Need clarity about relativity/magnetism explanation

Question

Please bear with me - this is not a duplicate question....

From what limited knowledge I have gathered, it seems that when two wires in parallel are carrying equal current, the electrons moving in them have a frame of reference where the stationary wires and protons in the wires are contracted due to relativistic contraction.And that this contraction causes a varying of charge density that causes the two wires to experience an attraction

Here is a picture I drew - The two boxes are sections of a long wire and I am only considering that section - if any electrons seem to exist outside the box, its because they are not in that section we are examining

Let's also dispense with the minute details of whether electrons actually move or that drift velocities are varying or that direction of movement is not exactly along the ideal straight wire. This is a thought experiment.

So here is the deal - from the wires viewpoint, the train of electrons moving w.r.t. to them are contracted. From the electrons frame of reference, the wires along with the protons in them are contracted When there is no current nothing is contracted from either frame of reference.

So obviously relativity causes the apparent charge density in the wire to seem different. But I fail to grasp how this can result in an attraction. If there are more protons per unit length in the other wire from the viewpoint of a moving electron, there are also the same number in the wire its moving through. So the electrostatic force should be balanced out - if anything "this" wires protons are much closer by an order of magnitude, so the "other" wires protons should barely have an effect

I can't get why the "other" wire seems to be different from "this" wire for an electron

Or have I got the whole explanation wrong?

Answer 1

If in one frame you have equal steady positive charges densities in the two wires and equal steady negative charge densities in the two wires but the negative charges are moving at a steady speed in the same direction in the two wires.

Then, in the frame of the negative charges, the two wires are positively charged. So yes, the negative charges feel a force from the net positive charge in their own wire and from the farther away wire. But in their own wire for each bit of net positive charge in front there is one equally far behind it that generates a force that totally and perfectly cancels it.

Whereas the forces from the other wire always have a component pointing to the other wire. The charge directly across pulls direct across the one in front of that and the one equally far behind that both have a component pulling directly across that add and the parts pulling back or forwards cancel.

It's just about adding forces as vectors.