Wire with current neutrally charged? 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?
 A: 
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. 

The analysis given by Veritasium originated from Purcell. My favorite presentation on the topic is here: http://physics.weber.edu/schroeder/mrr/MRRtalk.html
Unfortunately, your question is the most common confusion from Purcell’s idea and it is not addressed. 
The confusion comes from the natural assumption that the electrons form a rigid body with a defined and fixed proper distance between electrons. This is not the case. The proper distance between electrons is highly variable and may be larger or smaller based on the EM fields and other charges in the environment. 
In practice the spacing between electrons is controlled by the voltage on the wire and the self capacitance of the wire. A strong positive voltage on the wire produces a low density of electrons in the lab frame, a neutral voltage produces a density of electrons equal to the proton density in the lab frame, and a strong negative voltage produces a high density of electrons in the lab frame. 
Once the density is determined in the lab frame, then the proper distance can be determined. The normal rules of relativity apply, so the density in the electron’s frame will be lower than in the lab frame. So the lab frame spacing undergoes length contraction, as normal, but from a proper distance that produces the observed charge density in the lab frame. 
