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Moving charged particles in an external magnetic field (of course non-parallel to each other) get deflected into a spiral path.

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For a calculation of this phenomenon it is not sufficient to use the [Lorentz force] (https://en.wikipedia.org/wiki/Lorentz_force#History) equation since this equation does not take into account the loss of kinetic energy due to the radiation of the charge during deflection.
A good description what happens is this about Can the path of a charged particle under the influence of a magnetic field be considered piecewise linear?

So what is the equation of the spiral path of an electron under the influence of a magnetic field?

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The spiral is a consequence of the Lorentz force and energy losses to the surrounding medium.

There are multiple processes for energy loss (and which ones matter depend on the particle in questions, as it is not only electrons that can describe such paths), and some of the are abrupt enough that you notice the kinks, but simple ionization losses are effectively continuous and result in a smooth spiral.

As usual the [PDG chapter on "Passage of particles through matter"] is a good compact review, but you may find the treatment in a introductory text of experimental methods in particle physics (like, Leo, perhaps) to be more forgiving.

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