Question

Your goal is to accelerate the particles to kinetic energy k. what minimum radius r of the cyclotron is required? express your answer in terms of m, q, b, and k.

Answer 1

The magnetic force experienced by a particle in a cyclotron is
$F=qvB$
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:
$qvB= \frac{mv^2}{r}$
where m is the particle mass and r the radius of the orbit.
Re-arranging the equation,
$r= \frac{mv}{qB}$
But the speed v is related to the kinetic energy K by
$K= \frac{1}{2} mv^2$
And so
$v= \sqrt{ \frac{2K}{m} }$
And replacing v into the formula of r, we find
$r= \frac{ \sqrt{2Km} }{qB}$