Relativity, magnetic attraction, and comoving electrons....

[primefield]

How can relativity explain the magnetic attraction of two electrons (or two electron beams) comoving in a vacuum at some certain constant velocity?

It is well known (https://acceleratorinstitute.web.cern.ch/acceleratorinstitute/ACINST89/Schindl_Space_Charge.pdf) that two parallel electrons or electron beams at the same velocity will experience an attractive Lorentz force that increases in magnitude and eventually equals the repulsive Coulomb force at the hypothetical velocity of c (if you could get to that velocity that is).

It seems that relativity would suggest that comoving electrons would of course view each other as being at rest and hence their Coulomb fields and forces would not be Lorentz contracted. In addition, they would not view any created magnetic fields and hence no Lorentz forces. This would seem to suggest that the electrons, from the comoving or proper frame of reference, would merely repel each other according to Coulomb's law. But, it is experimentally known that two parallel electrons or electron beams do repel each other due to Coulomb's law, but, ALSO have a magnetic attraction due to the Lorentz force when moving at a given constant velocity which could be viewed as a lowering of the Coulomb force.

When there is no wires involved, there seems no way to 'introduce' via length contraction, time dilation, or otherwise, a means via relativity to explain the attractive Lorentz force that negates and or counters the Coulomb force for these comoving electrons.

Please note, I am specifically NOT talking about wires, so I kindly request no references or explanations based on a Lorentz contraction of the positive atomic lattice in said wires as is generally done. By the way, I do not dispute the wire-based answers at all as they are entirely logically consistent.

The only things involved in the question, are the comoving parallel electrons or electrons beams and a vacuum through which they are moving at some constant velocity.

By the way, in asking this question to another, I received the answer that two comoving electrons would experience a time dilation effect and would therefore move apart in a slower fashion than if they were at rest in the lab frame.

But, that explanation does not seem right, as that same explanation should then obviously be the sole explanation for the case in which the electrons are moving in parallel wires, instead of having to talk about length contracted positive ions in the wire.

I hope that all makes sense!

 

[mfb]

But, it is experimentally known that two parallel electrons or electron beams do in fact attract when moving at a given constant velocity.

No, they repel each other. Just less than they they would do without their motion. In the frame of the beams, they repel each other more, but if we look at the separation after some given distance in the lab (e. g. the beams hitting the wall), then in the beam frame they travel for a shorter distance and therefore a shorter time, leading to a smaller separation - the same reduction we get from the magnetic field component in the lab frame.

 

[primefield]

No, they repel each other. Just less than they they would do without their motion. In the frame of the beams, they repel each other more, but if we look at the separation after some given distance in the lab (e. g. the beams hitting the wall), then in the beam frame they travel for a shorter distance and therefore a shorter time, leading to a smaller separation - the same reduction we get from the magnetic field component in the lab frame.

Right, I wrote that in a way that sounds like it contradicts what I wrote just above it.

"It is well known (https://acceleratorinstitute.web.cern.ch/acceleratorinstitute/ACINST89/Schindl_Space_Charge.pdf) that two parallel electrons or electron beams at the same velocity will experience an attractive Lorentz force that increases in magnitude and eventually equals the repulsive Coulomb force at the hypothetical velocity of c (if you could get to that velocity that is)."

What I wrote there is what I mean, and that is that there is an attraction that 'lowers' the effective Coulomb force, but, overall, the beams will always repel each other.

But, that does not rectify the seemingly counter explanations given for electron currents/beams with and without wires.

Perhaps I can ask the question in the most direct way, is there anything 'different' with wires vs no wires in terms of overall forces with two hypothetical electron 'currents' flowing at say 0.5c, one set of parallel currents upon wires, and another set of parallel currents 'as beams' and hence without wires?

It just seems that the length contracted positive lattice ions explanation obviously disappears in the no wires example, and therefore would suggest the wire vs no wire examples are 'different' in some way.

Thanks.

 

[Dale]

But, that does not rectify the seemingly counter explanations given for electron currents/beams with and without wires.

If you actually work out the math it does rectify the situation entirely. If you transform the Coulomb force in the rest frame to the EM force in the lab frame you get complete agreement.

 

[jartsa]

If we ask the moving electrons, the charge density is decreased, and the Coulomb force is decreased because of that.

When we transform the force from the electrons' frame to another frame, the force decreases some more, because we divide the force by gamma, because that's the transformation.

So f`=f/gamma<sup>2</sup>
where f` is the force when the electrons move, and f is the force when the electrons do not move.

This is not from any textbook. The decrease of charge density is from the Bell's spaceship paradox, and the force transformation - I guess that is in textbooks.

 

[jartsa]

It just seems that the length contracted positive lattice ions explanation obviously disappears in the no wires example, and therefore would suggest the wire vs no wire examples are 'different' in some way.

Let's see what the effect of the positive lattice is on an electron moving relative to the lattice.

The electron says the positive charge density is increased, and because of that the Coulomb force is increased.

We divide the force reported by the electron by gamma, to get the force in our frame. Surprisingly the force is just the normal unchanged force.

(Or maybe it is not so surprising that there is no magnetic force, when one of the charges is not moving)

 

[jartsa]

No, they repel each other. Just less than they they would do without their motion. In the frame of the beams, they repel each other more, but if we look at the separation after some given distance in the lab (e. g. the beams hitting the wall), then in the beam frame they travel for a shorter distance and therefore a shorter time, leading to a smaller separation - the same reduction we get from the magnetic field component in the lab frame.

That makes no sense.

Let's consider just two electrons side by side. When both electrons start to move to the same direction, what does change in the electrons' frame? Answer: Nothing changes.

F_before = F_after , in electrons' frame.

It is possible to transform that F_after to the original frame like this: F`= F/ϒ (An astronaut in a fast moving spaceship throws a basketball on a wall. The force of the collision is small, because the collision lasts a long time, because of the time dilation, in a frame where there is time dilation.)

 

[mfb]

Let's consider just two electrons side by side. When both electrons start to move to the same direction, what does change in the electrons' frame? Answer: Nothing changes.

That is in perfect agreement with what I wrote. I don't see the problem.

Was that the source of confusion?

In the frame of the beams, they repel each other more,

"More" is here meant compared to the lab frame where the electrons move.

 

[jartsa]

That is in perfect agreement with what I wrote. I don't see the problem.

Was that the source of confusion?"More" is here meant compared to the lab frame where the electrons move.

Oh yes, now it makes sense. I just couldn't understand it correctly.

 

[Vitro]

@primefield, try a numeric example. First consider only the frame where the two electrons are initially at rest and calculate a series of events describing their repulsive motion, something like (x0, t0), (x1, t1), (x2, t2), etc. for each of the electrons. Then simply transform these events to any other frame moving relative to the first. There's no need to complicate things and worry about magnetic fields and Coulomb forces in the moving frame, just a straightforward Lorentz transformation.

This type of questions arise from forgetting (or mistrusting) the Principle of Relativity. Always pick the frame of reference in which the problem has the simplest form, analyze it there then transform the results to any other frame you like.