I thought I had the conceptual understanding of how magnetism between a charged particle and a current carrying wire can be seen from the point of view of relativistic effects, but after revisiting the concept, I got conceptually stuck.
The easiest way to show my issue is to reference this video (between 1:20 and 2:30): https://www.youtube.com/watch?v=1TKSfAkWWN0
The argument goes that in the reference frame shared by us and the stationary particle (stationary with respect to the wire and the positive charges in it), the positive charges in the wire are stationary, while the negative charges drift along with some velocity. The claim is that the two types of charge have similar charge densities, so the attraction and repulsion effects cancel, and we perceive no force on the test particle.
Next we have the test particle start moving with the same velocity v, so that now it's the negative charges that look stationary, and the positive charges are "moving". Because of relativstic effects of length contraction, the positive charges seem to get closer, while the negative charges spread out to their "true" separation as seen from their reference frame. The net effect is that the positive charge density is now greater, so the test particle will feel a force.
The story makes sense expect the first scenario has the negative charges moving with respect to the "stationary" test particle. So these charges need to have their length contracted relative to what we'd see if they were not moving relative to the test particle/charge.
!!! So my central issue is with the claim at the beginning that the positive and negative charge densities are the same and would cancel for the "stationary" particle. !!!
The way I see it, the test particle will see the moving negative charges as contracted and the positive stationary ones as not contracted, resulting in a higher charge density for the negative current. That forces me to conclude the positive test charge will then be attracted toward the wire.
I END UP WITH THREE SCENARIOS (Which I know are wrong):
First, we have a wire with positive charges and negative charges, and a test particle/charge. There's no current yet, so the positives and negatives in the wire are not contracted, and so have the same charge density. Nothing happens. Then we flip a switch and cause a current to flow.
- Test particle is stationary, sees stationary positive charges and moving + contracted negative charges. The negative charge density is greater, so the test charge feels a force toward the wire.
- Test particle moves in the direction of the negative current, but with half the velocity of the current. This setup results in the test particle seeing both the negative and positive charges in the wire appearing to move away from the test particle with the same velocity but opposite directions. So both would be equally contracted in distance, resulting in equal charge densities, leading to no force on the test particle.
- Test particle moves along at the same velocity as the negative charge, so this charge is not contracted, but now the positive charges in the wire are contracted and as much so as the negative charges were in the first scenario. So now the positive charge density is greater, and the particle feels a force away from the wire.
This is obviously NOT what occurs in reality, so I've reasoned myself into a corner...
Thank you for any suggestions!