Question

3) A cyclotron is needed to accelerate protons to (1/30) c( i.e., one-thirtieth the speed of light). The cyclotron magnetic field is 1 T.

(a) What is the minimum radius the cyclotron can have?

(b) What energy would an 'α' particle and a deuteron1have if produced from the same cyclotron?

Answer 1

Since generally we consider relativistic effects when the magnitude of the speed is greater than $c/10$, we can ignore them in this problem.

(a) Equilibrium of forces:

$$qvB=m\frac{v^2}{R},\\\space\\ R=\frac{mv}{qB}=0.104\text{ m}.$$

(b) At these speeds their kinetic energies would be

$$E_{K\alpha}=\frac12 m_\alpha v^2=2.071\text{ MeV},\\\space\\ E_{KD}=\frac12 m_\alpha v^2=1\text{ MeV}.$$