I Question about special relativity and magnetism

[curiousP]

TL;DR

Connection between special relativity and magnetism

I’ve watched many YouTube videos explaining how length contraction from special relativity (SR) accounts for magnetism. But there’s one point I still don’t understand. All of the videos assume that, in the lab frame, positive and negative charges move in opposite directions with equal speed. This supposedly causes equal length contraction for both, resulting in no net electric field for an external charge.

But isn’t it the case that, in the lab frame, the positive charges (in the nuclei) are stationary while only the electrons are moving? If that’s true, then only the electrons should undergo length contraction—leading to a higher linear charge density for them and thus creating a net electric field that affects a stationary charge outside the wire. But there should be no magnetic force on that charge at rest.

I’d appreciate any help in clearing this up. Thank you. Here are links to some of those videos:

 

[Ibix]

All of the videos assume that, in the lab frame, positive and negative charges move in opposite directions with equal speed.

No, they assume that the net charge is zero in the lab frame. So the spacing between electrons must be equal to the spacing between protons in this frame. That isn't because they're moving with opposite velocities, it's just the way the experiment is set up.

Because they have different velocities, though, you get different length contraction effects for the two species when you change frames, so in other frames you get different charge distribution around the circuit and an electric field arises.

 

[Dale]

TL;DR: Connection between special relativity and magnetism

But isn’t it the case that, in the lab frame, the positive charges (in the nuclei) are stationary while only the electrons are moving?

Yes, that is the case. I think you misunderstood the videos, but you correctly understood the physics.

leading to a higher linear charge density for them

The fact that the wire is uncharged in the lab frame is a given. The experimenter is free to charge the wire or not, and we are given the fact that they did not.

Maybe an analogy will help: an experimenter could (in principle) build a train to any length desired. So you could have a situation where you are given that a moving train and a stationary train are the same length in the lab frame. Only the moving train would experience length contraction, but we are given that the length contraction is such that the lengths are equal in the lab frame.

 

[PeroK]

TL;DR: Connection between special relativity and magnetism

I’ve watched many YouTube videos explaining how length contraction from special relativity (SR) accounts for magnetism.

Why isn't one video enough? Maybe two at most.

But there’s one point I still don’t understand. All of the videos assume that, in the lab frame, positive and negative charges move in opposite directions with equal speed. This supposedly causes equal length contraction for both, resulting in no net electric field for an external charge.

But isn’t it the case that, in the lab frame, the positive charges (in the nuclei) are stationary while only the electrons are moving? If that’s true, then only the electrons should undergo length contraction—leading to a higher linear charge density for them and thus creating a net electric field that affects a stationary charge outside the wire. But there should be no magnetic force on that charge at rest.

There are an equal number of positive and negative charges in the wire to start with, and motion of the electrons cannot change that overall number. After the electrons have started moving, they must have become further apart in their own rest frame. This is necessary to maintain neutrality of the wire in the lab frame. The wire cannot have an overall negative charge in the lab frame. Therefore, the spacing between electrons must remain the same in the lab frame when the current is flowing and when it is not flowing.

 

[A.T.]

TL;DR: Connection between special relativity and magnetism

But isn’t it the case that, in the lab frame, the positive charges (in the nuclei) are stationary while only the electrons are moving? If that’s true, then only the electrons should undergo length contraction

Yes

leading to a higher linear charge density for them

No. Just because in the lab frame the electrons (their fields) are contracting, doesn't mean that their spacing is contracting, when the current starts. They are still repelling each other, and thus stay as far as possible apart. This implies that they get further apart in their rest frame, as @PeroK writes above.

Here is a good explanation and diagram by @DrGreg :
https://www.physicsforums.com/threads/explanation-of-em-fields-using-sr.714635/post-4528480

Your misconception is very common, and those videos / texts should expect and preempt it, by explicitly addressing and clarifying it. You can find many previous threads on this by searching mentions of @DrGreg by me:
https://www.physicsforums.com/search/22203114/?q=DrGreg&c[users]=A.T.&o=relevance

 

[SiennaTheGr8]

TL;DR: Connection between special relativity and magnetism

I’ve watched many YouTube videos explaining how length contraction from special relativity (SR) accounts for magnetism.

I think it's worth mentioning that this statement isn't true. The "electricity + length-contraction = magnetism" concept works for some special cases, and you can use it to show that magnetism must exist (as Purcell famously does in the textbook Electricity and Magnetism), but you can't use it to "account for" magnetism in general. The magnetic field is no less "real" or "fundamental" than the electric field is. In fact, there are scenarios where changing reference frames can completely eliminate the electric field but not the magnetic field. And with light, neither the electric nor the magnetic field can be eliminated by boosting to another frame.