All Problems Motion of Charged Particles in Electric and Magnetic Fields
Problem 3.398
Singly charged ions \(\mathrm{He}^{+}\) are accelerated in a cyclotron so that their maximum orbital radius is \(r=60 \mathrm{~cm} .\) The frequency of a cyclotron's oscillator is equal to \(v=10.0 \mathrm{MHz},\) the effective ac celerating voltage across the dees is \(V=50 \mathrm{kV}\). Neglecting the gap between the dees, find: (a) the total time of acceleration of the ion; (b) the approximate distance covered by the ion in the process of its acceleration.
3.398. (a) \(t=\frac{\pi^{2} v m r^{2}}{e V}=17 \mu \mathrm{s}\) (b) \(s \approx \frac{4 \pi^{3} v^{2} m r^{2}}{3 e V}=0.74 \mathrm{~km}\) Instruction. Here \(s \sim \sum_{n=1}^{N} v_{n} \sim \sum \sqrt{n},\) where \(v_{n}\) is the velocity of the particle after the \(n\) th passage across the accelerating gap. Since \(N\) is large, \(\sum_{1}^{N} \sqrt{n} \approx \int_{0}^{N} V \bar{n} d n .\)