A Anisotropic pinch force in high energy electron beam

[kiwaho]

Here I am only considering 4 adjacent electrons in electron beam, especially the 2 electrons that are moving in tandem with velocity v.
Moving charge will generate circular magnetic current. you can imagine magnetic flux as current, just like electric current.
So, the question is same to find the magnetic attraction force between in-tandem electrons. (their induced magnetic currents are same direction, both clockwise or counter-clockwise dependent on observation direction).
As to other 2 electrons moving in shoulder by shoulder, it is easy to calculate the pinch force, but tandem mode may be not. Many textbook just simply treat it as parallel current lines.
I believe the magnetic forces different between tandem-pair and tandem-pair, that is why I claim anisotropic pinch force.
For high energy electron beam researchers, they may concern what is the minimal electron distances in accelerated packed electron bunch. We emphasize too much on plasma particle density, but neglect the attainable minimal electron distance, though we can deduce it from density by cubic root.
Temporarily ignore spin magnetic field.

 

[mfb]

I don't know what you mean by tandem and shoulder, but you can always go to the rest frame of the beam, calculate the forces there, and transform back. Much easier because you don't have to care about retarded potentials and so on.

 

[kiwaho]

I don't know what you mean by tandem and shoulder, but you can always go to the rest frame of the beam, calculate the forces there, and transform back. Much easier because you don't have to care about retarded potentials and so on.

Tandem mode means one electron moves in front of another electron, they are moving not only in same direction but also in a line.
Shoulder mode means 2 electrons moving in same direction, but the line that crosses the 2 electrons is orthogonal to the moving direction.