(i) Obtain the expression for the cyclotron frequency.
(ii) A deutron and a proton are accelerated by the cyclotron. Can both the accelerated with the same oscillator frequency ? Give greason to justify your answer.

Explanation

(i) In a cyclotron, centripetal force is provided by the magnetic field to a charged particle which helps the particle in moving in a circular path. Equating the two forces we have
`(mv^(2))/(r )= qvB`
`rArr v - (qBr)/(m)`
Period of Revolution `T = (2pir)/(v) = (2pir)/(qBr) .m`
`rArr T = (2pim)/(qB)`
The frequency of particle in a cyclotron,
`v - 1/T = (qB)/(2pim)` Ang. frequency `= omega = (2pi)/(T) = (Bq)/(m)`.
(ii) A cyclotron works only when the frequency of the revolution of particle is equal to the frequency of oscillation.
`:.` Mass of proton and deutron are different `(m_(3) = 2m_(P))` and `v alpha = 1/(m)`.
`:.` The frequency of revolution of proton and deutron is different.Hence, they can't be accelerated with the same oscillator frequency.