Cyclotrons and Synchrotrons — Question 3

Question 3

Successive outer orbits crowd together even when the energy gain per turn is constant

In a nonrelativistic cyclotron, a particle of charge q gains 2qV0 per full turn in a fixed field B. Its radius just before a turn is r and just afterward is r+delta r. Treat the energy gain as localized at the gaps.

1. Derive the exact radial increment between these two sampled orbits.

2. Find its leading form when the energy increment is small compared with the existing kinetic energy, and explain the extraction consequence.

Question 3 - Solution

Successive outer orbits crowd together even when the energy gain per turn is constant

(a) Method and calculation

$$K(r)=\frac{q^2B^2r^2}{2m},\quad K(r+\Delta r)-K(r)=2qV_0.$$$$\Delta r=\sqrt{r^2+\frac{4mV_0}{qB^2}}-r.$$

(b) Interpretation and further calculation

$$\frac{4mV_0}{qB^2r^2}\ll1\Rightarrow\Delta r\simeq\frac{2mV_0}{qB^2r}.$$$$\Delta r\propto r^{-1};\quad\text{larger-radius turns are more closely spaced}.$$

Physical check

Constant energy gain produces equal increments in radius squared, not equal increments in radius.