The magnetic force experienced by a particle in a cyclotron is
[tex]F=qvB[/tex]
where q is the charge, v the speed and B the magnetic field intensity.
since its motion is a circular motion, this must be equal to the centripetal force:
[tex]qvB= \frac{mv^2}{r} [/tex]
where m is the particle mass and r the radius of the orbit.
Re-arranging the equation,
[tex]r= \frac{mv}{qB} [/tex]
But the speed v is related to the kinetic energy K by
[tex]K= \frac{1}{2} mv^2[/tex]
And so
[tex]v= \sqrt{ \frac{2K}{m} } [/tex]
And replacing v into the formula of r, we find
[tex]r= \frac{ \sqrt{2Km} }{qB} [/tex]
