Cyclotrons and Synchrotrons — Question 2

Question 2

The outer dee radius sets a kinetic-energy ceiling in the classical model

A nonrelativistic cyclotron has fixed B and maximum usable orbit radius R. It accelerates positive particles of mass m and charge q.

1. Derive the maximum speed and kinetic energy in terms of B and R.

2. For the same m and q, compare doubling B with doubling R. State the consistency condition for using the classical result.

Question 2 - Solution

The outer dee radius sets a kinetic-energy ceiling in the classical model

(a) Method and calculation

$$R=\frac{mv_{\max}}{qB}\Rightarrow v_{\max}=qBR/m.$$$$K_{\max}=\frac{q^2B^2R^2}{2m}.$$

(b) Interpretation and further calculation

$$B\prime=2B\text{ or }R\prime=2R\Rightarrow v_{\max}\prime=2v_{\max},\quad K_{\max}\prime=4K_{\max}.$$$$qBR/(mc)\ll1\quad\text{is required for nonrelativistic consistency}.$$

Physical check

Increasing the geometric or magnetic limit can also invalidate the assumption of speed-independent cyclotron frequency.