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Re: Capacitor problem



Thus, in the two-capacitor case, an ideal switch when closed
(zero resistance) implies an equal potential difference on each side,
so one is trying suddenly to impose the condition of equal p.d. across
two capacitors which, by the initial supposition, are at a different
p.d. The interface is therefore undefined (what p.d. is it at?), and
the outcome is indeterminate (which side changes, by how much,
and in what way?).

This makes a good teachable moment for a very important point which is
too often insufficiently emphasized in introductory E&M courses.

The terms "potential energy", "potential difference" and "potential"
should never be applied in other than static systems. "Voltage" would
be better. It is certainly possible to have a voltage across a perfect
conductor in a nonstatic system. Consider, for the simplest example,
a conductor moving at constant velocity through a uniform magnetic
field. There exists a "motional emf" between points on the conductor.
There also exists a frame of reference in which this might appear to
be a fully static problem, i. e. the frame of the moving conductor.
It is not the case that the problem is static in that frame if the
source of the magnetic field is included in the system; that source is
in motion in the conductor's frame. If we take the field to be static
(neglecting its source) then we must acknowledge the presence of an
applied uniform electric field parallel to the conductor. This field
will be opposed by the field due to charges on the polarized conductor,
and the electrical potential is defined in this frame of reference.

Here we see Einstein's synthesis again affecting our sense of reality.
There should be no conceptual problem for those of us who accept the
idea of centrifugal force already; those who don't may have to
interpret the electric field in the frame of the moving conductor as a
fictitious field, however, since it doesn't exist in the frame of
reference in which the problem was originally proposed. Note that in
this case both frames of reference are inertial.

It appears to me that the improper use of "potential" frequently
leads to conceptual problems of this sort. Be more careful! I have not
read the article in question by O'Connor, but I frequently find
articles in TPT (to which I subscribe) lacking in physical insight and
rigor. They are not subjected to the sort of editorial scrutiny which
makes such errors somewhat less frequent in the peer reviewed
literature. Using a two-buckets-connected-by-a-pipe analogy for a
hypothetically resistanceless circuit shows a marked disregard for the
crucial aspect of this question in my view.

I don't think Donald's two capacitor problem is that hard to analyze.
See my analysis of yesterday - I'll flesh it out with more words if
they are needed, but I don't think they should be. I always do similar
analyses for my students. Ludwik's modification is certainly easily
explained in terms of a time evolving system as well. He should lose
those gratuitous caveats about fringing, however. Fringing exists; it
can't be ignored *a priori*. It must be shown to be ignorable in any
particular instance in which that assumption is made.

Leigh