Cyclotrons and Synchrotrons — Question 6

Question 6

Maintaining a fixed cyclotron frequency requires a radius-dependent relativistic field

An idealized isochronous cyclotron is designed to keep angular frequency omega0 fixed as energy rises. Use $r=\gamma mv/(qB)$ and $\gamma=(1-v^2/c^2)^{-1/2}$. Ignore focusing and edge-field requirements.

1. Derive B(r) in terms of $B_0=m\omega_0/q$.

2. Find B/B0 when omega0 r=.600c and explain the limiting radius in this model.

Question 6 - Solution

Maintaining a fixed cyclotron frequency requires a radius-dependent relativistic field

(a) Method and calculation

$$v=\omega_0r,\quad\omega_0=\frac{qB}{\gamma m}\Rightarrow B=\gamma B_0.$$$$B(r)=\frac{B_0}{\sqrt{1-(\omega_0r/c)^2}}.$$

(b) Interpretation and further calculation

$$\omega_0r=.600c\Rightarrow B/B_0=1/\sqrt{1-.600^2}=1.25.$$$$r<c/\omega_0;\quad B(r)\to\infty\text{ as }r\to(c/\omega_0)^-.$$

Physical check

This is a timing condition only; a complete accelerator also needs transverse confinement and practical magnet limits.