How To Consistently Explain Electromagnetism With Relativity

[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.

 

[A.T.]

the electrons start moving but their separation remains the same.

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:

https://www.physicsforums.com/threads/explanation-of-em-fields-using-sr.714635/page-2#post-4528480

 

[Dale]

Superconducting Ring

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

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

 

[Geocentricist]

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

This is correct. There is no reason why the electron separation should change, when they start flowing.

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

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.

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?

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

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

 

[Dale]

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

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

 

[Ibix]

Does relativistic electromagnetism create and annihilate protons and electrons?

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.

 

[Geocentricist]

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.

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.

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

Ok, thanks.

 

[A.T.]

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

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?

 

[Geocentricist]

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?

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?

 

[Ibix]

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?

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.

 

[A.T.]

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?

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.

 

[Geocentricist]

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.

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

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.

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?

 

[Dale]

What am I missing?

The magnetic force

 

[Geocentricist]

The magnetic force

What does it do?

 

[Ibix]

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

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.

 

[Ibix]

What does it do?

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.

 

[Dale]

What does it do?

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.

 

[jartsa]

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?

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.

 

[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?

 

[Dale]

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?

Yes.

 

[Geocentricist]

Yes.

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?

 

[Dale]

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?

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

Would this mean magnetic force is stronger than electric force?

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.

 

[Geocentricist]

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

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.

 

[Dale]

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

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).

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?

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.

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

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.

 

[Geocentricist]

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).

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?

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.

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).

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.

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.

 

[Ibix]

Two isolated electrons will always repel. Two isolated protons will always repel. But you can't treat a wire as isolated protons and isolated electrons because that would be ignoring the interaction between the electrons and the protons.

If you analyse a pair of parallel current carrying wires in their rest frame then you will see only a magnetic field which causes an attractive force. If you view them in any other frame you will see a (different strength) magnetic field which causes an attractive force, but also an electric field which causes a repulsive force. The attractive magnetic force will always be stronger than the repulsive electric force so the net force will always be attractive.

You can analyse this in terms of pairs of streams of positive and negative charges with different velocities and rest charge densities, but you must remember both the electric and magnetic effect on one stream due to all three of the others. Sometimes you can short cut it and note that some effect cancels with another one - for example in the rest frame of the wire the electric effect of a proton stream cancels the electric effect of its electron stream. But that does not hold in every frame.

 

[Dale]

But I thought you said earlier a pair of moving electrons attract?

No, I said that the magnetic force between a pair of electrons is attractive. The electrostatic force is repulsive. In this case the electrostatic force dominates and the net force is repulsive.

How am I going to explain why the wires on the left attract then?

Through the magnetic force. In this case the magnetic force dominates and the net force is attractive.

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).

I am not contradicting what I said. You are asking different questions about different situations and they have different answers. As I said above, whether the magnetic force or the electric force is stronger depends on the situation. In the case of two co-moving charges of the same sign the electric force is stronger and the net force is repulsive. In the case of two parallel current carrying wires the magnetic force is stronger and the net force is attractive.

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.

That should be pretty clear, I don't know how you are at a loss on that one. The wires are neutral, so the electric force is 0. The wires are carrying parallel currents so the magnetic force is attractive. In this case the magnetic force is clearly the dominant one, so the net force is attractive.

 

[jartsa]

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?

View attachment 215426

Well there a is really simple thing that you are ignoring:

The positive thing at the top left corner of the picture attracts the negative thing at the bottom left corner of the picture.

 

[Geocentricist]

Okay, thanks for all the responses. I've figured out that in the first situation, all forces cancel except for the magnetic attraction between the two moving electrons. But in the second situation, I'm still working on it. The two protons in each wire repel each other electrostatically, but also attract magnetically, right? But since electrostatic force is stronger than magnetic, this would mean there's a net repulsion yet this is not true so I must be wrong somewhere.

By the way, if the electrostatic force is 1, what would the magnetic force be?

 

[Ibix]

But since electrostatic force is stronger than magnetic

This statement is not true in general. It depends on the situation.

By the way, if the electrostatic force is 1, what would the magnetic force be?

In what setup, in what frame, between what objects?

 

[jartsa]

Okay, thanks for all the responses. I've figured out that in the first situation, all forces cancel except for the magnetic attraction between the two moving electrons. But in the second situation, I'm still working on it. The two protons in each wire repel each other electrostatically, but also attract magnetically, right? But since electrostatic force is stronger than magnetic, this would mean there's a net repulsion yet this is not true so I must be wrong somewhere.

Well now the two positive thighs at the top right corner of the picture attract the one negative thing at the bottom right corner of the picture. Just normal electrostatic attraction. Oh yes, the two positive things would be farther apart if they we not moving.If we make the positive things move together very fast, we can ignore net force between them, because the net force becomes very small. If the fast motion causes more positive things to get into the picture ... then it isn't so simple.

 

[Geocentricist]

In what setup, in what frame, between what objects?

In the frame of two wires with moving electrons, if a moving electron repels another moving electron with an electrostatic force of 1 then how strong is the magnetic attraction between them?

Well now the two positive thighs at the top right corner of the picture attract the one negative thing at the bottom right corner of the picture. Just normal electrostatic attraction.

Okay, but doesn't the top proton pair repel the bottom proton pair with double the force that it attracts the bottom electron? So the top wire is still, overall, repelling the bottom wire, contrary to the result I'm aiming for (the reality that the two wires attract).

Also I wanted to bump one of the original questions I had in the OP about Veritasium's video, which is, are they mistaken in portraying the electron-frame electron spacing different at different times?

 

[jartsa]

Okay, but doesn't the top proton pair repel the bottom proton pair with double the force that it attracts the bottom electron? So the top wire is still, overall, repelling the bottom wire, contrary to the result I'm aiming for (the reality that the two wires attract).

No. There's the 'magnetism' thing. Parallel currents attract.

But now that I thought about it I noticed that the contraction thing and the 'magnetism' thing cancel out in the electron frame. No matter how fast the protons move the net force between them is the always the same - which means it's the same as when the speed is zero. This is in the electron frame.

 

[Geocentricist]

No. There's the 'magnetism' thing. Parallel currents attract.

I know that if I put two wires with identical current side-by-side they will attract. But I want to know why this is by understanding the electrostatic and magnetic forces between the protons and electrons. That's why I'm confused as to how they can attract given they are both net positive (they each have 2 protons per electron). So there must be some magnetic force that overcomes this repulsion. Since the electrons aren't moving, they can have nothing to do with this magnetic force as it only involves moving charges. So does the magnetic attraction between two co-moving pairs of protons overcome their mutual electrostatic repulsion?

But now that I thought about it I noticed that the contraction thing and the 'magnetism' thing cancel out in the electron frame. No matter how fast the protons move the net force between them is the always the same - which means it's the same as when the speed is zero. This is in the electron frame.

I'm confused. When the protons don't move there's only one proton in the wire as opposed to two. So how could the net force between them be the same?

 

[jartsa]

I'm confused. When the protons don't move there's only one proton in the wire as opposed to two. So how could the net force between them be the same?

No motion - no contraction.
No motion - no 'magnetism'.