In this video How Special Relativity Makes Magnets Work Veritasium explains phenomenologically how the Electric and Magnetic force are really just the flipside of the same coin, connected via special relativity. Now I just have a quick question concerning his statement about the wire that is neutral while a current flows through it.
Essentially a wire (not moving) with no current is neutral, since the negative and positive charges balance each other. Now lets say a current is switched on, leading to a flow of negative charges with some velocity $v$. This decreases the spacing between the electrons, therefore it should increase the negative charge density, while the positive charge density stays the same.
Effectively the wire with some current should be negatively charged to the external observer, or what?