question_answer

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  used in the cyclotron and the kinetic energy (1C) of the proton beam, produced by it, are given by                                                                                 [AIPMT (S) 2012]

A)  \[B=\frac{mv}{e}\] and \[K=2m{{\pi }^{2}}{{v}^{2}}{{R}^{2}}\]

B)  \[B=\frac{2\pi mv}{e}\] and \[K={{m}^{2}}\pi v{{R}^{2}}\]

C)  \[B=\frac{2\pi mv}{e}\] and \[K=2m{{\pi }^{2}}{{v}^{2}}{{R}^{2}}\]

D)  \[B=\frac{mv}{e}\] and \[K={{m}^{2}}\pi v{{R}^{2}}\] Correct Answer: C Solution : [c] Frequency \[v=\frac{eB}{2\pi m}\]

\[KE=\frac{1}{2}m{{v}^{2}}\] and radius \[R=\frac{mv}{eB}\]

Here, velocity \[v=\frac{\pi R}{T/2}=\frac{2\pi R}{T}=2\pi Rv\]

\[\therefore \] Radius \[R=\frac{m(2\pi Rv)}{eB}\]

Magnetic field \[B=\frac{2\pi mv}{e}\] Kinetic energy

\[K=\frac{1}{2}m{{(2\pi Rv)}^{2}}=2m{{\pi }^{2}}{{v}^{2}}{{R}^{2}}\]