Do we measure a voltage across a conductor moving at a constant velocity in a uniform magnetic field with a moving voltmeter?

Question

Image courtesy of @Alfred Centauri from Is there an EMF in a conductor moving at constant speed across the uniform magnetic field

This is what we observe from the rest frame. But were we in the frame of the conductor i.e. the voltmeter were attached to either end of the rod and moving with it, would it detect a voltage?

I guess even from the rest frame looking at the voltmeter we would see zero voltage, because now there is a closed loop of wire moving through the uniform magnetic field (closed by the voltmeter) at constant speed.

So then the only way we could really observe a voltage on the voltmeter would be to have a stationary (relative to us) voltmeter consisting of two rails on which the bar slides (like a rail gun) --> law of induction

Is that so? It would seem in some ways disappointing, in others interesting. Sorry for my naivite I am in no way a physicist!

Answer 1

Despite your modest disclaimer, I think you have more than a little of the physicist in you! What you say is correct. There are at least two ways of looking at why the voltmeter in the moving loop would read zero. For the sake of simplicity, let the loop, abcd, be rectangular with the direction of motion parallel to sides bc and da.

(1) Free electrons in ab and cd will experience forces urging them upwards in ab and in the same direction in cd. [These forces, arising from the electrons moving in a magnetic field (because the wire is moving) are called motor effect forces or magnetic Lorentz forces.] It's easy to calculate the so-called emfs (voltages) arising from these forces in ab and cd. But these emfs are in the opposite sense when we take a trip right round the loop, so they will cancel.

(2) If you have learned about magnetic flux, and if you know that the (total) emf in a moving circuit is equal to the rate of change of the flux through the loop, then it's obvious that there will be no emf in this case, because if the loop is moving through a uniform field the flux through it will never change. On the other hand, with your rod-sliding-on-rails set-up you will constantly be enlarging your loop or diminishing it, so the flux through it will be changing and there will be an emf.