# Relativistic charge density in a closed loop

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

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.

• – user4552
May 2 '13 at 12:43
• They speak about charges in the corners. I think that they discuss the imbalance between dense horizontal currents and unaffected vertical ones. I would explain it using relativistic field of a moving charge, en.wikipedia.org/wiki/…, which says that fields also shrink so that one charge may carry different charge in different directions and everything is conserved. Thanks, I think that I have also resolved my question: I have forgot that not only density increased but horizontals also contracted. Very simple!
– Val
May 2 '13 at 13:28
• This runs the risk of falling into the Ehrenfest paradox. May 3 '13 at 15:01
• @EmilioPisanty: I don't see how the Ehrenfest paradox is related. The Ehrenfest paradox is about rigid rotation. This isn't rigid rotation.
– user4552
May 3 '13 at 15:07
• @BenCrowell: Why should the body must be rigid? It is originally rigid because it is difficult to rotate liquids. But, here I show that it is possible and, furthermore, contraction is impossible because we see that the "rigid walls of the pipe" are not contracted. So the question is how do you contract the liquid of charge inside a rigid pipe? The answer ought to answer whether it is exactly Ehrenfest paradox or what is the difference.
– Val
May 5 '13 at 12:05

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):

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


After reading several textbook examples relating to this matter I see how you question arises. Most textbooks simply start with a current in a (squared) ring, and assert that the ring is neutral in the lab frame. Then they view the ring from the moving frame of the testcharge.

None of the textbooks care to explain the internal workings of this trivial neutrally charged, current carrying ring. Hence your question, I presume...

By the way, someone else came with a similar question, but also did not get a fully satisfying answer. http://van.physics.illinois.edu/qa/listing.php?id=2358

You are correct to assume that the solution you suggest, ie. only moving electrons, and a neutral wire, will lead to a logical absurd.

Let me therefore suggest another option that you might embrace: What if a current is not merely electrons moving one way, but also positively charged holes the other way at the same velocity? Both would contract in the same way, and the ring could therefore stay neutral (in the lab frame only).

Another solution might be that when a current (in this case only moving electrons) starts, it will not start instantaneously (the electromagnetic field needs time to propagate) along the whole wire : The electrons that are closest to the positive pole of the battery will start moving first, and thus at first their density will be a bit lower. Like cars after a traffic congestion, their distances increase. Maybe they increase just as much as the relativistic contraction does contract, keeping the total density the same. (I've not taken the effort to do this in formula's)

Let me know what you think about it.

I was told in the paradox of conductor's neutrality and this is confirmed in Follow-Up #2: relativity and charge density that in addition to contraction, mentioned in the relativistic electromagnetism tutorials, the electrons of conduction experience an expanding force, when accelerated. That is they see that the distances between them stretch out in their proper frame, as they experience the acceleration. The expansion is natural for non-rigid bodies, as explained in the Bell's spaceship paradox.

I emphasize this since it is absolutely counter-intuitive for any layman like me, who are taught that moving train is contracted w.r.t. stationary observer. It is underspoken rigidness of the train that prevents it from expansion in its proper frame, so the train shrinks indeed in the stationary frame when accelerates. But, spacing between electrons of conduction is not tied rigidly, so, when electrons experience acceleration, they also see that the distances between them increase. The expansion factor is exactly Lorentz $\gamma$, which exactly compensates the contraction seen from the stationary station. The net effect is that distance between electrons is not changed in the lab frame. So, maintaining neutrality also eliminates the paradox of wire neutrality.

Though I still do not understand the source of the expanding force (which energy does support it?), the paradox vanished. Might be this also partially answers the Relativistic origin of magnetic field.

However, one thing is still not clear. If electrons see their chain expanded while loop contracted then how do they fit into the loop? I guess that it's a famous ladder paradox. Yet, it is in a loop now and since my question is exactly about this case, I would like to resolve it also.

• I was told... that the electrons of conduction experience an expanding force - I was never saying that. I was just saying that electrons lack experiencing any contracting force, which would contract an accelerated rigid body. Besides that, they experience only an accelerating force. The expansion is natural for non-rigid bodies - nothing is natural for non-rigid bodies, and strictly speaking, they are not bodies at all - just bunches of particles. May 11 '13 at 12:08
• if the loop has contracted from the electrons' point of view, then, how does the expanded chain of electrons fit into it? - From the (one!) electron's point of view, the chain of electrons is not expanded uniformly. Some parts of it are expanded, and some are contracted even more, because electrons from the other side of the loop move with the speed $2v$ with respect to the observing electron. May 11 '13 at 12:10
• 1. You are right. I am not accurate. I was not sure whether it is train that experiences contraction or these are electrons, who experience expansion. 2. You explain the electronic balance. That was another topic. Here, I wanted to know how contracted volume of electrons fits the wire (expanded) space. The fact that electric charge density is balanced by another side (btw incompletly, because we are in a moving frame), does not explain how the ladder of electrons fits into garage of the stationary nuclei.
– Val
May 11 '13 at 12:57
• To any questions about "how it fits" (be it length, volume or what). Learn SR geometry. The lengths, volumes and such are relative quantities, and thus "unphysical", in sense they do not depict some reality. They only depict some view on reality. And the reality itself consists of space-time geometric figures and quantities: 4-volume, 3-cross-section of such volume, 3-surface, 4-length (called interval), and so on. What fits and what doesn't - is decided by means of these concepts. A particle becomes a world-line. A linear body - a world-plane or sheet. Simultaneity change solves many prbl May 11 '13 at 13:24
• I know that simultaneity can resolve the ladder paradox. But, as this problem of electron acceleration demonstrates, it is insufficient to analyze the worldlines to draw them. Before analyzing, I had to draw them in the first place. And wikipedia nor feynman nor anybody do explain why they draw them that way. Quite the contrary. Looking at the dozens of spacetime analyses I was drawn into delusion that everything boosted into another frame is seen contracted in the original frame.
– Val
May 11 '13 at 14:52

The problem of charge conservation is much simpler than stated, if you consider that current always needs a closed circuit and that the electron gas is compressible. Firstly you will note that all charge disparities in a current carrying wire observed by a moving observer belong to an electric dipole field like that of a rectangular circuit where its upper section might charge up positive and its opposed lower section negative summing up to Zero total charge difference. As to the Lorentz-contraction of an electron gas drifting truogh a current carrying ring it is obvious that the solid state structure of the ring is capable to inhibit that Lorentz contraction mechanically, which however results in a very small mechanical compressive stress appearing within such ring. If you look at the equations of state of an electron gas you can calculate the amont of stress needed to inhibit Lorenz-contraction. Of course the Lorentz-contraction of any elastic body can be inhibited by a counter-acting mechanical stress as given by Hookes Law.

We have to be careful using our classical intuition when the problem isn't classical in nature. The electrons in the conductor are in delocalized states, that are distributed over the conductor. In this picture, it makes no sense to speak about the distance between the electrons. To find the charge in a given region, we should instead sum the probability distribution of each electron for this region. In the rest frame of the conductor the electrons are at rest in average. They have an angular momentum, but their center of mass doesn't move. In the wire electrons are circulation in both directions, even when there is no current. A current arises when more electrons are in states circulation in one direction than the other. The situation is much as an electron in an atom. I think this is a better picture to hold in mind.

The question then is, how these electrons states transform when viewed in a different frame? Unfortunately, I have not yet found a proper analysis of this. But we know for sure that an electrical force will occur in the rest frame of a charge, that moves through a magnetic field. I believe, however, it is mistaken to think this force arises due to length contraction alone. It is an electrodynamic effect, and not an electrostatic. See my questions about this:

Is magnetism an electrostatic or electrodynamic effect in the rest frame of the affected charge?

How do you model the electric field of an electron in motion in a conductor? Is the field given by static or dynamic equations?

I think your question is very relevant.

I think I can put it in a wider perspective.

In Minkowski spacetime when you close a loop interesting things happen.

I believe the ring configuration that you point out exposes a severe weakness in the usual exposition of magnetism-as-relativistic-side-effect-of-the-coulomb-force.

I will abbreviate this 'Relativistic Charge Density Contraction' as 'RCDC'.

The RCDC explanation does not pay attention to relativity of simultaneity. That is a perilous simplification. It may be that the RCDC explanation contains errors that fortuitously drop away against each other. While the RCDC explanation produces the sought after result, it may nevertheless be unsound.

About loop geometry in special relativity:
I find the article Sagnac effect, twin paradox and Ehrenfest paradox by the physicist Olaf Wucknitz very interesting. Wucknitz explores what you get when you close a loop in Minkowski spacetime. It may help you understand why the ring configuration that you point out poses a challenge to the RCDC explanation.

• This has nothing to do with the question. The Ehrenfest paradox has nothing to do with the question.
– user4552
May 6 '13 at 23:32
• @BenCrowell Your accusation is unfounded. What I stated in my answer is that the ring configuration that is pointed out in the question is a case of loop geometry in Minkowski spacetime. I pointed out the article by Olaf Wucknitz because of the theme of loop geometry. The article by Olaf Wucknitz is a general discussion, and among other things it also touches on the Ehrenfest paradox, but the theme is the case of a closed loop. My response mentions 'ring configuration', and *not*> 'Ehrenfest paradox'. I agree of course that by itself the Ehrenfest paradox is unrelated. May 7 '13 at 21:10