All Problems Motion of Charged Particles in Electric and Magnetic Fields
Problem 3.397
Protons are accelerated in a cyclotron so that the maximum curvature radius of their trajectory is equal to \(r=50 \mathrm{~cm}\). Find: (a) the kinetic energy of the protons when the acceleration is completed if the magnetic induction in the cyclotron is \(B=1.0 \mathrm{~T}\) (b) the minimum frequency of the cyclotron's oscillator at which the kinetic energy of the protons amounts to \(T=20 \mathrm{MeV}\) by the end of acceleration.
(a) \(T=\frac{(e r B)^{2}}{2 m}=12 \mathrm{MeV}\) (b) \(v_{\min }=\frac{1}{\pi r} \sqrt{\frac{T}{2 m}}=20 \mathrm{MHz}\)