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There are many videos explaining how special relativity causes magnetism, with the most relevant part being that from the reference frame of the electrons, the positively charged ions in the wire have gotten closer together, and the negatively charged electrons have spread apart, a non-zero electric charge in a wire. We can determine that this must happen using the fact that in a lab reference frame, the wire is electrically neutral, and applying a Lorentz transformation to the electrons and ions. However, the electrons must see a different mechanism causing the space between them to increase. They need to see something pushing them apart. They don't know that the wire needs be electrically neutral in another reference frame. What's this mechanism that they see? Does it have anything to do with the period of time when they're accelerating?

I saw a bunch of questions that seemed similar to this one, but I didn't find any of the answers to be satisfactory. They all seemed to just explain how we know that the wire must be charged in different reference frames, using the Lorentz tranformation.

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In the accelerating frame of an electron accelerating in an uniform electric field there is:

  1. a Coulomb-force
  2. an inertial force (not real force)

Those two cancel out, so that the electron does not accelerate in the accelerating frame.

But some other electron accelerates in said frame. The force that causes the acceleration must be the sum of the Coulomb-force on that electron and the inertial force on that electron.

Now if we assume that the Coulomb-force on all electrons is the same (according to electrons), then we know that according to some electron electrons ahead of it have less inertia and electrons behind of it have more inertia.

The location dependent inertia is related to location dependent speed of light, which can be demonstrated in an accelerating rocket by measuring the two way speed of light of middle-rear-middle path and middle-front-middle path, the latter path will be faster).

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  • $\begingroup$ Can you point me to a source on location-dependent inertia so I can look into it more? $\endgroup$
    – zucculent
    Oct 27, 2021 at 22:57
  • $\begingroup$ @zucculent I managed to find this: physicsforums.com/threads/… It maybe says that there is no such dependency - have to read it myself. $\endgroup$
    – stuffu
    Nov 21, 2021 at 9:16
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I don't think there's any local explanation of this. In principle you can add/remove charge carriers from the whole circuit, leaving it with a net electric charge, and the remaining carriers will still be evenly spaced in the steady state, so it isn't any particular density in any frame that they prefer, but just a lack of inhomogeneity. The way in which inhomogeneities disappear is complicated, with transient voltage spikes bouncing around the circuit until they thermalize.

If you imagine a current being spun up in a circular wire by a highly symmetric emf, so that there are never any inhomogeneities, then the local explanation in any small part of the wire would be that the electrons in front are accelerated slightly sooner than the electrons in back (relativity of simultaneity).

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The Lorentz transformation does not model some mechanism actually increasing or decreasing distances - it is just the expression of a symmetry in the laws of physics, where there is no absolute time and no abolute space, and so in our case here no absolute measure of distances.

Meaning there is no "one true" distance between the electrons, that the electrons in their own frame would have to consider and use as a reference in order to decide whether they have been pushed apart (which is exactly what you hint by saying that the electrons do not need to know that there is a frame of reference where the wire is neutral).

The electrons see whatever spatial distances they see in their own frame, and the lab observer sees its own corresponding distances. The Lorentz transformation just connects the two points of view.

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