Re: Toward the equilibrium

[Ludwik Kowalski <KowalskiL@MAIL.MONTCLAIR.EDU>]

In his first reply Bob Sciamanda wrote:

> Using Maxwell's equations, the continuity equation , and the definition of
> conductivity, texts routinely derive an exponential decrease of rho in a
> region of constant conductivity.  The relaxation time is shown to be
> epsilon/conductivity.  If memory serves, this is about 10^-19 sec for
> copper.  This is the simple, classical model, and even ignores mean free
> path considerations.

Bob was responding to this:

> > Suppose that at t=0 a charge of 1 microC is deposited at one
> > point of a copper sphere whose diameter is one meter. I used
> > to say that "the equilibrium is going to be established very
> > quickly, perhaps in a couple of nanoseconds or so. Why?
> > Because it can not be faster than  3.3 ns; the time light
> > needs to cover the distance of 1 m in air.
> > . . .
> > 2) How to calculate the time needed to establish equilibrium?
> > I suspect it is much longer than ~3 ns but I do not know how
> > to estimate it realistically.

I got nearly the same (unrealistically small) order of magnitude on
the assumption that the charge equilibration takes place through the
volume (rather than along a very thin skin). It goes like this:

1) The capacitance of an isolated sphere is about 50 pF (assuming
the nearest wall is very far away)

2) The resistance is roughly the same as for a one-meter copper
cube, or about 2*10^-8 ohms.

3) Therefore, the RC constant is roughly 10^-18 seconds.

Note that the answer does not depend on the size of the sphere
because C is proportional to the radius while R is inversely
proportional to the radius. This is very unrealistic; how can
the equilibration time be the same for one meter and for 1000
km? Something is basically wrong. Do we have to use special
relativity to calculate small time constants?

Will I be able to read e-mail from where are am going today?
Ludwik Kowalski