Protons are accelerated in a cyclotron so that the maximum curvature radius of their trajectory is equal to `r = 50 cm`. Find:
(a) the kinetic energy of the protons when the acceleration is completed if the magnetic induction in the cyclotron is `B = 1.0 T`,
(b) the minimum frequency of the cycloroton's oscillator at which the kinetic energy of the protons amounts to `T = 20 Me V` by the end of accelearation.

Explanation

(a) From `(mv^(2))/(r) = Bev`, or, `mv = Ber`
and `T = ((Ber)^(2))/(2m) = (1)/(2) mv^(2) = 12 MeV`
(b) From `(2pi)/(omega) = (2pi r)/(v)`
we get, `f_(min) = (v)/(2pi r) = (1)/(pi r) sqrt((T)/(2m)) = 15 MHz`