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Ref: How many volts ?



Date: Mon, 24 Mar 1997

In responding to the "how many volts" problem both John and Mark used an
implied assumption that distributions of charges on positive and negative
parallel plates are uniform when the distance between them is very small
in comparison with diameters (or other sizes). This is an accepted truth
when |Q1| and |Q2| are equal. But I am not sure that the distributions of
charges are uniform when |Q1| and |Q2| are very different. Gauss's law is
very general but the uniform distribution of charges is not. In fact, it
may be an exception valid only when the net charge on two plates is zero.

In my opinion charges are not spread equally when plates are far away
from each other. They would be distributed uniformly on spheres (away
from walls, floors, etc.) but on thin circular plates, for example, they
must be uniformly distributed along the rims. For the same reason (mutual
repulsion to minimize potential energy) a single rectangular plate will
concentrate its electrons (or "holes") near corners. Thus an isolated
circular plate of very large radius will create the field which is the
same as that due to a uniform circular loop (a thin wire).

We bring two such loops toward each other. One has much larger lambda
(linear charge density) than the other. Think about two circular "loop-
plates" on a common axis, to simplify analysis. The process of bringing
them toward each other will result in induction and in a redistribution
of charges. But how do we know that distributions will become uniform
when d becomes very small in comparison with plate diameters?
Ludwik