# Classical physics vs relativity: parallel electron beams

Can someone explain to me how two parallel electron beams can be attracted to each other according to special relativity.

Classical physics says each beam will generate a magnetic field so the beams will be attracted to each other.

Relativity says the magnetic field only exists for observers moving relative to the electron beam, hence there is no magnetic field if the beams are moving in parallel and at the same speed.

I have been searching for an explanation, here is the closest I got to a discussion of this issue and the so-called expert doesn't even understand the question!

http://en.allexperts.com/q/Physics-1358 [Broken] ... -LAW-2.htm

I am not concerned with relativistic speeds - relativity should explain magnetism at any speed.

There are numerous other discussions explaining how conducting wires are attracted (ie Ampere's experiment), due to the electrons moving relative to the positive metal atoms in the other wire, but there is no equivalent conductive medium in an electron beam!

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Classical physics says each beam will generate a magnetic field so the beams will be attracted to each other.

Not exactly....the force between two charges is not only due to the magnetic field but also from the coulomb field...and it is given by the Lorentz force eqn......
F = qE + qv X B

Thus the force on each charge (or line of charges) is composed of two components, one of which is velocity dependent.

Creator

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When you are riding on the beams they repell one another by coulomb forces.

When you are standing still the beams still repel one one another but just a little less. In order for the forces to balence and have zero attraction the electron beam velocities have to approach c.

This is the clearest example I know of that shows the link between electromagnetism and relativity. Because time appears to run slower for the beam particles as they zip by, their mutual acceleration toward one another must be slowed by something. In our frame it is the magnetic field which reduces the coulomb forces between the moving beams.

When masses do this instead of charges we say they "got heavier". Since charge is invariant, you need a way to reduce the forces.

Cleonis
Gold Member
[...]
This is the clearest example I know of that shows the link between electromagnetism and relativity. Because time appears to run slower for the beam particles as they zip by, their mutual acceleration toward one another must be slowed by something.
[...]

I'm always uncomfortable with usage of the expressions such as 'time appears to run slower'.
No doubt the reason that you expressed it that way was just for brevity; we know that relativity is not about deceptive appearance.

- When the physics is represented in the coordinate system that is comoving with the parallel electron beams then the coulomb repulsion is the only factor in the equation.

- When the physics is represented in a coordinate system that has a velocity relative to the electron beams (parallel to the beams) then in accordance with the time dilation factor the mutual acceleration is different than in the co-moving coordinate system. As you point out it follows that there must be an additional factor in the equation, accounting for the difference in mutual acceleration.

There is the case of two parallel wires, conducting current, and the case of two parallel electron beams, and clearly those cases are not comparable.

In the case of two parallel wires, conducting current, the explanation of the magetic effect that arises is dynamical in nature. The electric current in a wire is a case of charge carriers having a velocity relative to a background structure consisting of the opposite charge, and due to the velocity difference there is a length contraction difference.

In the case of two parallel electron beams the way the difference is accounted for is kinematic in nature. If the coordinate system has a velocity relative to the represented electron beams then in the equation there is a magnetic term that arises from the coordinate velocity.

In the lab reference frame, the Coulomb force between two charged particle beams is opposite to and greater than the magnetic force, except when the beams are extremely relativistic, in which case the two forces cancel. For the Coulomb force, the electric field of one beam acts on individual charges in the other beam. For the magnetic force, the magnetic field of the current in one beam acts on individual charges in the other. See attachments. If Lorentz-transformed into a moving frame where the two beams are at rest, the force is pure Coulomb. In the case of two parallel wires, there is no Coulomb force in the lab frame.

Bob S

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• Beam_beamRev3A.jpg
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• Beam_beamRev3B.jpg
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