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>

 > Ok, if I understand right you're using a tubular ring, rather than
 > a flat tape ring.  It seems to get you into the right ballpark.

Yes. A thin tubular ring.

 > Yes, try it for discs too, because there you have an edge to deal
 > with.

I didn't try disks yet, but I can make a half sphere, that has also
an edge. I verified the behavior of the obtained capacitances of the
rings, and observed the problem with oscillations near the edges
that you mentioned. I noticed then that if I reduce the thickness
of the tubes for the Pjj terms to about 1/3 of the distance to the
adjacent ring the oscillations disappear, and I obtain very good
results with the rings right at the surface. I then adjusted the
tube distance to a value that gives the exact value for the 90x30
toroid with 100 rings.

 > Tssp's tcap program gives, for the toroid above,
 >
 >    40.2 pF using 10 rings.
 >    40.4 pF using 20 rings,
 >    40.50 pF using 40 rings,
 >    40.54 pF using 80 rings,
 >    40.563 pF using 160 rings,
 >    40.586 pF using 320 rings,

I get now (C exact: 40.583973 pF):

10 rings: 40.5487 pF
20 rings: 40.5797 pF
40 rings: 40.5835 pF
80 rings: 40.5839 pF

For some other structures:

Sphere with 90 cm of diameter:
Exact:         50.0692525224 pF
With 20 rings: 50.0681169556 pF
With 80 rings: 50.0692908972 pF

Open Half sphere with 90 cm of diameter.:
Exact:         40.9721643329 pF
With 20 rings: 40.7035446304 pF
With 80 rings: 40.9047733616 pF

A very interesting effect happens when I specify a toroid that overlaps
itself, as a 90x60 toroid. The program then places some loops inside
the figure. The capacitance continues to be calculated correctly, and
the capacitances of the internal rings result as almost zero, as should
be due to the shielding caused by the external rings. The choice of
the radius of the tubes also affects the results for these rings. A
wrong value causes oscillations and negative capacitances there.
90 x 60 toroid with 80 rings: 46.0583086244 pF

 > With more than 300 rings, the result begins to deteriorate, and the
 > software would need a little fine tuning to do better than this.
 > Surprising how well the method goes with just 10 rings, isn't it?

Enough for practical purposes. Also indicates that a toroid made with
a cage of tubes has really practically same capacitance of a solid one.

There is a discussion about grids and plates in Maxwell's book.

 >  > http://www.coe.ufrj.br/~acmq/programs/inca.zip
 >
 > I've no means of running this program.  You might want to consider
 > making the source available.

Windows (Delphi) programming forces the source code to be split in
many parts. I will try to make a Linux version.
I added a plot with the shape and position of the rings.

 >  > The thin ring approximation is a first step. The integral that
 >  > would give the inverse of the capacitance of a ring with uniform
 >  > charge seems solvable, but not trivial.
 >
 > Yes, perhaps someone has worked this out long ago - buried in an
 > old textbook somewhere - that often seems to be the case!  It would
 > sure speed up the calculations.

They calculated the potential and capacitance for a toroid, that is
more complicated. A belt is a simpler problem, it seems.
The toroid formulas appear in a paper by W. M. Hicks, in the Proc.
of the Royal Society of London, 1, 31, 1881. The paper mentions older
works, however. The paper is very difficult to follow.

 > A challenge facing us is to combine the axial-symmetric model with
 > a full 3d model of a curved streamer.  I'm considering going back
 > a step: ditching the rings, and doing the entire thing in 3d with
 > multipole approximations for the more distant elements of [P] - along
 > the lines of the program 'fastcap'.

A single point breaking the symmetry would cause the charges in the
rings to become assymmetrical, complicating the formulation.

 > BTW, for your test toroid above, I get the highest surface field
 > strength in the plane of the toroid (no surprise there!) and the
 > value is 0.035 volts per metre, per volt on the toroid.  I've
 > taken to calling this figure the 'specific surface field', for want
 > of a terminology.

My ring formulation adds the difficulty that the electric field
can't be evaluated at the surface, at least where the rings are.

 > And, to those of you who sat through math lessons thinking - "what
 > use is all this boring stuff about matrices?"  Well, now you know!

An essential tool.

Antonio Carlos M. de Queiroz