the ps tools
for download too.

 > While we're on the subject of postscript files, here's another one.
 > Pages 2 and 3 give a concise account of the boundary element method
 > and show how the equivalent charge formulation can be fitted in to
 > the overall scheme to deal with dielectrics:
 >
 >   http://faculty.smu.edu/tausch/Papers/mtt1.ps.gz

I will see. I found this site:
http://hermes.phys.uwm.edu/~russell/projects/masters/index.html
It has the derivation of the potential of a ring, exactly as I
obtained it. Shows also the potential of a belt, but doesn't solve
the integral. The references and the other methods may be of interest
too.

 > gcc is the GNU C compiler, which is available for almost all systems,
 > except windows and DOS.

There is a gnu C for DOS: http://www.delorie-dot-com/djgpp
Works with Windows too. If you want a graphical user interface for it,
I have one: http://www.coe.ufrj.br/~acmq/xview-pc.html

Antonio Carlos M. de Queiroz