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By the Lenz's law, when a charged particle goes through a coil it generates a magnetic field. This field generates a current in the coil, slowing down the particle. But by Special Relativity (STR), the velocity of the electromagnetic interaction may not exceed $c$.

Let the electron flies between two reflectors with $2L$ distance between them. The velocity of the particle is such ($V = L c / R$) that particle flies through the coil radius $R$, reflected from the reflector and come back exactly the time $2R / c$. Then, instead of slowing down, magnetic field accelerated particle.

How can I prove that this scheme of perpetual motion would not work?

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