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Having read that question: Magnetic force as a relativistic effect?

And quoting from the answer: https://physics.stackexchange.com/a/143901/7743

If you want to analyze things in the electrons' rest frame in wire A (whose electrons move at less than half the speed, relative to the wire, as the electrons in wire B), you have to take into account not just the electric force on the electrons in the wire A, which will be repulsive as you say, but also the electric force on the positive charges (the ions that have lost electrons) in wire A, along with the magnetic force on the positive charges in wire A, which are moving in this frame. When I did this in a numerical example below, I did find that the net force on wire A was attractive despite the fact that the electric force on the electrons was repulsive.

So basically the Lorentz force from the viewpoint of the electrons outside the wire can be explained by the difference of apparent charge densities in the wire due to the length contraction of charges moving in different speeds in the wire.

Now what if we remove the positive charges from the picture and deal with a pure electron beam? Could there be an attraction or will it always repel?

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  • $\begingroup$ Did you calculate the velocity the electrons would need to have in order for the attraction to be positive? $\endgroup$ – CuriousOne Jun 11 '15 at 13:49
  • $\begingroup$ Self-magnetically insulated transmission lines are well known in the pulsed power community. Is that sort of what you are looking for? $\endgroup$ – Jon Custer Jun 11 '15 at 13:55
  • $\begingroup$ @CuriousOne Nope. Would it be faster than light? $\endgroup$ – Calmarius Jun 11 '15 at 13:58
  • $\begingroup$ Intuition would suggest so. Intuition can be wrong, of course, so you might want to check against the actual physical equations of motion. You need to use the relativistic equations in en.wikipedia.org/wiki/Biot%E2%80%93Savart_law. $\endgroup$ – CuriousOne Jun 11 '15 at 14:03
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It will all depend on the energy/momentum of the spectator electrons. When there is a beam, as in an accelerator, of charged particles there are ambient low momentum electrons

Electron clouds are created when accelerated charged particles disturb stray electrons already floating in the tube, and bounce or slingshot the electrons into the wall. These stray electrons can be photo-electrons from synchrotron radiation or electrons from ionized gas molecules. When an electron hits the wall, the wall emits more electrons due to secondary emission. These electrons in turn hit another wall, releasing more and more electrons into the accelerator chamber.

Here is what is happening with a positron beam , ie positive charge at velocities near light and interaction/radiation produced electrons in the vacuum of the beam pipe.

Negatively charged electrons liberated from the accelerator walls are attracted to the positively charged beam, and form a "cloud" around it.

This means that an electron beam will be repelling these spectator electrons.

Now for higher energy spectator electrons a calculation has to be done, as advised in the comments, to see at what parallel speeds the attractive force overcomes the repulsive from the charges.

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