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Similar, but not the same as: Magnetic field inside a charged stream because I am interested in what happens inside a conductor, but the behaviour in a free beam can help clarifying what I am asking.

Let's consider a current of electrons (locally) limited to a cylindrical boundary of section A, from the point of view of classical (non quantum, non relativistic) electrodynamics Supposing the current density is constant throughout the section, the Ampere Maxwell law tells us the B field goes linearly with r, inside the section itself.

In an electron beam in vacuum, the force on the electrons that move with velocity v parallel to the stream will be directed towards the center of the beam. This will result in a tendency to shrink the beam (that won't win against the repulsion and the tendency to diverge) and/or a rotation of electrons around their otherwise linear trajectory (depending on their velocity this might even make the beam 'rotate' as a whole). Is there experimental evidence of any rotation of electron beams?

Now, in a conductor in classical ED the flow of a current is represented by a stream of ordered electrons flowing along the circuit with drift velocity vd. In this case as well the field inside the stream goes linearly as r (and as 1/r outside of it). But, to my knowledge, we do not observe a shrinkage of the stream (an anti-skin effect, so to speak - but at DC). Can this be satisfactorily explain by the observation that in this case the velocity vd is only an average velocity while the magnetic field acts on the actual velocity of electrons between collisions? Each electron deviates in a different direction not only because the magnitude of their velocity is mostly random, but also because the time between collisions is random and this would destroy any correlation between deviations, so the stream does not show neither shrinking nor rotation. Is this a reasonable explanation (in the domain of classical ED) or is this another case where classical ED cannot explain what we observe?

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    $\begingroup$ > rotation of electrons around their otherwise linear trajectory (depending on their velocity this might even make the beam 'rotate' as a whole) Why would there be "rotation"? Around what center would the rotation happen? Which force would cause it? Do you just mean deflection of the electrons towards the center line? $\endgroup$ Commented May 19 at 22:58
  • $\begingroup$ @JánLalinský rotation around the axis of the beam or around the original trajectory. vx B should be directed towards the axis, so the force will make the electron deviate towards the axis, but then its velocity changes direction and the new v' x B' changes as well. I tried to see in three steps with pencil and paper and it looked like the beam would twist around the axis (but I may be mistaken).while looking for a picture on the web I found references to electron beam vortexes (which are another thing but the idea of a vortex is more or less what I had in mind). $\endgroup$
    – Peltio
    Commented May 20 at 7:26
  • $\begingroup$ I just found this abstract (I haven't read the article) ieeexplore.ieee.org/document/108421 that mentions a rotation and shear of the beam due to self-field. And another peper which I cannot find now that states that rotations can be explained by quantum mechanics and not classically. But I'm in dire need of sleep, I might have misunderstood this last one. $\endgroup$
    – Peltio
    Commented May 20 at 7:30

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Parallel currents attract each other by their magnetic field. As you said, this should lead to an electron beam narrowing. In plasma discharges this leads to the so-called pinch effect. Very large currents in a tube-shaped metallic conductor produce a compression of the tube. In a cylindrical solid metallic conductor, the electrons should also experience a compressive force towards the axis, which should lead to an electron accumulation of negative charge near the axis and an electron depletion on the periphery of the metallic cylinder. This charge carrier redistribution creates a counteracting radial electrical field and diffusion current of the electrons which, when reaching a stationary state, exactly balances the magnetic force current. This is similar to the creation of a balancing electric field by surface charge carrier accumulation and depletion due to an external magnetic field in the Hall effect. You need only consider the average electron current densities described by electron concentrations and electron mobilities (and diffusion coefficients) due to their collisions with the phonons and defects of the metal crystal lattice.

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