[chris (OP)]

I was asked today by someone what would happen to two electrons moving, side by side? The query related to the fact that they will feel an attractive magnetic force pushing them together. So what would be the ultimate outcome?

Chris

[JP]

Are both electrons moving at the same velocity?  If so, you could fly alongside them and take the measurement.  In this case, they're not moving relative to you, so you'd see no magnetic field.  With no magnetic field, you'd expect the electrons to move apart (though calculating exactly how they do so could get tough, as they radiate when they start moving apart).   

Since anyone moving at a constant velocity should agree on the basic physics happening (this is a basic postulate of both relativity and Newtonian mechanics), the electrons should appear to move apart even if you're standing still and watching them fly by. 

[DoctorBeaver]

I thought electrons didn't like each other due to the negative charge. Wouldn't that push them apart? (although they'd have to be pretty damned close together for it to have any effect)

[graham.d]

They will move away from each other. Maxwells equations will work in any frame of reference (see Lorentz transformation) so, if you pick a frame moving with the electrons they would be two stationary electrons which would simply repel each other. Picking a frame so that they are moving would mean there would be a magnetic field as seen by an observer but the coulomb repulsion is much greater than the magnetic attraction. The change in the relative effect of the two forces would only be observable at relativistic speeds, but the motion of the two electrons would then need to take account of the changes in how the observer does the measurements from his frame. It gets quite complicated mathematically.

[lyner]

Keeping electron beams tightly together is a real problem in many vacuum tubes. The electric repulsion is the dominant effect, whatever else is going on.

[lightarrow]

The real original "paradox" is this: in the ref frame co.moving with the electrons there is only electric repulsion, while in a ref frame where the two electrons are moving, we observe the electric repulsion AND the magnetic attraction, so the effect (as total force) should be different from a frame to another.

The answer is that in the frame where the electrons are moving the electric field generated from the two charges IS NOT the same as that generated from the two STILL charges, it transforms relativistically, so it's not the coulombian field anylonger.

Furthermore, to compare the two results we have to choose two different time intervals because if the experiment lasts for one second in a frame, then, because of time dilation, it dosn't last for one second in another frame.

[chris (OP)]

Ah, thanks lightarrow, that must be what the person was referring to; he did say it was some kind of paradox. I have to say I don't completely understand what you've said, but that's my fault not yours. Can you, or someone else, turn what you've said into idiot-proof language for me?

Cheers

Chris

[lyner]

If you are traveling alongside the two electrons then you will be in the same reference frame as them. (Just as if you were all stationary). Time and distances will be the same for you as the electrons. You will see them move together at a rate which is 'explained' by electric attraction, alone.
If you are in another frame - let's call it 'stationery' - and they are moving relative to you. then you will also see them move together but 'your time' will be different from theirs. Their time will appear to be 'going slower' than yours so they will appear to be moving together slower than you might have expected. One way of explaining this is to say that they have a magnetic interaction, too, to explain the apparent difference in the force.
It is worth mentioning that, if you assume the electrons are stationary  and you are moving past them, you get the same result.
It is quite another matter as to whether there 'really are' forces acting in these situations. We only use the concept of forces to explain and predict things.

[JP]

To rephrase what everyone else has said (broken down into steps)

(1) Special relativity tells us that if put an experiment on a train and run it past our observer at high speeds, he sees the experiment happening more slowly than it would appear to if the train is still.

(2) In order to test this paradox, we put two electrons on a table on the train some distance apart.  To someone onboard the train with them, they appear to repel entirely because of electrical forces since they're sitting still onboard the train.  They therefore move apart.

(3a) To an observer outside the train, because the experiment appears to slow down (special relativity), the electrons appear to move apart more slowly.  However, he can explain this entirely by understanding the electric repulsion on the train and then applying relativity to slow the effects down.

(3b) If the outside observer doesn't know about relativity (and electromagnetism was formulated before relativity), he can explain the "slowing down" as a result of the magnetic fields/forces that are generated by the motion of the electrons which oppose the electric "push" apart.

(4) The paradox is: if you use the magnetic field explanation instead of relativity, why does one observer see a magnetic field and the other not?  How can a field just vanish like that?  The resolution is that magnetic fields have a deep connection to relativity, and that you can describe the situation here without them if you understand the relativity going on.

[DoctorBeaver]

*looks at Chris*

[lyner]

jp

That's a fine explanation. I just couldn't afford the train fare!

[Ron Freimann]

It's an old topic, but maybe someone will answer.

So everybody here said that electrons will move apart and blamed that on special relativity as those two electrons would be standing still in their frame of reference. But, how would you explain the phenomen that two wires carrying current in the same direction don't repel but attract each other? Electrons in that wire would be traveling at same speeds (ofcourse not exactly because of brown movement, but this can be ignored for time being) so their frame of reference would also point out that they are standing still compared to each other.

[Atomic-S]

Well, for one thing, in a wire the electrons' charges are counterbalanced by the protons charges.  When there is no current, that does away with all net electomagnetic effects as pertains to outside macroscopic observations. When the electrons move but the proton's don't, there is no longer a perfect balance because the fields of the two different types of particles are not in the same reference frame.