# How would I go about calculating the energy input needed for a cyclotron reactor?

I'm trying to do some math regarding cyclotron reactors. I've figured out how to calculate the radius of a particle's motion in the cyclotron, a rough way to calculate energy output, but I have looked in many, many places, and nowhere have I found a way to determine how much energy I have to put into the reactor based on, say, the desired radius or speed of the particles. I know that a typical cyclotron runs on a radio frequency alternating voltage, but what I'm interested in here is a way to determine a value for this.

• Curious, you say cyclotron reactor. And you tagged this fusion. Are you asking what sort of energy you'd need to make a cyclotron beam-target or beam-beam fusion system? – Maury Markowitz Oct 10 '19 at 20:17

Ah, I think I may have found a solution here: http://physicstasks.eu/551/cyclotron

We can calculate the energy gained by a particle with charge e accelerated by voltage U:

$$E_k = eU$$

$$E_\mathrm{max} = nE_k$$

Based on the energy gained by each turn, you can calculate the cyclotron motion radius with:

$$r = mv/qB$$

You can then simply calculate the Coulomb barrier for fusion for whatever particles are in question. Then you can set the max energy to the Coulomb barrier calculated, and from there it's a relatively simple task to increase or decrease the turns needed to achieve max energy, which will affect energy/rotation, which then affects the voltage.