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So,

I understand relativity makes speedy electrons in a coil produce what we call a magnetic field by apprearing closer together via length contraction. My question is basic and stems from that; if we had a circular superconductor and using some external magnetic force speed up the electrons inside of it, what would the effect be?

My first train of thought was the fact length contraction only happens in the direction of motion, which in the case of electrons in a circular path would be every direction. So if the electrons appear closer together, does the ring appear smaller? What would this look like?

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  • $\begingroup$ The drift velocity of the electrons isn't speedy at all. A snail crawls faster. It's just that the electromagnetic force is so strong, even that minuscule length contraction is sufficient to create noticeable effects at macroscopic scale. $\endgroup$ – Phil Frost Jun 29 '18 at 11:42
  • $\begingroup$ I say speedy sparingly, as it's all relative (ha-ha). As well thank you for the insight, I hadn't thought of the strength of EM. In regards to the situation mentioned, what do effects do you think we'd notice? I am at a loss as to what length contraction would do to a circular path. $\endgroup$ – Seth Weinstein Jun 29 '18 at 14:46
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There's a similar paradox called the Ehrenfest paradox, which is basically the same question but with a spinning disk instead of electrons. Of course we'd like this disk to fit some definition of "rigid".

It's not one of those paradoxes where it turns out you were wrong, and leave with a deeper understanding. Instead the paradox ends with:

Those kinds of rigid bodies aren't possible, sorry ¯\_(ツ)_/¯

It's pretty unsatisfying, but I feel OK about considering rigid bodies weren't physically realizable anyhow.

I'm going to make a guess here, and say you are thinking as the electrons accelerate the space between them must decrease as if they were a train.

Consider: if the average distance between the electrons contracted as their drift velocity increased, there would have to be more of them per length of wire. The unmoving protons suffer no such contraction, and so the wire (which could be a simple electromagnet) would acquire a net negative charge.

Of course it's demonstrable by simple experiment that an electromagnet does not acquire an electric charge, at least in the laboratory frame of reference.

It seemed odd the first time I thought about it, but it's really no different than asking 10 people spread over 100 meters to start running, then asking them individually to speed up or slow down to maintain 10 runners in 100 meters from your stationary reference frame. In the case of moving electrons, the feedback comes from their desire to maintain equilibrium with the protons.

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