This has been answered a lot of times, and I have been reading about it on many websites, including PSE, but I still don't get it. So please don't mark it as a duplicate unless it really is one.
Two parallel wires at rest don't attract or repel each other because both of them have roughly the same amount of electrons and protons, so their net charges are mostly zero.
If there's a current (in both wires, same direction), the attraction between the wires can be calculated
- using Ampère's force law
- using the Maxwell equations (the current creates a magnetic field)
- using Special Relativity
I want to understand the third case.
Take the reference frame of the moving electrons. Due to relativistic length contraction, the protons get closer together, so the number of protons per length unit is higher than that of the electrons. So far so good.
At this point, all answers to this question just go like
Because wire 2 has more protons than electrons (per length unit), the electrons in wire 1 are attracted to wire 2, so the two wires attract each other.
But wire 1 has more protons too! Both wires become effectively positively charged due to the movement of the electrons. Because both wires are positively charged, they should repel each other.
Yes the electrons in wire 1 are attracted to wire 2 because of wire 2's net positive charge. But the protons in wire 1 are repelled from wire 2 because of the same reason. And there are more protons (per unit length) in wire 1, so the repellent force is stronger than the attraction.
So, again: Why do the wires attract each other?
EDIT: Maybe it has to do with the fact that the electrons can "re-arrange" themselves to counterbalance the positive charge? But then both wires would again have zero net charge.
EDIT 2: Here are some PSE posts and websites that try to answer this question:
However, as I said, they don't go into my interpretation that both wires should have the same net charge and thus should repel each other.