This is a well known though experiment, which Richard Feynman used to emphasize
- infinite wire with balanced positive and negative charges, so no net charge, but a net magnetic field
- a single charge moves parallel to wire at the same speed as the electrons
In the above experiment:
- from the wire frame, the charge feels electromagnetic force, because it is moving and there is a magnetic field
- from the single charge frame, there is still magnetic field (positive charges are moving), but the body itself is not moving, so there is no force!
The solution to this problem is length contraction: the positive charges are length contracted and the moving electrons aren't, and therefore they are denser and therefore there is an effective charge from that frame.
This is also mentioned at David Tong www.damtp.cam.ac.uk/user/tong/em/el4.pdf (archive) "David Tong: Lectures on Electromagnetism - 5. Electromagnetism and Relativity" "5.2.1 Magnetism and Relativity".