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Re: Capacitor problem



Replying to my first message yesterday Donald Simanek, the initiator of
the capacitor problem, asked me to summarize the cited article of O'Connor
in Physics Education (32, 2, March/97) about the 'lost' energy by joining
two capacitors. I will try it. But because my possibilities to formulate in
English aren't the best and will cost me much time, I'll do it mostly in the
words of O'Connor and you will forgive me, when I do so without
quotation marks. Furthermore I will try another type on my German
keyboard for the apostrophe - ' -, hoping it will now be transmitted
without any change or transforming.


First O'Connor explains the problem with the two capacitors. Then
after mentioning parallel situations as that with the two buckets on a
table or the spring example, where a compressed spring is combined
with an uncompressed, and at last the case of inelastic colliding
masses, he formulates a "law", which as an idea should has been first
formulated by two friends and former mentors of him. His version for
this law sounds: "Energy of a particular kind, stored in one part of a
system, cannot transfer, without loss, to energy of the same kind
stored in another part of the system, unless the transferred energy is
converted to another kind of energy at an intermediate stage in the
process."

The problem with the lost energy arises, as O'Connors says,
because an attempt is made to join two initially independent energy
storage systems. At this joining instant each of the two formerly
independent systems undergoes a change at one of its defining
bounderies, so that each is required to become a boundary for the
other. Here a problem arises when, on each side of the divide, (a) the
same kind of variable is being constrained (b) in the same way (c) to
a different value and (d) with no clear mechanism for resolving the
conflict.

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?).

O'Connors then refers that in all the ' trouble making' examples we
are dealing with models of reality which idealize and simplify. We can
avoid such problems either by demanding that all models obey certain
rules of good modelling which are based on physical laws and which
prevent such 'impossibilities', or by adding extra model elements to
represent other physical effects which we know will always be present.

For example, a basic rule for electrical circuit models could be that
Kirchhoff's circuit laws be obeyed at all times. This is only possible if
we add a different circuit element, such as a resistor R, to the circuit.
And O'Connors states, that with this resistor to model energy loss the
system is made completely determinate when the switch is closed
and by defining precisley the right amount of current which must flow
at every instant during the transient stage then the total dissipated
energy will sum exactly to the calculated 'missing' final energy.
There is no prove of that statement in the article and I didn't
calculate it, too, but I think it is true.

O'Connors continues and says, that also by adding non-energy dissipating
elements as an inductor the initial boundary incompatibilities can be
resolved. The energy is then changing from electric to magnetic and
vice versa. All kind of stored energy, he summarizes, can be thought of
as 'potential', in the sense that they all imply the potential to do work.
What the 'law' is saying is that this potential to do work can be realized
only by doing work, not by passing on the stored energy directly. So the
capacitor does work on the inductor, which in turn can do work on the
second capacitor. And in fact this is just what happens when energy is
transmitted by waves: the energy is continually changing between two
forms and the process can indeed be lossless.


Sorry, the summary has became a little long, but I hope you will pardon
me that. I for my part can follow the arguments of O'Connor and find
them problem clearing. Whether the referred facts should be called a
'law', as O'Connors sees it, I don't know. I would hang it a little bit lower.

Sincerely

Herbert Moedl
D-73525 Schwaebisch Gmuend
Germany