Magnetic forces in a rotating frame

[Xynon]

Imagine we had two line currents (ignoring the rest of the circuit, return paths and the voltage source) of same direction and rotating in a common x axis with equal angular velocities. Assume that the distance between them is much larger than the wire lengths.

 

If we were sitting on a wire and rotating with it, in the S’ frame they would appear stationary to each other so there is no time delay and they would be exerting the full magnetic force on each other.

But in the S frame, they would look rotating and their EM fields arrive at each other with a phase delay, which means each one should be “seeing” the other as slightly backwards in time (and angular position). There should be a little less magnetic force on each other due to the slight angular difference/delay between them.

Or amI ill-defining the S' frame as keeping it's orientation with the wire? Maybe the wires would be looking as rotating around S' frame (around a point on a wire)? Then in both S and S' frames, there would be a decrease in the magnetic force, once the wires started rotating.

I would be grateful if anyone helps.

Edited May 22, 2017 by Xynon

[swansont]

Imagine we had two line currents (ignoring the rest of the circuit, return paths and the voltage source)

Can you do this?

But in the S frame, they would look rotating and their EM fields arrive at each other with a phase delay, which means each one should be “seeing” the other as slightly backwards in time (and angular position). There should be a little less magnetic force on each other due to the slight angular difference/delay between them.

What is the field due to the rotation? The current flow will look different than in S'. The rotation looks like a circular current, which gives you an axial field. That part will look static.

[KipIngram]

Can you do this?

 

What is the field due to the rotation? The current flow will look different than in S'. The rotation looks like a circular current, which gives you an axial field. That part will look static.

 

I'm not sure - that definitely something to be careful about in problems like this. I could imagine the return circuit as comprised of wire segments that are perpendicular to the ones shown, running away from the other wire (I'll call those perpendicular returns) and then a piece parallel to the wires shown (I'll call that the parallel return. In the static reference frame the Biot Savart law says that the perpendicular returns make fields lines in the plane of rotation of the shown wires, so the shown wires wouldn't cut those lines. The paralle return will make a field that gets cut, but we could put those as far away as we wished.

 

In the other frame, where the shown wires are rotating, I'm not as sure that the perpendicular return fields won't matter - the same reasoning the OP uses (i.e., the distance causes a time delay such that you're seeing a field corresponding to an earlier position) applies here too. I'd be very hesitant to neglect that contribution without working out the details more rigorously.

[swansont]

I'm not sure - that definitely something to be careful about in problems like this. I could imagine the return circuit as comprised of wire segments that are perpendicular to the ones shown, running away from the other wire (I'll call those perpendicular returns) and then a piece parallel to the wires shown (I'll call that the parallel return. In the static reference frame the Biot Savart law says that the perpendicular returns make fields lines in the plane of rotation of the shown wires, so the shown wires wouldn't cut those lines. The paralle return will make a field that gets cut, but we could put those as far away as we wished.

 

In the other frame, where the shown wires are rotating, I'm not as sure that the perpendicular return fields won't matter - the same reasoning the OP uses (i.e., the distance causes a time delay such that you're seeing a field corresponding to an earlier position) applies here too. I'd be very hesitant to neglect that contribution without working out the details more rigorously.

 

Right. A wire segment with current flowing violates some basic physics, and you have to be sure that this violation doesn't have unintended consequences.

[KipIngram]

Also, if an observer sees only a magnetic field in one reference frame, won't they see both a magnetic field and an electric field in a frame that's moving wrt the first one? Maybe that doesn't matter - maybe there's no net charge in the line segment anyway, but there are just a lot of details to make sure you've gotten right here. Especially if you're trying to show the breakdown of a very fundamental physical principle.