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

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

Original poster: "Gerry Reynolds" <gerryreynolds-at-earthlink-dot-net> 


 > Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk>
 >
 > Gerry wrote:
 >
 >  > Im planning on decomposing the toroid into tiny surface patches
 >  > described in (R, phi) coordinates for major dimensions and
 >  > (r, theta) coordinates for minor dimensions
 >
 > That seems to give you one too many coordinates - isn't the R co-
 > ordinate constant for a given toroid - the radius of the tube
 > center line?

Yes,  R and r are constants (major and minor radius of the toroid) but still
needed to translate to x,y,z coordinates for purposes of determining
separation distances.


 >  > charge density approximation will initially be constant over
 >  > the entire surface and the total toroid charge will be Q.
 >
 >  > First, I need to redistribute the charge in theta (taking
 >  > advantage of symmetry) until the summation of all forces on each
 >  > patch charge is zero tangental in minor curvature.
 >
 > Ok, and symmetry takes care of the major.
 >
 > When you say 'redistribute the charge', do you mean some kind of
 > relaxation algorithm?

Probabably so.  I'm new to the symatics.  I'll invent some form of
redistribution algorithm.  I doesn't have to be perfect if I iterate.  I
need to determine the criteria for bailing out of the iteration.


 >
 >  > Second, I need to compute the integral (from infinity to the
 >  > toroid surface)(E.dl).
 >
 > I don't think you'll need this step.  The topload potential should
 > pop out of the first step, in units of Q.  Hmm, where is the E going
 > to come from in the E.dl?

I'll get the charge distribution from the first step but feel I will need to
compute E.dl to get the potential for purposes of scaling.  But maybe what
you are suggesting is computing the field from  "gradient of the
displacement field = change density"??  This would give me the field
gradient for the assumed charge but again I assumed a charge and need to
determine the voltage the charge corresponds to in order to scale.  Maybe I
need to think about alternatives more.  Anyway there may be more than one
way to skin this cat.

The E field will need to be computed from Coulombs Law for each point on the
path of E.dl


 >
 > I would advise to stay as far as possible with just the potentials
 > during the calculations, rather than bringing in the field strengths.
 > Only at the end compute E where you need it.

I think this is what I'm planning to do.


 >
 >  > I hope this will be sufficient for my purposes in determining
 >  > the "reach" various toroid sizes have.
 >
 > I'm not sure it will, but you're embarking on the essential first
 > step.  I've a horrible feeling that we're going to have to
 > abandon the exploit of axial symmetry in order to solve for the
 > charge distribution (and the field) in the presence of an
 > asymmetric streamer load, eg a single streamer starting out from
 > the rim.

I would expect this.  But I want to understand what I can before a streamer
breaks out.  I guess I'm thinking about the field gradient across horizontal
space as a precondition for streamer growth.


 >
 >  > (I don't have a C compiler on my windows98 PC).  I may need to
 >  > get my hands on a unix machine for this purpose.
 >
 > It's hard to get any work done with windows - ditch it for a proper
 > OS!  For C compiling, install gcc.

tell me more about gcc and does this work in windows or DOS.

Gerry R
Ft Collins, CO