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Motion of Charged Particles in Electric and Magnetic Fields

Problem 3.398

Singly charged ions He+\mathrm{He}^{+} are accelerated in a cyclotron so that their maximum orbital radius is r=60 cm.r=60 \mathrm{~cm} . The frequency of a cyclotron's oscillator is equal to v=10.0MHz,v=10.0 \mathrm{MHz}, the effective ac celerating voltage across the dees is V=50kVV=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.

Reveal Answer
3.398. (a) t=π2vmr2eV=17μst=\frac{\pi^{2} v m r^{2}}{e V}=17 \mu \mathrm{s} (b) s≈4π3v2mr23eV=0.74 kms \approx \frac{4 \pi^{3} v^{2} m r^{2}}{3 e V}=0.74 \mathrm{~km} Instruction. Here s∼∑n=1Nvn∼∑n,s \sim \sum_{n=1}^{N} v_{n} \sim \sum \sqrt{n}, where vnv_{n} is the velocity of the particle after the nn th passage across the accelerating gap. Since NN is large, ∑1Nn≈∫0NVnˉdn.\sum_{1}^{N} \sqrt{n} \approx \int_{0}^{N} V \bar{n} d n .
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