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Consider a conducting rod moving in a uniform magnetic field $\vec{B}$ with a uniform velocity $\vec{v}$. According to the theory involved, the electrons experience the magnetic force ($-e\cdot\vec{v}\times\vec{B}$) and will shift to one end of the rod, creating a positive charge on one side and negative on another.

This causes the existence of Electric Field and hence the potential energy associated with it. $\vec{B}$ does no work on the electrons or any of the charged particles. Therefore no work has been done in moving the charges to the places they have moved. But the charge separation has a potential energy which is greater than when the conductor was neutral. Therefore, it seems now that without doing any work, we have "created" energy in the form of potential energy. Where is the fallacy in this thinking?

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  • $\begingroup$ "... B⃗ does no work on the electrons or any of the charged particles. Therefore no work has been done" that is invalid reasoning. Magnetic has not done work, that's true, but the agent that made the rod move did work on the rod and the rod did work on the charges. $\endgroup$ – Ján Lalinský Jul 20 '15 at 19:10
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I would say that we're putting in work to set the rod in motion, but the argument could be made that it could already have been in motion. So, consider a rod moving at constant velocity, in vacuum in a magnetic field B, no force would have to be provided and but there would still be an electric field due to charge separation. However, unless you stop the rod, or otherwise change its motion, you would not be able to 'tap into' the potential energy of that field because that is the balanced configuration of the system (attraction between -ve and +ve charges offset by magnetic force due to B. In fact, if it weren't for the electric field, there would be unbalanced forces on the rod!) Simply connecting a wire between the two points wouldn't result in a current, because the system is already in equilibrium, so it isn't really potential energy! So without doing some external work to the rod, (either by stopping it, changing its velocity or breaking it in half to get two separately charged pieces, etc) you haven't actually created any new potential energy, although it might seem like it to you, when observed from rest.

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