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I agree with Nick that the situation can be turned into a paradox. One
person, "she", observes no emf accross the rod while the other, "he", does
observe it. He (the observer for whom v is not zero) strongly believes
that + and - charges are separated. He concludes that the VOLUME DENSITY
of free electrons is excessive at one end and deficient at the other. But
she (the observer sitting on the rod) does not detect any emf. Thus she
concludes that the volume densities of free electrons are identical at
two ends. ...