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In this ( https://www.youtube.com/watch?v=1TKSfAkWWN0 ) specific video uploaded by Veritasium on YOUTUBE , we come to know about how special relativity makes magnets work. We observe that when a moving proton is brought near a magnetic field produced by electric current, the proton is repelled. Obviously it is supposed to happen.

But suppose that the proton is not moving. So now in that proton's frame of reference , that proton and the protons of the wire are motionless whereas the electrons are moving. According to special relativity the distance between two electrons will be contracted whereas the distance between two protons will expand. So the wire becomes negatively denser which means the proton will feel an attractive force.

But we know this is not possible. Because F = qvB sin(theta) and here v=0. But according to the explanation given by Veritasium, the motionless proton should feel an attractive force. I know I'm missing a point somewhere. Can anyone please point that out for me ? I really appreciate any help you can provide.

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  • $\begingroup$ Protons do experience a net force in a non-constant magnetic field because they have a finite magnetic moment. This even happens for resting protons. $\endgroup$
    – CuriousOne
    Commented Oct 31, 2015 at 20:23
  • $\begingroup$ Aat the atomic and molecular level the wire has one quantum mechanical solution and the electrons in the conduction band are on orbitals, not trajectories, and on the average move very little with a drift velocity and the wire will remain neutral in all frames as the current electrons are replaced by the battery. $\endgroup$
    – anna v
    Commented Mar 15, 2017 at 6:46

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When you say

in that proton's frame of reference , that proton and the protons of the wire are motionless whereas the electrons are moving. According to special relativity the distance between two electrons will be contracted whereas the distance between two protons will expand. So the wire becomes negatively denser which means the proton will feel an attractive force.

you're assuming the wire is CHARGED in the proton's rest frame. Thus, in the proton's rest frame, there is not just a magnetic field. There is also an electric field, which is what does the attracting.

The case you should think about is when the proton is stationary AND there is no net charge on the wire in the protons frame. Then clearly the electric field is zero, and the proton is not attracted to the wire.

Note that in both of these cases, there is no magnetic force.

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