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= (mv^2)/(r)`
where m is the particle mass and r the radius of the orbit.
Re-arranging the equation,

`r= (mv)/(qB)`
But the speed v is related to the kinetic energy K by

`K= (1)/(2) mv^2`
And so

`v= \sqrt{ (2K)/(m) }`
And replacing v into the formula of r, we find

`r= ( √(2Km) )/(qB)`