# SR for Moving Electrons in Wire

Let's say we have a metal conducting wire with delocalized electrons and positive metal ions. We apply a potential difference across it. Now, the electrons will start flowing in one direction due to the electric field. If we observe from the reference frame of one of the moving electrons, won't the space in the wire between the positive ions be contracted (length contraction) and the density of positive charges would be more than that of negative charges (electrons) from the electron reference frame? Wouldn't this net positive charge attract the electrons moving through the wire?

Also, isn't this what a magnetic field due to moving charges is? (electric field from a different frame of reference). Or is that different..

You're completely correct to say in the reference frame of the electrons, the charge density of positive ions increases by a factor of $$\gamma$$. However if we're in the middle of the wire, there'll be a roughly equal amount of charge either side of the electrons, so no net force overcoming the electromotive force. Even if there was some net force on the electrons from these positive ions, it would be swamped by the force from the potential difference, otherwise we would have had no current to begin with. You could maybe even argue that whatever force is pulling the electric current along (e.g. a charged plate at the end of the wire) would be closer to the electrons by the same length contraction, so this might counter your increased force from a larger positive charge density.

Your other point about magnetic fields coming from moving charges is completely correct too. In the frame of the electrons they're not moving so just produce aan $$\bf E$$-field. In the lab frame, they're moving and making a $$\bf B$$-field. Relativity handles this by unifying $$\bf E$$ and $$\bf B$$ together in a field strength tensor, $$F_{\mu\nu}$$.

• Thanks very much for answering. I have a doubt: >> there'll be a roughly equal amount of charge either side of the electrons, Wouldn't the part behind the electrons be elongated and thus reduce the charge, in fact doubling the effect?
– AVS
May 10, 2022 at 15:44
• You might be confusing length contraction with the doppler effect. Lengths always contract from their rest length. The perceived "colour" of the ions may indeed be different in front and behind, but this won't affect the forces.
– Garf
May 10, 2022 at 16:05
• Ohh.. thanks a ton, really helped me out. Appreciate it!
– AVS
May 10, 2022 at 16:09
• @AarnavSood The effect is doubled because there there are not only more protons everywhere, but also less electrons everywhere, according to the moving electron. The effect being the wire being positively charged. (This is a slow current approximation. If other electrons are already far away according to an electron, then other electrons getting even more far away has almost zero effect.) May 12, 2022 at 7:49

Wouldn't this net positive charge attract the electrons moving through the wire?

According to the electrons, yes.

According to the ions, no.

Also, isn't this what a magnetic field due to moving charges is? (electric field from a different frame of reference). Or is that different.

Those moving electrons are currents. Those currents attract each other, as we know from old textbooks about magnetism. This attraction is equivalent to the repulsive Coulomb forces between the electrons being reduced. Those repulsive Coulomb forces are the forces that are preventing the electrons from being pulled to the center of the wire by the positive charges. I encourage the reader to think what happens when the forces that are preventing the electrons from being pulled to the center of the wire decrease.

Note carefully that the pull of positive charges has not changed. (Except if we ask electrons, they would say that the pull has increased. We can explain why electrons feel that the pull has increased by the effects that the motion of the electron has on the electron's views or opinions or measurements, relativity considers just that kind off stuff all the time)