Re: is induced emf a frame dependent quantity

The formula for induced EMF works if the magnetic field, B, is measured in the same frame of reference that the velocity of the rod is measured in. The (good) question you raise is what if we make measurements from the frame of the rod instead? Then the velocity of the rod is zero, so whatever the B field (whether it is frame dependent or not) it looks as if we would expect zero EMF, as you say.

However, there is a very important and surprising subtlety. Let's imagine the B field is created by a pair of electric coils above and below, in a conventional way. From the frame of the rod, these coils move during the course of the experiment. Even if the velocities involved are small, there will be an extremely tiny effect of special relativistic length contraction of the coils from the rod's perspective. Now, at first sight this should just change the shape of the coil cross-section from circular to slightly elliptical. However, if you consider the moving electrons within the coils, on one side of the coil their motion will be in the same direction as the coils are moving, and on the other side it will be in the opposite direction. In the former case the relative velocity should be a little higher, and the tiny length contraction a little bit larger than the latter case. So from the rod's point of view there will be a higher density of negative charges on one side of the coils than on the other. Thus the rod will feel an EMF due to conventional Coulomb force, which exactly matches the EMF we ascribe to magnetic induction in the lab frame.

This consequence of Special Relativity is very profound, and shows that the unification of electromagnetism achieved by Maxwell has an even deeper explanation: magnetism turns out to be just the Coulomb force manifested in a moving frame of reference. It is particularly surprising that it should be seen at low velocities, and that is a consequence of the fundamentally very strong nature of the electromagnetic force.