Electromagnet as a consequence of special relativity

Question

We all have been given examples and the visualisations that electricity and magnetism are the same thing viewed from different references and that one can be transformed into the other. In relation to this view, I have seen in many places that the electromagnet is described in terms of special relativity. It says that, suppose there is a metal wire, it has tons of positive nuclei surrounded by a lot of negative free electrons(the wire is metallic). Suppose a positive charge is brought nearby, from common sense there will not be any force on positive charge since the wire is over neutral.

Now suppose a current is set up in the horizontal wire in the left direction, still there is no force on our positive charge(experimentally it can easily be seen). But now the positive charge starts to move in the right direction, and as we know that a current carrying wire applies a force to moving charges but not stationary charges. Now a very simple explanation for this phenomenon is by assuming that a magnetic field is set up due to current in the wire and that applies a force on the moving charge. And according to Fleming's left-hand rule the positive charge will be repelled from the wire. But if we think it in the relatively sense, we are given the explanation that when the positive charge is in motion, special relativity comes into play(even though the results are tiny due to small velocity) and from charge's frame of reference the protons(positive charged nuclei) are in left direction motion with relative velocity greater than that of elections in reference of the positive charge, so they will be ever so slightly more compressed than the electrons, and thus there will be a net increase of positive charge density in the wire and as a consequence the positive charge is said to be repelled and that is exactly what we get from experiment. Also if now the current is set up in right direction and same way the positive charge is given velocity at right , from Fleming's left-hand rule we find now the charge is attracted towards the wire. So this time from relativity point of view the electrons in the wire are moving in leftdirection to cause a current in right direction , and for the charge moving in right direction the relative velocity of electrons is more that that of protons, the electrons will appear to the charge to be more compressed than the protons and thus there will be a net increase in the negative charge density in the wire and the positive charge will be attracted, which is what experiment concludes.

Now my question is very simple! The moment the current is set up in the wire, and the positive charge is at rest. From experiments we see that there is no force on the positive charge whatsoever. But can't it be imagined that in this case only electons are moving with respect to the charge and so they will be slightly appear to be compressed and so there should be a net increase in the negative charge density in the wire and thus the positive charge should be attracted. But this does not happen experimentally.

Can anyone kindly explain why this relativistic explanation of magnetism fails here, or is there a way to correctly apply that even in this case so that the theory of relatively comes in agreement with the observed phenomenon???

Answer 1

But can't it be imagined that in this case only electons are moving with respect to the charge and so they will be slightly appear to be compressed and so there should be a net increase in the negative charge density in the wire

Yes, you are exactly correct here. They are in fact slightly compressed and with a resulting increased charge density compared to the electron’s rest frame. So the question is what is the charge density in the electron’s rest frame?

Naively you might think that the charge density of the electrons in the electron’s frame is the same as the charge density of the protons in the proton’s rest frame. However, there is no physical reason why that should be the case, the electrons are not rigidly bound to each other at a fixed distance, and in fact their proper separation is easy to change, it is essentially a free parameter which can be adjusted to meet the conditions of the scenario.

So what is the relevant condition? It is the fact that the wire is uncharged in the proton’s frame. The separation in the electron’s frame is the distance necessary so that the wire is uncharged in the proton’s frame. Thus, when the separation is contracted and the charge density increased compared to the electron’s frame, the force on the external proton at rest remains 0, in accordance with observation.

It may seem bizarre that the charge density in the electron’s rest frame just happens to be the right amount to have 0 net charge in the wire frame. Why not some other value than 0? It turns out that this is not a general fact of nature, but is under experimental control. If the wire is kept at a high positive voltage then due to its self capacitance the net charge will be greater than 0, and vice versa if the wire is at a strong negative voltage.