# Electromagnet as a consequence of special relativity

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???

• I'm not sure about this, but the electrons are always moving, even before you turned on the current. Get a neutral wire and there is no change in situation before and after current is switched on. Aug 2, 2019 at 17:30
• But when there is no current the motion of electrons is random and they move about in any direction so the effect can be assumed to be cancelled out. But when there is current there is uniform motion, I mean in any defined direction of all the electrons. Aug 2, 2019 at 17:34
• It's hard to make out the details of what you're describing, especially without the relevant diagram. However, it sounds like you're imagining that every situation involving magnetism can be reduced to an electric one by a change in the frame of reference. That's not true, and nobody claims that. The actual claim is much weaker: that if we knew about SR and electricity but not magnetism, we would be forced to invent magnetism, and that a certain mixture of E and B in one frame is a different mixture in a different frame.
– user4552
Aug 2, 2019 at 17:53
• @BenCrowell Yes your reasoning seems very satisfying. But I'm left with this doubt that what if we apply SR in the above case of the wire and charge, I mean there seems nothing wrong with applying this concept and it describes quite well the two cases when the charge is in motion in two different directions. And if we think in the same way, how can we explain this rest case in SR terms, or if we can't , then why does SR fails in this case? What is the wrong supposition or wrong frame of reference that we have chosen? Aug 3, 2019 at 5:04

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.

• Hey Dale, thanks for the explanation. I understood that you mean to say that as the electrons are not bond to each other physically and are free to change their relative separation from each other, so they adjust in such way that the charge density doesn't change and thus no force on positive charge. Can you please explain why applying a very high positive voltage or vice versa will lead the charge to be attracted or repelled? It means that if i set up very high voltage in a wire and keep an electrically charged body nearby, then the wire will magically attract it? Aug 3, 2019 at 4:54
• It is not magic, this is called self capacitance. I have updated the answer, and here is a link en.m.wikipedia.org/wiki/Capacitance#Self_capacitance
– Dale
Aug 3, 2019 at 11:07