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The Casimir effect happens due to the difference in field on both sides of the slabs. And they are brought together by this potential difference. Doesn't that mean we extracted a work out of the vacuum fluctuations?

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The precise interpretation of the Casimir effect is somewhat controversial. Casimir's original derivation was via van der Waals forces, not vacuum energies. See "The Casimir effect and the Quantum Vacuum" by R. Jaffe for a strong proponent of the idea that the Casimir effect doesn't really have much to do with "zero point energies" at all.

The point of contention here is that the Casimir force becomes equal to the value one computes from the "vacuum fluctuation" idea only in an idealized limit where the fine structure constant $\alpha$ is large against certain other parameters so that we can take $\alpha\to\infty$, which corresponds essentially to modelling the plates as perfect conductors. The real, non-approximative Casimir force depends on both $\alpha$ and details of the materials of the conducting plates and is not equal to the value one obtains from the computation via vacuum energies (however, in practical experiments the difference is usually very small).

That the simple result you get from the vacuum energy computation cannot be the correct model of the Casimir effect can be argued without explicitly doing the computation for realistic plates: The vacuum energy result for the Casimir force does not depend on $\alpha$ or any other quantity related to electromagnetic coupling, but clearly if we turned off electromagnetic interactions we should not expect to see a Casimir effect at all (since then the plates would no longer enforce any boundary conditions on the electromagnetic field). Discontinuous results in physics, however, are always suspicious: The idea that the strength of the Casimir effect should be constant as long as the electromagnetic coupling is non-zero, however small, is rather absurd - in general an extremely small coupling should be distinguishable from no coupling only by extremely small effects.

Hence the realistic Casimir effect is just a macroscopic result of van der Waals forces and not related to zero point energies, we're not "extracting energy from the vacuum" here, it's just ordinary interactions between charges. The only version of the Casimir effect that is equivalent to doing something with vacuum energies is one between idealized conductors that doesn't happen in reality.

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Good question.

Yes, you could extract work as a "one-off" burst if you connect a spring (say) to one of the plates and use the energy "released" from the plates' attraction to store energy in that spring.

But you cannot build an engine out of it. Because the Casimir force is conservative. Once the energy stored in the spring is released, in order to recharge it, you'd have to re-separate the two plates. And you'd have to do work do that, against the very same attractive force you had exploited before. This work equals the work released in the reverse process (because it's conservative). So there is no net gain.

See here for interesting discussion. Also here for a patent that was submitted for a "vacuum fluctuation battery".

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