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

Answer

A

Explanation

In cyclotron frequency, `v=(eB)/(2pim)`
or `B=(2pir*v)/(e)`
Also, relation for radius (R) is
`R=(mv)/(eB)orv=(eBR)/(m)`
`:." "KE=(1)/(2)mv^(2)=(1)/(2)mxx(e^(2)B^(2)R^(2))/(m^(2))=(e^(2)B^(2)R^(2))/(2m)`
`(e^(2)R^(2))/(2m)xx(4pi^(2)m^(2)v^(2))/(e^(2))=2pi^(2)mv^(2)R^(2)`