[Physics] Relativistic charge density in a closed loop

electricityelectromagnetismspecial-relativity

When charges of conductance are at rest, there is an average distance between them. The relativistic origin of magnetic field says that distances between electrons shrink when they are set into a motion (we get current). This means that the electron density is increased while total charge is conserved. I just try to relate the contracted distance between electrons with the conductor, which stands still and does not contract.

If we had infinite-long wire, we could borrow any amount of charge from the edges of Hotel Infinity to increase the electron density without increasing it. But, real wires are of limited length. So, condensing charge -*--*--*--*--*--*- => -----*-*-*-*-*--- cannot make one segment more dense without depleting density in the other parts. I cannot make the whole wire more charged (and keep it neutral in the lab frame, at that). The same applies to the loop. 

 * - * - * - *      
 |           |      * * * *
 *           * ==>  *     *
 |           |      * * * *
 * - * - * - *

Here alternative physics draw a nicer picture to ask the same question
enter image description here

The image says that the loop of electrons shrinks as they start to flow. Yet, mechanically, electrons do not escape the solid wires, which stand still and do not contract!

That is the question: how do inter-electron distances shrink without leading to logical absurd?

This question stems from understanding why loop with current stays neutral, despite the electron density increase. I first guessed that the conductor stays neutral because positive charges flow symmetrically in opposite direction so that relativistic charge density increase in both directions compensate each other and keep the loop neutral. In this case the conducting loop would contract proportionally with the loop of electrons but it would not stay at rest in this case. Yet, the problem is that the loop of wire stays at rest and does not contract.

Best Answer

Distances are shrunk, but not in all directions equally! One should read about addition of relativistic velocities for non-parallel directions.

In the case of a linear wire, in which the electrons move in the same direction (and even same speed) as the test charge things are simplified a lot so that equations are relative easy.

However for non-parallel (or non equal) velocities things are just that bit different.

Your loop will look somewhat like this (depending on the orientation and position of the loop with regard to the velocity vector of the testcharge):

--+-+-++
-      +
-      +
-      +
-      +
--+-+-++
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