# Relativistic effect in a current carrying wire

In many places I read that in the rest frame of a current carrying wire (electrons in motion), the spacing between the positive ions is length contracted and equal to the length contracted spacing between electrons, and the wire is said to be neutral. Cannot understand why. Why is the spacing between the positive ions length contracted too? For me, if there’s no current, both spacings are the same. And if there’s a current, then - the spacing between the electrons is length contracted in the rest frame of the positive ions and - the spacing between the positive ions is length contracted in the rest frame of the electrons.

It should be symmetric, no?

• In many places I read that in the rest frame of a current carrying wire (electrons in motion), the spacing between the positive ions is length contracted and equal to the length contracted spacing between electrons, and the wire is said to be neutral. This is just wrong. Either you're misreading the statements or these are statements by people who don't know what they're talking about.
– user4552
Feb 18, 2019 at 15:03
• For instance here :web.hep.uiuc.edu/home/g-gollin/relativity/… Feb 18, 2019 at 16:04

## 1 Answer

In the rest frame of the wire, the positive ions are not moving, but the electrons are moving. You can think of the wire as an infinite rod-shaped region of a certain negative charge density (i.e. with a certain number of electrons per meter) superimposed on an infinite rod-shaped region of positive charge density. There are two important bits here:

1) Since the electrons are moving, the rod of electron density is length-contracted in the wire's rest frame. Length contraction of an infinite rod increases the apparent density of the rod.

2) The wire is assumed to be electrically neutral in its rest frame. This means that the positive charge density is equal to the length-contracted negative charge density.

Now we transition to a frame where the electrons are at rest (meaning that the positive ions are moving). Two things happen during this transition:

1) The electrons aren't moving in this frame, so the rod of negative charge density is no longer experiencing length contraction. This means that the apparent negative charge linear density decreases relative to the wire's rest frame.

2) The positive ions are moving in this frame, so the rod of positive charge density is experiencing length contraction. This means that the apparent positive charge linear density increases relative to the wire's rest frame.

Both of these effects work together to give the wire an apparent net positive charge. Ultimately, the thing that breaks the symmetry between frames is the assumption that the wire is electrically neutral in one of them.

• OK I follow you at each step but one. This one : The wire is assumed to be electrically neutral in its rest frame. Feb 18, 2019 at 16:05
• @Anarchasis That assumption is supported by observations that a current-carrying wire at rest does not attract or repel an electrically charged object at rest. Feb 18, 2019 at 16:09
• The wire is already electrically neutral with no length contraction when the electrons are NOT in motion with respect to the wire. If they are in motion with respect to the wire, how can it continue to be neutral ?? Feb 18, 2019 at 16:16
• @probably_someone there are no such observations, there are observations to the contrary: wires do attract electrically charged objects. However, this is due to electrostatic induction, not necesarrily because the wire is charged (which it could be). Feb 18, 2019 at 18:44
• @Anarchasis the neutrality is just a simplifying assumption that is not strictly obeyed in reality, because there can and usually are some charges on the surface of the wires that push the current through the metal. However, for a given current, the better the conductivity, the lower the amount of charges on the surface needed, so in the limit of ideal conductor, none are needed and we can have electrically neutral piece of current-carrying wire. Feb 18, 2019 at 18:47