Re: Toward the equilibrium

[John Denker <jsd@MONMOUTH.COM>]

At 08:45 AM 1/6/01 -0500, Ludwik Kowalski wrote:

> 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?

Ludwik has posed a well-pointed question.  It is smart to check whether a
calculation gives sensible asymptotic behavior.

> Something is basically wrong.

Indeed something is basically wrong.  The problem is that inductance has
not been mentioned.  Inductance grows in proportion to the size.

> Do we have to use special relativity to calculate small time constants?

Yes and no.  Indirectly yes, in the sense that to calculate the real
physics we need to use Maxwell's equations (or the equivalent), which are
relativistically correct.

===========

Modelling the sphere as just a resistor and capacitor is an unsatisfactory
approximation to Maxwell's equations, especially for short times and large
spheres.

OTOH if you do it right, the system (including electromagnetic radiation)
can be modelled rather well using a network of capacitors, resistors, AND
inductors.