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In a comment, Elio Fabri says:

A well-known case is the famous paradox of the two capacitors, one charged, the other uncharged. If you connect them, charge is shared, and energy appears not being conserved. But if the connecting wire has a non-null resistance $R$, you can see that the lacking energy is dissipated as Joule heat in $R$. The paradox is that the energy dissipated does not depend on $R$, so that the limit $R\rightarrow$ 0 would still give the same value.

I haven't thought about it very deeply, and I'm sure others have analyzed this carefully, but it seems to me that for small values of resistance, this circuit would behave like an $LRC$ circuit and exhibit oscillations, and in the limit $R\rightarrow0$, radiation would become a more efficient dissipative mechanism than resistive heating. Is this a correct analysis?

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    $\begingroup$ Duplicate physics.stackexchange.com/q/35843/104696 $\endgroup$
    – Farcher
    Commented Oct 28, 2018 at 18:35
  • $\begingroup$ Voting to close my own question as a duplicate. $\endgroup$
    – user4552
    Commented Oct 28, 2018 at 21:26

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in the limit R→0, radiation would become a more efficient dissipative mechanism than resistive heating. Is this a correct analysis?

That would depend on the physical construction of the circuit, how big a loop the wires formed, etc.

The well-known result essentially assumes you build the circuit small enough that radiation is not significant. This is a very common assumption in circuit theory, called the lumped circuit approximation.

In this case, we can still find a mechanism to explain the difference in capacitively stored energy after connecting the two capacitors, so we don't need to go look for radiation as an explanation.

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I'm sure others have analyzed this carefully

This recently updated problem and solution, A Capacitor Paradox, by Kirk T. McDonald is well worth a read. I've referenced it in a few answers (for example, this one) and comments related to questions regarding the two capacitor missing energy 'paradox'. Quoting McDonald:

Indeed, for low Ohmic resistance, the current in the circuit would perform a damped oscillation with nominal angular frequency $\omega_0 \approx \sqrt{2/LC}$, and the associated electric and magnetic dipole radiation would have power well described by $P_{rm}(t)=I^2(t)R_{rad}$ where $R_{rad}$ is a constant with dimensions of electrical resistance.

So, your intuition is in agreement with McDonald's analysis.

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  • $\begingroup$ My reference to the condensers' paradox was absolutely incidental - I only meant to cite it as an instance of fallaciuos limit behaviour. I was well aware of the long discussion on the paradox, even if I'm far from having read all papers. But I know McDonald's and think your quotation doesn't give a good synthesis of it. I can't exhaust the argument in a comment. Since question is closed, I'm going to put another question. $\endgroup$
    – Elio Fabri
    Commented Oct 29, 2018 at 14:14