Electron beams and an inertial frame problem

[Hamish]

I wasn't sure whether to post this in the classical or relativity section. While pondering something I found an embarrassing gap in my knowledge. I can't quite get my head around this problem and I need a fresh perspective.

Consider two beams of electrons moving parallel to each other. The electrons are monoenergetic (all moving at exactly the same velocity, v). Each beam, being a flow of charge, carries a current I.

Now,in the laboratory rest fram, an observer sees two current carrying streams of particles which will tend to attract each other due to their magnetic field. There would appaer to be a force acting parallel to the direction of motion of the electrons.

In the rest frame of an electron in the beam, none of the other electrons are moving, so the electron sees no current and hence, no magnetic field. The only forces are electrostatic.

So do the beams attract each other or not? Why does there appear at first glance to be a transverse force which can be transformed away?

I'm sure I've made a trivial oversight somewhere. Help.

 

[Hamish]

Ok,I think I worked it out.

The observer in the laboratory rest frame sees the electron beam moving and as such witnesses Lorentz contraction of the distance between the electrons. This means a relativistic correction to the charge density per unit length. This increases the repulsive force between the two beams by (and I did work this out) exactly the amount by which they were attracted by the magentic field. So the only force between the two beams is equal to the electrostatic force between them in the electrons' rest frame.

Or something like that.

I brought this up because it seemed counterintuitive that two current carrying beams would have no magnetic attraction between them. But it seems they don't if there is only a single, monoenergetic charge species in the beam.

Was I right?

 

[DW]

Originally posted by Hamish
I wasn't sure whether to post this in the classical or relativity section. While pondering something I found an embarrassing gap in my knowledge. I can't quite get my head around this problem and I need a fresh perspective.

Consider two beams of electrons moving parallel to each other. The electrons are monoenergetic (all moving at exactly the same velocity, v). Each beam, being a flow of charge, carries a current I.

Now,in the laboratory rest fram, an observer sees two current carrying streams of particles which will tend to attract each other due to their magnetic field.

Actually he would reckon two contributions to the net force per length on a beam. He would find an attractive Lorentz force and a repulsive Coulomb force. Added together the Coulomb force dominates for your scenario where there are no protons. So he agrees with the rest frame observer that they should repel and the amount that they should is just a matter of a frame transformation going between the two perspectives.

Now in the case of current carrying wires the net result is actually attractive. From the wire frame perspective the reason that the net result is attractive in current carrying wires is that according to the wire frame there is no net charge to produce a repulsive Coulomb force, but there is a net current in each wire to produce and respond to magnetic fields. From the electron frame perspective the protons are more closely spaced in a beam and there is a Coulomb force repulsion between the protons of one beam and the protons of the other, but the presence of the electrons contributes to a Coulomb attraction between the beams that lowers the Coulomb repulsion to the point that the attractive Lorentz force between the protons dominates and the net result is attractive. Once again the amount of attraction is a matter of a frame transformation going between the two perspectives.

There would appaer to be a force acting parallel to the direction of motion of the electrons.

No there wouldn't. Did you mean to say perpendicular? In reality there is only one real force involved which is the electromagnetic force. This can be broken into two incomplete force contributions, the Coulomb force and the Lorentz force. Each of these is perpendicular to the direction of electron flow for this gedanken.

 

[Hamish]

Originally posted by DW
No there wouldn't. Did you mean to say perpendicular?

Yes, I did. Sorry, late night posting. Thank you for your reply.

 

[JonathanTheology]

Inquiry

If one were to take a color CRT (Cathode Ray Tube/picture tube), cut off the front (side the screen is on) part down to near the end of the narrow part of the tube, take off the deflection plates, and allow the three electron beams to proceed out of the glass tubing would the three electron beams likely experience a force of attraction to each other?

Jonathan

 

[selfAdjoint]

If one were to take a color CRT (Cathode Ray Tube/picture tube), cut off the front (side the screen is on) part down to near the end of the narrow part of the tube, take off the deflection plates, and allow the three electron beams to proceed out of the glass tubing would the three electron beams likely experience a force of attraction to each other?

Jonathan

Repulsion, since they have the same charge. They would spread apart.

 

[Calculex]

Repulsion, since they have the same charge. They would spread apart.

What if the electron beams flowed in opposite directions?

 

[JonathanTheology]

Consider

Repulsion, since they have the same charge. They would spread apart.

Read above what "DW" wrote; he wrote this:

"From the electron frame perspective the protons are more closely spaced in a beam and there is a Coulomb force repulsion between the protons of one beam and the protons of the other, but the presence of the electrons contributes to a Coulomb attraction between the beams that lowers the Coulomb repulsion to the point that the attractive Lorentz force between the protons dominates and the net result is attractive."

Jonathan

 

[JWWells]

a question about history...

I believe as the electron beam speed approaches c, the magnitude of the magnetic (attractive) force approaches the electrostatic (repulsive) force. Correct?

But the interesting question to me about this example, which someone out there may know the answer to, is the following: in pre-relativistic days, did anyone notice that this example provides a clear contradiction between Maxwell's equations and Galilean relativity? Without the benefit of the Lorentz contraction, I don't see an obvious way to make the magnitude of the net force on the electrons - and their resultant acceleration - in the laboratory frame equal to the magnitude of the net force in the electron frame. Or is there a classical explanation that I'm missing?

I know that Einstein used the example of the relative motion of a magnet and a loop of wire to point out the inconsistency of Maxwell's equations and Galilean relativity, but I'm not aware of the use of this example.

 

[Chris Hillman]

Unintended consequences?

JW, you nearly caused an emergency reboot, since I didn't immediately realize that you resurrected a thread from 2004!

Probably unneccessary explanation: in deciding whether to read a thread, I always check to see if anyone who posts good stuff has contributed. I was briefly startled to see selfAdjoint's username, since, darnitall, he died in Dec 2006 https://www.physicsforums.com/showthread.php?t=160724 post #15.

The special shock of unexpectedly encountering on-line someone who has died should probably be a named phenomenon in psychology. It seems to involve thinking (or feeling?) about a webforum such as PF as a conversation in a living room rather than a wall where one reads possibly ancient graffiti.

We now, uh, return you to the original thread...

 

[wisp]

Ok,I think I worked it out.

The observer in the laboratory rest frame sees the electron beam moving and as such witnesses Lorentz contraction of the distance between the electrons. This means a relativistic correction to the charge density per unit length. This increases the repulsive force between the two beams by (and I did work this out) exactly the amount by which they were attracted by the magentic field. So the only force between the two beams is equal to the electrostatic force between them in the electrons' rest frame.

Or something like that.

I brought this up because it seemed counterintuitive that two current carrying beams would have no magnetic attraction between them. But it seems they don't if there is only a single, monoenergetic charge species in the beam.

Was I right?

Yes.

Other thoughts to consider are:

If one of the beams were in a wire, the positive ions in the wire would neutralize the electrostatic repulsion effects. And then the velocity of the electrons in the wire would create a magnetic field, which would cause the two beams to attract.

Another interesting point is that the electrons in a single beam travel along like little bullets in a line, but when two beams travel together they strike a distant target as if they were made from wave parts not bullets.