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In Purcell's Electricity and Magnetism, Section 7.2, he talks about the field of a conducting rod moving through a uniform B-field. He then goes on to say the E-field and B-Field in the reference frame moving with the rod is related by E'=v x B'. In the previous chapter he mentioned that in the special case where in one reference frame E=0, then the field in the moving reference frame can be given by E'=v x B'. But, the problem I am having is that in the frame where the rod is moving with velocity v, the electric field is not zero because the magnetic field causes the charges to separate in the rod and hence an electric field appears. So why is it that we can use E'=v x B'?

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It's just an opinion, but if you are in the rod reference frame your velocity say $v$ is zero, and therefore there's no electric field generated in the rod.

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