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?

• Why do you think that the spacing between electrons decrease? For example, water flows in a pipe keeping the same density. Feb 9 '20 at 21:58
• Within special relativity? I imagine it as follows: Consider the electrons evenly spaced on the x-axis at the integers when there is no current. They are all attached to a long cord. Now the cord starts moving along the x-axis and pulls the electrons i.e. we have a current. Since the cord is moving with respect to an external observer which notices the current, the distance between the electrons should decrease by lorentz-contraction. PS: In your water example, is it still the same density when it moves close to the speed of light? Feb 9 '20 at 22:30
• I mean lets consider a box a water which moves in x-direction. Since the dimensions in x-direction decrease by a factor of $1/\gamma$, the volume decreases by that amount for the same matter inside and therefore increases the density by a factor $\gamma$. However, additionally the mass of the matter increases by a factor of $\gamma$ due to it moving, so actually the mass density of water should increase by $\gamma^2$, or? Feb 9 '20 at 22:46
• The length of the wire doesn't change, (it is at rest). The number of electrons in the wire keeps the same. The charge of the electron is invariant. So the density of negative charge doesn't change. Feb 9 '20 at 23:39

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.

• Why is the density of electrons in the lab frame small when the voltage is positive and high when the voltage is negative? Afterall positive and negative voltage just changes the direction where the electrons move and the situation should be symmetric!? Feb 14 '20 at 16:00
• @Diger When the voltage is positive then the wire has a positive net charge. Since the distance between the positive charges is fixed, a net positive charge means there is a low density of negative charges. When the voltage is negative then the wire has a negative net charge and hence a high density of negative charges. I am not sure why you think the two situations should be symmetric. Can you explain?
– Dale
Feb 14 '20 at 16:50
• Maybe I missunderstand what you actually mean but "voltage". I was thinking about a wire where you apply a voltage at both ends, but from your explanation it sounds more like you talk about the voltage of the electric field that is generated by the wire due to positive/negative charges (charge imbalance experienced by the observer in the labframe) inside the wire. Feb 14 '20 at 16:54
• @Diger I am talking about applying a voltage to both ends also. I am not sure where the confusion is. If we have a wire where the left end is at +1000 V and the right end is also at +1000 V (wrt ground) then there will be no current through the wire but a net positive charge on the wire (low density of electrons).
– Dale
Feb 14 '20 at 17:00