Question

Answer the following questions:

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

Answer 1

$\left(i\right)$ In cyclotron, magnetic field is always ${\perp }^r$ to the motion of particle and provides necessary centripetal force.

$\frac{m{\vartheta }^2}{r}=q\vartheta B$

$\therefore \vartheta =\frac{qrB}{m}$

and hence period of revoluation

$T=\frac{2\pi r}{\vartheta }=\frac{2\pi r}{\frac{qrB}{m}}$

$T=\frac{2\pi r}{qB}$

$\therefore$ cyclotron frequency $\frac{1}{T}=\frac{qB}{2\pi m}=\vartheta$

$\left(ii\right)$ Proton Deutron

mass=m mass=2m

charge=q charge=q

$\therefore {\vartheta }_p=\frac{qB}{2\pi m}$ ${\vartheta }_d=\frac{qB}{2\pi 2\left(m\right)}==\frac{qB}{4\pi m}$

as cyclotron frequency in both case is different hence same oscillator frequency can not be used to accelerate both particle.