GATE 2025 PH – Question 59
Cyclotrons accelerate deuterons and alpha particles in the same magnetic field to energies 10 and 20 MeV, respectively. Their masses are $M_d=2000$ MeV/$c^2$ and $M_\alpha=4000$ MeV/$c^2$. The extraction-radius ratio $r_\alpha/r_d$ (in integer) is _____
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Correct answer: 1
Explanation
In a cyclotron the particle moves on a circle in the magnetic field $B$ with radius
$$r=\frac{mv}{qB}=\frac{\sqrt{2mE}}{qB}\quad(\text{non-relativistic},\ E=\tfrac12mv^2).$$
**Deuteron:** $q=e$, $m=2000$ MeV/c², $E=10$ MeV:
$$r_d\propto\frac{\sqrt{2\times2000\times10}}{1}=\sqrt{40\,000}=200 .$$
**Alpha particle:** $q=2e$, $m=4000$ MeV/c², $E=20$ MeV:
$$r_\alpha\propto\frac{\sqrt{2\times4000\times20}}{2}=\frac{\sqrt{160\,000}}{2}=\frac{400}{2}=200 .$$
$$\frac{r_\alpha}{r_d}=\mathbf{1}.$$