Cyclotrons and Synchrotrons — Question 4

Question 4

Acceleration time and extracted beam power count different kinds of current

Singly charged ions enter a cyclotron at 40.0 keV and leave at 2.04 MeV. They gain 20.0 keV per gap crossing, cross twice per turn and circulate at a constant 10.0 MHz. The extracted beam current is 2.00 microampere. Ignore losses.

1. Find the number of turns and acceleration time.

2. Find the RF power transferred to the extracted beam above its injection energy, and the total output kinetic power.

Question 4 - Solution

Acceleration time and extracted beam power count different kinds of current

(a) Method and calculation

$$\Delta K=2.00\,\mathrm{MeV},\quad\Delta K_{\mathrm{turn}}=40.0\,\mathrm{keV}\Rightarrow N=50.0.$$$$t=N/f=50/(10^7)=5.00\,\mathrm{\mu s}.$$

(b) Interpretation and further calculation

$$P_{\mathrm{RF}}=\frac{I_b}{e}\Delta K=I_b(2.00\times10^6\,\mathrm V)=4.00\,\mathrm W.$$$$P_{\mathrm{out}}=I_b(2.04\times10^6\,\mathrm V)=4.08\,\mathrm W.$$

Physical check

Extracted current counts particles delivered per second; it is not the current associated with one circulating particle.