Modell of a Cyclotron (non relativistic)

Setup:

Main Parts of a Cylotron are two hollow D-shaped sheet metal electrodes called "Dees". To the Dees a high frequency alternating voltage is applied. This voltage charges the Dees alternately positive and negative. So in the narrow gap between the "Dees" an electric field emerge. Above and below the "Dees" two strong electromagnets are placed. They produce a static magnetic field perpendicular to the electrode plane. In one "Dee" a particle source is placed. It emits charged particles in the gap between the "Dees".

Function of a Cyclotron:

The particle source T emits for example protons (positive charged) with an initial speed v0 in the gap between the dees. If dee 1 is charged negative and dee 2 positive, the particle is accelerated in the gap because of the electric field. When the particle enters dee 1, it leaves the electric field. In the dee is a static magnetic field perpendicular to the electrode plane. So a Lorentz force acts on the particles which causes the particles path to bend in a circle. While the proton is in the dee, the charge of the dess changes - the electric field in the gap changes the direction. So, after a semicircle the protons get accelerated again in the gap between the dees. In dee 2 the Lorentz force causes the particles path to bend in a circle again. Because of the now higher speed of the particle the radius of this semicircle is bigger as in dee 1. Increasing distance and higher speed lead to a constant cycle time.
After serveral times the velocity of the protons gets near 10% of the speed of light where relativistic effects become relevant. Prior that, in a non-relativistic cyclotron a defelction plate directs the particles out of the cylotron and on any target.

Restrictions:

This type of cyclotron can be used only for particle speeds up to 10% of the speed of light. When the particles are getting faster relativistic effects must be considered. These effects lead to an increasing mass. So the cycle time T is no more constant. So the frequency of the acceleration voltage does not fit to the cycle time of the particles anymore.
Because of that, only heavy particles as protons or deuterons are accelerated with such a cyclotron.
Iin a typical cyclotron with acceleration voltages of several hundred volt protons reach energies of 10 MeV after 50 circulations.