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Imagine we have a hollow metallic toroid, with copper wire winded around it, which carries electric current. That implies we have magnetic field inside the hollow toroid. The toroid has vacuum inside. We have a set up of a high voltage supply and an electron gun that takes the free electrons from the metallic toroid and shoots them inside it. The velocity of electrons shot inside the toroid is low enough so the magnetic field will bend its trajectory in a complete circular path within the boundaries of the toroid. Now the electrons are flying in circular trajectory inside the toroid. But this situation can't be hold forever, as the electrons are centripetally accelerated the irradiate photons and thus loose kinetic energy. As they loose kinetic energy and velocity, at some point they will stop orbiting and will stay still. But unless they still exactly in the geometrical center of the section of the toroid (which is highly unlikely), they will be attracted to the boundary of the toroid due to Coulomb forces. And as they move towards the boundary, they will regain velocity and start orbiting again. Resuming, the electrons will loose energy due to their accelerated motion, then regain energy, and then loose it again. Apparently this cycle will repeat endlessly, meanwhile they will radiate photons as they loose kinetic energy. If my analysis is correct, how the energy conservation principle will be applied here? Radiating photons endlessly means giving endless energy.

The second part of my question is as follows. Suppose the electrons doesn't radiate photons, due to some arbitrarily stated postulate (like the Bohr's explanation on why electrons doesn't fall on the nucleus of an atom). Apparently there is no obstacle to hold an infinite amount of electrons inside the toroid. The only limiting factor will be the amount of the voltage applied to fire new electrons inside the torus, as previously fired electrons will create repelling Coulomb force for the new incoming electrons. But there will not be such thing as "dielectric rupture" as in the case of an ordinary capacitor, so hypothetically an infinite amount potential difference can be set between the hollow toroid and its inside. Is this assumption correct?

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    $\begingroup$ An electron moving in 2D in a homogeneous magnetic field is a solved problem (was solved by Landau in the early days of quantum theory). There are stable orbits that don't radiate in this model. There is a Wikipedia page about it. $\endgroup$
    – Marius Ladegård Meyer
    Commented Jul 30, 2023 at 17:57
  • $\begingroup$ And just like in other parts of quantum physics, the Pauli principle limits the number of electrons in a given quantum state, so you cannot hold an arbitrary number of electrons there. $\endgroup$
    – Marius Ladegård Meyer
    Commented Jul 30, 2023 at 17:59
  • $\begingroup$ The electric attraction from the wall doesn‘t starts only after the standstill. It will influence the trajectory of the spiralling electron from the beginning of its fly. So the impact to the wall will happens long time before any standstill. $\endgroup$
    – HolgerFiedler
    Commented Aug 1, 2023 at 3:18
  • $\begingroup$ @HolgerFiedler I agree that it will be attracted to the wall from the very moment it enters the torus, but I don't agree it will hit the wall. This is why. In plasma physics there is something called "plasma drift", that is when positively charged nuclei or electrons in a plasma, gyrating due to an applied magnetic field, are subjected to a force parallel to the plane of their gyration, they won't move in the direction of that force , but perpendicular to it. $\endgroup$
    – Eduard Oganesian
    Commented Aug 1, 2023 at 23:25

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