In the common configuration of the Hall Effect, why wouldn't the conductor move? It follows the same principle of the Lorentz force or basic electric motor doesn't it?
I can't seem to find the difference between them...
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There is, of course, a force on the conductor. It would move if it could. But, in the Hall experiment it is held still by mechanical forces. Actually, that is not quite true. There is a force on the conductor, but (in the usual experiments) the conductor can not move very much because it is a solid wire, attached to other solids all of which have very high elastic moduli. The electrical forces are quite small, so the motion of the conductor is negligible.