# B How To Consistently Explain Electromagnetism With Relativity

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1. Nov 21, 2017

### Geocentricist

Superconducting Ring

In a superconducting ring does the contraction of space between electrons cause them to move inwards? Like in this animation.

Force Between Parallel Wires With Current

In the proton frame of two parallel wires with identical current, I've been told they attract and this is because the motion and length-contraction of the electrons increases the negative charge density. Is this correct? Because it seems this explanation accounts for the protons in one wire being attracted to the excess electrons in the other wire, but ignores the excess electrons being repulsed by the excess electrons in the other wire.

This animation shows a neutral wire becomes negatively charged when a current runs through it. So it's easy to see how two such wires would repel each other.

Veritasium's Video On Electromagnetism

I'm referring to this video:

Notice the separation in the electrons' rest frame at 1:17. Twelve electrons fit on the screen.

At 1:28 the electrons start moving but their separation remains the same.

At 2:08 the electrons' rest frame is again shown but the separation has increased so only 8 of them fit on the screen, and no explanation is provided for why this is.

Did Veritasium make a mistake with this video?

Another thing is at 1:28 when the electrons start moving, it seems their density should increase, and attract the cat-ion. Yet this density increase is not shown. In fact the narrator specifically says the density doesn't change.

I've posted all these questions to Reddit but response time is very, very slow there.

2. Nov 21, 2017

### A.T.

This is correct. There is no reason why the electron separation should change, when they start flowing. Even though the fields of the electrons are contracted, they are still repelling each other, so they still spread as far apart as possible within the same wire length.

Here is a good explanation by DrGreg:

3. Nov 21, 2017

### Staff: Mentor

No, there is essentially nothing correct about that animation. In superconductivity electrons are no longer spatially localized.

4. Nov 21, 2017

### Geocentricist

Okay. But what about the video showing electron spacing change while the electrons are at rest? Is that correct?

Wouldn't "as far apart as possible" be contracted?

Would you happen to have a similar graphic explaining the force between two parallel, current-carrying wires?

If the blue spheres represent electron probability clouds instead of actual electrons, would the animation be correct?

5. Nov 21, 2017

### Staff: Mentor

No, the animation is flat out wrong. The electrons are not localized, that means that their probability cloud is spread out throughout the entire superconductor. They don’t have a location, they don’t move around the wire, they don’t contract, and they certainly don’t jump off the inside of the superconductor! The animation is about as wrong as it can possibly get

6. Nov 21, 2017

### Ibix

No. But you can't get a completely coherent picture if you ignore the return leg of the current loop. If you examine the whole loop you'll find that the electron and proton numbers are equal between frames. The balance of electrons between the out and return arms turns out to be different, as DrGreg's illustration shows.

7. Nov 21, 2017

### Geocentricist

Thanks for pointing this out. I actually noticed this after I posted and edited that part of my post out, but you seem to have caught it before I did so.

Ok, thanks.

8. Nov 21, 2017

### A.T.

The number of electrons doesn't change when the current starts.
The length of the wire doesn't change when the current starts.
Why would the maximally possible distance between them change?

9. Nov 21, 2017

### Geocentricist

I see what you're saying now. I guess I didn't before. But don't protons repel each other just like electrons? Why do they get to squeeze closer together when they move, since electrons don't?

10. Nov 21, 2017

### Ibix

Because the electrons are the ones accelerating when the current turns on. The protons never change their state of motion. They're doing different things, so their behaviour is different.

11. Nov 21, 2017

### A.T.

Unlike the free electrons, the protons are fixed in the lattice and have therefore constant proper distances (the distances in their rest frame). The proper distances of the free electrons can change, while their distance in the wire frame stay constant.

12. Nov 21, 2017

### Geocentricist

This explanation doesn't seem to work after the acceleration is over.

This also feels unsatisfactory for some reason but I will just accept it for now.

Moving on, here is an animation of two parallel wires with identical currents. In the proton frame, both wires are neutral, but in the electron frame, both wires are positive. This seems a contradiction to me because it seems like there would be no force between the wires in the proton frame but a repulsion in the second. What am I missing? If I performed this experiment in real life, what would actually happen?

13. Nov 21, 2017

### Staff: Mentor

The magnetic force

14. Nov 21, 2017

### Geocentricist

What does it do?

15. Nov 21, 2017

### Ibix

Yes it does.

When there is no current flowing, the electrons and protons are at rest with respect to one another. The electrons have the same spacing as the protons, and all frames agree on this although they will not agree on what the spacing is - frames where the wire is moving will see a smaller (length-contracted) spacing.

When the current is flowing, the wire remains uncharged in its rest frame. So the spacing of the electrons in this frame must be the same as the spacing of the protons. But this is not the rest frame of the electrons any more - we accelerated them. So this spacing must be a length-contracted version of the spacing in the rest frame of the electrons. But we haven't done anything to the protons. This is why the result is different for the electrons and the protons - we changed what the electrons are doing.

We are currently in the rest frame of the protons. If we change to any other frame, the spacing between the protons will length contract. But we are not in the rest frame of the electrons, so the spacing between them will either further length contract or will un-contract, depending if the frame change is to a frame closer to the electron rest-frame or further from it.

16. Nov 21, 2017

### Ibix

Makes the two wires attract, more strongly than the magnetic force in the wire rest frame. The like-charges-repel effect counters that a bit, but the net interaction is still attractive.

17. Nov 21, 2017

### Staff: Mentor

It makes it so that the net electromagnetic force is attractive.

The reason the Purcell example (the one described by the Veritasium video) is chosen was to simplify the scenario and avoid electric forces in one frame and magnetic forces in the other frame. If you choose a different example then it won’t simplify the same way. With your example the magnetic force cannot be neglected in either frame.

18. Nov 21, 2017

### jartsa

The proton frame is the lab frame.

1: In the lab frame the forces between all protons are unchanged when current changes, obviously.

2: In the lab frame forces between protons and electrons are unchanged when current changes.

3: In the lab frame forces between the current carrying electrons are changed when current changes. The forces are decreased when the directions of the currents are the same.

Those are correct statements in the lab frame. Any questions about them?

Let me guess: "But in number 2 the electrons see a contracted proton formation. And what is the reason for number 3?"

I can answer the first part. Electrons' opinion about the force between the electrons and the protons is just an opinion. The opinion changes when the velocity of the electrons changes. Same logic applies to protons, their opinion about the force between the electrons and the protons does not change, as the velocity of the protons does not change. And the proton frame is the lab frame, so therefore in the lab frame there is no change of force between the electrons and the protons when the current changes.

Last edited: Nov 21, 2017
19. Nov 21, 2017

### Geocentricist

Okay, putting off replying to some comments for a bit while I learn about this magnetic force. I found this image from this Quora answer. Am I interpreting this correctly that an electron moving to the right at V will feel a magnetic force attracting it to another electron beside it also moving to the right at V?

20. Nov 21, 2017

### Staff: Mentor

Yes.

21. Nov 21, 2017

### Geocentricist

Okay, so if I consider the lab frame and simplify each wire to just one proton and one right-moving electron each, I can see how the magnetic force causes the two wires to attract. Both electrons move to the right, so they attract each other with a magnetic attraction greater than their electric repulsion, and I just ignore the protons since they aren't moving.

But what about in the electron frame, if I consider each wire to be two protons moving left and one stationary electron? Does a pair of left-moving protons attract to another pair of left-moving protons with an attraction strong enough to overcome their electrostatic repulsion? Would this mean magnetic force is stronger than electric force?

22. Nov 21, 2017

### Staff: Mentor

No, the net force is always repulsive for a pair of protons.

There is no universal answer to that question.

The quantity $E^2-B^2$ is an invariant. If that quantity is negative then in a sense the magnetic field is stronger than the electric field, and there is a frame where electric field is zero. If that quantity is positive then in the same sense the electric field is stronger, and there is a frame where the magnetic field is zero.

For a pair of charges the quantity is positive, and for a pair of current carrying wires the quantity is negative.

23. Nov 21, 2017

### Geocentricist

So a pair of electrons moving together attract but a pair of protons moving together doesn't?

Then I'm back to having two left-moving protons and an electron, and another two left-moving protons and an electron. Since each of these two clusters are net positive, and therefore an electric force repels them, how do I explain why they actually attract?

I didn't comment on your answer to my second question because I don't understand it.

24. Nov 21, 2017

### Staff: Mentor

For both a pair of electrons and a pair of protons the electric force is repulsive and the magnetic force is attractive and the net force is repulsive (the quantity is positive).

Through the magnetic fields! In this case the magnetic force is stronger than the electric force (the quantity is negative). Yes, there is a net charge on the wire, but it is small compared to the current.

To simplify, the question was if the electric field was stronger or the magnetic field. The answer is: it depends. In the case of two comoving charges the electric field is stronger. In the case of two parallel currents the magnetic field is stronger.

25. Nov 21, 2017

### Geocentricist

But I thought you said earlier a pair of moving electrons attract? Now you say they're repulsive? How am I going to explain why the wires on the left attract then?

I'm so confused because now it seems you're contradicting what you just said, that a pair of a pair of protons repulse (the net force is repulsive because the quantity is positive).

Okay so two comoving electrons, their electric force (repulsion) wins. So I'm at a loss as to how the wires attract in the lefthand scene, and I had thought I got that part down.