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Wed, 23 Jan 2008 A couple days ago I gave a talk at the company's technical symposium on a power converter. On one slide, I discussed the classic conundrum of energy loss in charging a capacitor. In my discussion of this, I claimed that even if I have a perfect conductor, the energy loss can be understood as having been radiated away from the circuit because of the (infinite, in the ideal case) speed at which the capacitor E fields change. A member of the audience came up to me after my talk and told me that he disagreed with my summary of the situation, and that instead one need only postulate a resistance in series with the capacitors the limit of whose value approaches zero. I replied that while that model of the circuit seemed a valid one, it did nothing to invalidate my explanation. After a bit of thinking, some digging through my old E&M textbooks, and a bit of calculation, I concluded that I was right, and was determined to prove it. Gautham and Ion suggested that I might also check if anyone had published a paper on this question, since it is such an interesting one, and lo and behold, someone had! (see links below) Here, then, is the response I wrote to the disbelieving fellow: D___ (and cc: recipients, who had a stake in at least parts of the conversation): After our discussion yesterday about the problem of cap-to-cap energy transfer, I took some time to sort out the problem and have concluded that, in essence, we are both correct (literally, the full answer is a linear combination of our claims, though with a rather dissatisfying caveat). The best treatment of the subject in the literature I've been able to find on short notice is from Mita and Boufaida: K. Mita and M. Boufaida. "Ideal capacitor circuits and energy conservation." American Journal of Physics, Volume 67, Issue 8, pp 737-739, August 1999. It's not available for free from their website, but I was able to obtain a copy (along with a note in response and a quick erratum) via the MIT Libraries: http://web.jfet.org/~kwantam/AJP000737.pdf http://web.jfet.org/~kwantam/AJP000576.pdf (quick erratum; see p. 3 of the PDF) http://web.jfet.org/~kwantam/AJP000668.pdf (beginning on p. 3 of the PDF, see comment) In brief, your solution to the problem is one possibility, mine is an alternative one, and they can be combined into a third. The solution comes down to this (and here is the disappointing caveat): in order for the question to be coherent, there _must_ be a resistance or inductive reactance in the circuit; no solution is possible otherwise (note, however, that while there is in fact energy associated with the acceleration of the electrons from one plate to the other, it does not serve to explain the energy loss). So we have two choices: (1) assume electro- and magnetoquasistatic behavior (viz., no magnetization or displacement current), in which case we insert an infinitesimal resistance in series with the capacitors and all energy is dissipated therein, or (2) instead insert an inductance into the circuit, in which case the energy is still fully accounted for, though the resultant waveforms are dependent on the inductance (i.e., the area of the enclosed loop) and its Q. A few comments on what I proposed yesterday, which amounts to the second case: First, a finite inductor Q can be explained either by resistance in the windings or by flux linkage to "the outside world." You'll note that the former is simply a generalized combination of (1) and (2), whereas the latter admits the possibility of a cap-cap loop with precisely zero resistance but which is either magnetically coupled to a parasitic loop or via a lossy medium. If we combine these ideas, we arrive at a fully generalized picture of the circuit: two capacitors, a real resistor, and an ideal transformer with a nonzero magnetizing inductance the secondary of which is terminated on a resistor. We can vary any element with impunity as long as we are careful to keep some nonzero impedance in series with the capacitors. Second, in retrospect I was too quick in agreeing to your claim that as R->0 we can always claim that all of the energy is dissipated therein. While I made some small noise about quasistatic approximations and problems with KCL/KVL, I wasn't rigorous enough in my objection. More properly stated, it is this: as R->0, we have to be careful about when KCL and KVL break down. If R*Ctot becomes very fast compared to l/c (l = length of the loop, c = speed of light), then the resistor no longer explains the power dissipation because at this point we can no longer neglect the inductance associated with the loop (and thus we end up with the generalized combination of (1) and (2) as stated above). So the "discontinuity" at R=0 in my explanation does not exist! Rather, the loss from the resistor is simply dominated at smaller scales by the inductance inherent in the loop. Finally, I claim that while both of us are correct, my answer is more satisfying given the way I posed the question, to wit, a perfectly lossless capacitive circuit unfettered by the restrictions of any quasistatic approximations. In this case, we cannot possibly get around the fact that the loop has some dimension associated with it (else we wouldn't have capacitors, epsilon*A/l et cetera), but I can easily appeal to a "perfect conductor" (rho=0, E=0, H=0) without any restriction on the length of the wires. In a sense, inductance is just more fundamental to Maxwell's equations than resistance. Best regards, -=rsw You should note that the last PDF above also has an interesting writeup called "Fourier transform solution to the semi-infinite resistance ladder." [ permalink | 2 comments ] |
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