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>

 > Nice work.  It hasn't taken you long to get this up and running!

Thank you. I had already all the routines in the Inca program.

 > At some tube radius, presumably, the self potential of the tube
 > ring becomes equal to the self potential of the tape ring, so I
 > guess you've found the correct 'equivalent tube' in this instance.
 >
 > Wonder if the 1/3rd is generally valid?  One might expect it to
 > depend on curvature of the surface in some way.

I verified what is the spacing that results in the right capacitance
with a large number of rings. For the values in the last post I
used 31.8372% of the maximum radius (that would make the rings touch),
that is the value that results in the right capacitance for the 90x30
toroid with 100 rings. For a sphere with 200 rings, the value is
31.83135%.
You could do the same "cheating" with the small plate method, adjusting
the Pjj so the correct capacitance of a standard object with a limited
number of elements is exact.

 > Still with the same tube_radius/spacing = 1/3 for that one?

As I mention above, the ideal spacings are a bit different.

 > Yes, you can use this method for looking at some quite general
 > arrangements of terminals... the 'test page' for Geotc has quite
 > a few examples, eg of spheres inside toroids, etc.
 >
 > One problem is that if you succeed in completely isolating one
 > terminal inside another, the inversion of the potential matrix
 > will fail - the matrix equation becomes singular because some
 > objects are decoupled from others.   In practice (with my code at
 > least) there is always enough 'leakage' to ensure that the [P]
 > stays non-singular.

But the objects are always coupled. The Pij terms never disappear.
The nulls only appear when I calculate all the charge in the internal
rings.

 > I have a recent enquiry about just this situation, but I've been
 > unable to make an informed reply.  The 'C' of a skeleton tube topload
 > is not the problem, but predicting the effect on breakout could be.
 > But I don't have the techniques to persue that issue (yet!).

A simple method is to expand the rings in small toroids, made of
rings too.

 > You can surely estimate the charge per unit area, which relates
 > directly to the surface field strength of the modelled smooth
 > surface, ie not the rings themselves.

Gauss' law. Really, I don't have to calculate the electric field.

 > Eg you could say that a length L of tube represents an area
 > L * 1/3 of the modelled surface, and you know the charge Q on that
 > bit of tube.  So the surface flux per unit area would be
 > 3*Q/L, and therefore the surface field (if you assume that all
 > the flux radiates outwards from the closed object) is just 3*Q/(L*e0).

I tried the following:
Each ring has a charge qi and a length 2*pi*ri. A length dl has then
a charge dq=qi*dl/(2*pi*ri). The spacing of the rings is a*dx, where
dx is the angular spacing and a is the minor radius of the toroid.
Each segment has then an area a*dx*dl and a charge dq. The charge
density is then p=dq/(a*dx*dl)=qi/(2*pi*ri*a*dx), and the electric
field is E=p/(2*e0). I found the division by 2 strange, but then
found a book about electromagnetism that says that the division
is really necessary. The obtained values are practically exact for
a sphere:
A sphere with radius=1 meter shall have a surface field of 1 V/m for
each Volt of voltage. With 11 rings I get 0.999... for all rings,
except for the rings nearest to the poles, where the approximation
fails. I get there 0.919 V/m/V. With more rings, the precision
increases,
but the smallest rings continue at this value.

For the 90x30 toroid, I get, with 200 rings:
At the largest diameter:   1.7670568794 V/m/V
At the center of the hole: 0.2392683872 V/m/V
The breakout voltage would be 3e6 V/m divided by 1.767= 1703 kV.

Comparing with some simulations that I run with the
Bela simulator for toroids, the results don't look very nice...
(I run these simulations for Dr. Resonance some time ago, and run
some of them again now with greater resolution):
Sizes in inches:
Size       Simulated     Calculated (200 rings)
12 x 3:    302 kV        501.28052 kV
16 x 4:    386 kV        668.37402 kV
20 x 5:    458 kV        835.46753 kV
26 x 6:    532 kV       1043.91679 kV
34 x 8.5:  795 kV       1420.29479 kV
48 x 12:  1353 kV       2005.12206 kV (Csimulated=94.8 pF, Cexact=52.4
pF)
12" sphere: 850 kV       914.39958 kV

There was a room around the toroids in the simulation, but some
differences are too big. The capacitances also look wrong by
large margins. In the largest toroid, the room was -very- large.
Something is wrong.

Antonio Carlos M. de Queiroz