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Re: Capacitance problem



Donald Simanek wrote:

... Why does the result (half the energy goes
elsewhere) *not* depend on these particular details? If the only loss were
radiative, you'd get 1/2 for the energy loss. If the only loss were
resistive, you'd get 1/2. If it is a combination of both, you'd get 1/2.
I chose identical capacitors, which is one reason you get 1/2, but that
was only for convenience of discussion. The principle (if there is one)
could be applied to any pair of unequal capacitors, and the resulting
fractional energy loss would be a different value.
...

I thought I gave the explanation for this in my previous post. What didn't
you like about it? The reason for the particular fixed fraction of energy
loss (which becomes 1/2 when the 2 capacitors have identical capacitances)
is the inevitable result of three physical facts: 1. the potential energy of
each capacitor is an increasing quadratic function of a dynamical parameter
(i.e. charge) which gives the potential energy a quadratic minimum, 2. the
composite system of 2 capacitors is coupled in such a way that the sum of
values of this dynamical parameter is constrained via a conservation law to
be fixed, and 3. the system equilibrates (as a consequence of the 2nd law) by
minimizing its energy. Condition 3. will automatically obtain as long as
the system is somehow (weakly) coupled to many other degrees of freedom
(i.e. the internal microscopic degrees of freedom of the conductors via
resistive friction, or the propagating EM modes via radiative interactions,
or both). If no such coupling to many other degrees of freedom is present
then the system conserves its energy and any motion of the dynamical
parameter will be permanent undamped oscillations if the dynamical degree of
freedom had an inertial kinetic term in the system's Lagrangian.

All three of these above conditions are needed for the result to follow. For
instance, if condition 1. was not met the result would not follow. If the
potential energy function was not quadratic in the charge or if it did not
have a minimum, then the result does not follow. For example, if the
potential energy was a *quartic* minimum then the energy retention fraction
(for the 2 identical capacitor case) would not be 1/2 but 1/8. In general if
the potential energy near its minimum varied as the 2*p power of the
dynamical parameter then the energy retention factor would be 2^(1 - 2*p).
For p=1 (quadratic case) the factor is 1/2, and for p=2 (quartic case) the
factor is 1/8. I also tried to explain that the mere existence of a (smooth)
energy minimum automatically made it quadratic in the usual case, at least in
a neighborhood of the minimum. At a minimum the first deriviative vanishes,
thus the leading nonzero/nonconstant term in a McLaurin expansion of the
potential energy would tend to be the quadratic term -- thus giving the
quadratic minimum. If an energy minimum did not exist at all then the
composite system could not equilibrate into any kind of balanced arrangement
at all.

If condition 2 (constraint of "charge" parameter conservation) did not
hold then the system would just equilibrate by having both parts
(capacitors) loose *all* of the initial potential energy (above the minimal
value except for k*T/2 worth) rather than just 1/2 of it -- assuming the
system was somehow coupled to many other degrees of freedom with which the
system has to share its initial energy as it equilibrated as a consequence of
the 2nd law. Actually, because of thermal exitation at finite temperature
even when the charge is conserved, the average energy of the equilibrated
coupled system will be k*T/2 larger than the value which precisely minimizes
the (charge-conserved) potential energy. Since the energy of the composite
system is assumed to be macroscopic the microscopic value k*T/2 of the noise
fluctuations is negligible by comparison however.

If condition 3. did not hold the system would not be able to equilibrate and
the dynamics would conserve the macroscopic energy of the system effectively
preventing any energy loss, let alone 1/2 of it.

Maybe you (Donald) wanted *just one* principle rather than the above 3
conditions as a general explanation of the energy loss effects(?).

...
Well, yes, this is an example of the sort of general principle I was
fishing for. I'm not sure how we'll apply it to the capacitor problem.

Another example would be the Thevinin and Norton theorems which tell us
that a circuit will behave in a certain manner with respect to two
terminals *no matter what combinations of emfs and resistors* the circuit
is made up of.

The example previously given of impedence matching is not quite the same kind
of phenomenon since in that case a fixed initial amount of energy is not
redistributed (with losses), but rather a steady-state power (energy flow)
condition is established where energy is being continuously pumped through
the system in such a way that half of it is consumed by the input circuit.
The factor of 1/2 for the energy loss is -- even in this somewhat different
situation -- also a consequence of the fact that the power consumption in
both the input and output circuits is a quadratic function of a quantity
(i.e. potential difference or current) and the impedance plays the role of
a coefficient for that quadratic dissipation function.

I was really hoping that someone more math savvy than I knew already what
principle might apply to the capacitor problem, and jump in and enligten
us all. Maybe there is no such principle, and this thread will die.

So you really only want just 1 overriding principle here? It seems to me
that we necessarily have more than one such simultaneous principle (i.e. 3).
I guess that the fact that these systems have a quadratic function to be
extremized somehow is a common thread among all the systems considered.

It's one of those things that tantalizes, making one think that when a
precise result is *so independent of specific details*, then there *ought
to be* a general principle predicting that result. Maybe that's the
clearest statement of my concern that I've been able to state so far. When
I posted this I fully expected someone to say "What's the matter, Donald,
didn't you learn Schwartzengruber's Dissipative Energy Particition
Principle in college"? (I made that up.)

The closest such *named* principle which has some applicability here is the
*Fluctuation-Dissipation Theorem* which relates the second derivatives (i.e.
the suseptibilities, compressibilities, specific heats, polarizabilities,
etc.) of the system's free energy to the coefficients of energy transport via
dissipation (diffusion constants, friction coefficients, resistance, acoustic
absorption, viscosity, etc.) via linear response theory. (Here the response
is linear in the disturbance from equilibrium and the damped dissipation
rate (as well as the free energy shift) is quadratic in the stimulus (and
hence also in the response).
--
David Bowman
dbowman@gtc.georgetown.ky.us