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I know that charging a capacitor only moves particles from one plate to the other, so the total amount of charge in the capacitor does not change, nor does the total number of particles.

However, the charged capacitor does have electrical energy that the uncharged capacitor does not have, and energy has a mass equivalence according to $e=mc^2$, so wouldn't the capacitor weigh a little bit more when charged?

A 470 μF capacitor charged to 3.3 V, for instance, would have a stored energy of 2.6 mJ, which has a mass equivalence of 0.03 femtograms, which is the weight of ~1,416,666 carbon atoms?

Likewise, a superconducting loop inductor would weigh slightly more when it is energized (has a current flowing through it) vs unenergized because of the mass equivalence of the energy stored in the magnetic field?

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  • $\begingroup$ physics.stackexchange.com/a/466278/176 $\endgroup$
    – endolith
    Jun 10, 2020 at 14:46
  • $\begingroup$ why do you doubt your calculations ? The football example is not the same. It depends on the inertial frame, whereas what you calculate does not. $\endgroup$
    – anna v
    Jun 10, 2020 at 15:02
  • $\begingroup$ @annav Because of the comment that I linked to $\endgroup$
    – endolith
    Jun 10, 2020 at 15:35
  • $\begingroup$ I had read the question. only. With special relativity any stored energy in principle should show. Think of it as the difference in the individual four vectors making up the capacitor plates, adding up to a change in the invariant mass. $\endgroup$
    – anna v
    Jun 10, 2020 at 18:18

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Your reasoning is correct. If you store energy in a system, mass-energy equivalence provides the conversion for calculating the matter equivalent which, if you had a weighing device sensitive enough, you could (in principle) measure.

For things like charged capacitors and stretched springs and pressurized gas tanks and cans filled with gasoline, the mass difference is too small to measure. Even in the most energetic reactions we can create (nuclear reactors, atomic bombs, fusion) the mass equivalence of the energy release is of order ~a fraction of a percent.

The most efficient method for wringing energy out of mass occurs when mass spirals into a black hole. Theoretically, it is possible (in an idealized process) to extract up to ~40% of the mass as energy release in this case.

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