Question

A cyclotron is used to accelerate protons to a kinetic energy of $$5 \,MeV$$. If the strength of magnetic field in the cyclotron is $$2T$$, find the radius and the frequency needed for the applied alternating voltage of the cyclotron.
(Given: Velocity of proton $$= 3 \times 10^7 \,m/s$$)

Answer 1

$$KE = \dfrac{1}{2} mV^2$$
$$= 5 MeV$$
$$= 5 \times 10^6 \times 1.6 \times 10^{-19} J$$
$$= 8 \times 10^{-13} J$$
by force balance
$$\dfrac{mV^2}{r} = qVB$$
$$\dfrac{1}{2} mV^2 \times \dfrac{1}{r} = \dfrac{qVB}{2}$$
$$K.E. \times \dfrac{1}{r} = \dfrac{qVB}{2}$$
$$r = \dfrac{2 \times K.E.}{qUB}$$
$$= \dfrac{2 \times 8 \times 10^{-13}}{1.6 \times 10^{-19} \times 3 \times 10^7 \times 2}$$
$$r = 0.166 m$$
$$V = r w$$
$$w = 2 \pi = \dfrac{qB}{m}$$
$$V = \dfrac{qB}{2 \pi m}$$
$$V = 3.18 \times 10^7 Hz$$