Re: relaxation method(s) .. was flat conductors

["John S. Denker" <jsd@MONMOUTH.COM>]

Ludwik Kowalski wrote:
...
> 2) Last year, while arguing with Leigh Palmer, you were
> saying that it is possible to use the relaxation method do
> calculate densities of space charges at various locations.

-- In general, sure.  You can presumably use some sort
of relaxation method to solve any problem you can think
of.
-- In particular cases, one would need to know more
about what the problem is before writing down the
solution.  Laplace's equation is particularly simple;
it says del squared phi equals zero.  I know exactly
what zero is.  Now you tell me you want del squared
phi to be nonzero.  Well, there are a lot of nonzero
things, and I don't have any idea which one you want.

To illustrate, let me answer a question slightly
different from the one that was asked.  Suppose we
were given, a priori, an assigned distribution of space
charge.  Then it would be super-easy to write down
the algorithm to solve for the fields.  We have a
Poisson solver instead of a Laplace solver, but the
idea is the same.
 -- The update-step in the Laplace solver looks
at the local del squared, and if it is nonzero
it takes a step in a direction that will make it
more nearly zero.
 -- The update-step in the Poisson solver looks
at the local del squared, and if it does not equal
the assigned value, it takes a step in the direction
that will make it more nearly so.

If you want to calculate the space charge as a
function of something else, you need to specify
what the something else is.

> > > Specifically, a disturbance in the charge pattern with
> > > wavelength lambda will die out exponentially with a
> > > time constant tau, where
> > >         tau = (eps0 lambda^2) / (k L)
> > > where 1/k is the resistivity of the paper
> >
> > I considered a disturbance of the form
> >         psi(t, x) = exp(-t/tau) sin(2pi x/lambda)
> > verified that it satisfied the equation of motion,
> > (not too near the boundaries) and solved for tau
> > as a function of lambda.
...
> I do not know how to implement the above formula.

You don't need to implement it.  That is something
that should come _out_ of a detailed simulation
of the time-dependence, not something that goes in.
And it's approximate, anyway.
All that went in to deriving it was things like
 -- the idea of a capacitance
 -- the idea of a resistivity
which are the sensible inputs IMHO.

That formula was intended to answer a particular
narrow qualitative question, namely what happens
if you change gradually from a high-conductivity
medium such as resistor paper to a tremendously
low-conductivity medium such as, say, rhombic
sulfur.  The answer is that the timescale for
the medium to settle down on "right answer"
diverges, and the formula tells you qualitatively
what the timescale is.

(Also note that the demands on your voltmeter,
such as input impedance, also diverge.  I suspect
lots and lots of students have gotten bogus
results from using junky voltmeters.)