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Discussion Forum



1)

An alternating electric field of frequency v is applied across the dees (radius R) of a cyclotron to accelerate protons  (mass m). The operating magnetic field B used and KE of the proton beam produced by it are respectively(where, e= charge on proton).


A) $\frac{2\pi mv}{e},2\pi^{2}mv^{2}R^{2}$

B) $\frac{2\pi^{2} mv}{e^{2}},4\pi^{2}mv^{2}R^{2}$

C) $\frac{\pi^{} m^{}v^{}}{e^{}},\pi^{2}mv^{2}R^{2}$

D) $\frac{2\pi^{2} m^{2}v^{2}}{e^{2}},2\pi^{2}mv^{2}R^{2}$

Answer:

Option A

Explanation:

 In cyclotron frequncy, $v=\frac{eB}{2\pi m}$

 or   $B=\frac{2\pi m.v}{e}$

 Also, relation for radius (R)is

$R=\frac{ m.v}{e B}$ or $v=\frac{eBR}{m}$

$\therefore$    $K E =\frac{1}{2}mv^{2}=\frac{1}{2}m\times\frac{ e^{2}B^{2}R^{2}}{m^{2}}=\frac{ e^{2}B^{2}R^{2}}{2m}$

 = $\frac{ e^{2}R^{2}}{2m}\times\frac{4\pi^{2}m^{2}v^{2}}{e^{2}}=2\pi^{2}m R^{2}v^{2}$