Where does the energy come from, by only using a capacitor and the lorentzforce?

Question

Me and the boys can't explain the following construction:
A capacitor with a hole in both plates is charged and an electron is accelerated in it. So the electrons potential energy should be converted in kinetic energy. Once the electron leaves the capacitor we ony use magnets to redirect the electron via the lorentzforce back into the capacitor. Shouldn't it be accelerated again? Where does all that energy come from? Because otherwise it would be a perpetual motion machine ;D
So we need to find our silly mistake.
Thanks in advance
Kindest regards me an the boys

RogerJBarlow · Accepted Answer

More subtle than it looks...
The electric field (lines of force) is not confined to the gap between the plates. It spreads out sideways.
So during the dashed-green part of the trajectory the electron feels not only the Lorentz force from the magnetic field but also a decelerating electric field.
Yes, as you get away from the capacitor plates the electric field drops off.    But it is also more spread out. So if you make the return loop larger it gives a weaker force  - but over a longer distance. These effects (no surprise) exactly cancel.
So your electron ends up back where it started - with the same velocity.

HolgerFiedler · Answer

... we only use magnets to redirect the electron via the Lorentz force back into the capacitor.

The path that the electron follows is not a circle. As you may remember, the deflection of the photon is an acceleration, and every acceleration of an electron is accompanied by the emission of photons. The recoil of the emitted photons takes some kinetic energy away from the electron, and instead of a circular path, the electron's path is a spiral path. In the end, the electron comes to a standstill in the middle of the spiral (in a model without considering the earth's gravity, of course).
Do not be disappointed. You will often hear the statement that the Lorentz force leads to a circular path. If you consider the spiral path of the electron in an external magnetic field in your model, it is clear that there is no such perpetum mobile.

Where does the energy come from, by only using a capacitor and the lorentzforce?

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