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. `B = (2pi mv)/( e) " and " K = m^2 pi vR^2`
B. `B = (2 pi mv)/(e ) " and " K = 2mpi^2 v^2 R^2`
C. `B = (mv)/(e ) " and " K = m^2 pi v R^2`
D. `B= (mv)/(e ) " and " K = 2m pi^2 v^2 R^2`

Answer

B

Explanation

`because` proton will change its velocity oppositely in time equal to half time period of AC source .
` therefore v =(piR)/(T//2) rArr v = (2pi R)/(T) rArr v = 2pi Rv`
` because R = (mv)/(eB) rArr R = (m (2pi Rv) )/(eB) rArr B = (2pi mv)/(e ) `
`K = 1/2 mv^2 = 1/2 m (2pi Rv)^2 = 2mpi v^2 R^2`