A cyclotron's oscillator frequency is `10MHz`. What should be the operating magnetic field fro accelerating protons? If the radius of its dees is `60cm`, what is the kinetic energy (in `MeV`) of the proton beam produced by the acceleration?
`(e-1.60xx10^-19C,m_p=1.67xx10^-27kg, 1MeV=1.6xx10^-13J)`

Explanation

The oscillator frequency should be same as proton.s cyclotron frequency.
Using equations `r= mv/ qB` and `omega = 2piv = qB/ m `we have `B=2 pi mv/q = 6.3xx1.67xx10^(-27) xx10^(7)/(1.6 xx10^(-19))`=0.66T
Fint velocity of protons is
`v=r xx2piv=0.6mxx6.3xx10^(7)=3.78xx10^(7)=3.78xx10^(7)m//s`
`E=1/2 mv^(2)=1.67xx10^(-27)xx14.3xx10^(14)/(2xx1.6xx10^(-13))=7 MeV`