An alternating electric field, of frequency v, is applied across the dees (radius=R) of a cyclotron that is being used to accelerate protons (mass=m) the operating magnetic field (B) used in the cyclotron and the kinetic energy (K) of the proton beam, produced by it, are given by:

Options

A. `(2pi mv)/(e), 2pi^(2)m v^(2)R^(2)`
B. `(2pi^(2)mv)/(e^(2)), 4pi ^(2)mv^(2)pi^(2)`
C. `(pimv)/(e),pi^(2)mv ^(2)R^(2)`
D. `(2pi^(2) m^(2)v^(2))/(e), 2pi^(2)m^(2) v^(2)R^(2)`

Answer

A

Explanation

Data
Now radiusof circular path `r= (mv)/(qB)`
`B=(mv)/(qr) =m/q .omega " "(because v=r omega)`
`B= m/q 2piv= (2 pim v)/(e) " "(because q=e)`
`K.E. =1/2 mv ^(2) but v = (2 pi r)/(T) =2 pi rv`
` therefore K.E. =1/2 m 4 pi^(2) r^(2) v^(2) =2pi ^(2) mr^(2) v^(2)`