Question
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 $$\left( B \right)$$ used in the cyclotron and the kinetic energy $$\left( K \right)$$ of the proton beam, produced by it, are given by :
A.
$$B = \frac{{m\upsilon }}{e}\,{\text{and}}\,K = 2m{\pi ^2}{\upsilon ^2}{R^2}$$
B.
$$B = \frac{{2\pi m\upsilon }}{e}\,{\text{and}}\,K = {m^2}\pi \upsilon {R^2}$$
C.
$$B = \frac{{2\pi m\upsilon }}{e}\,{\text{and}}\,K = 2m{\pi ^2}{\upsilon ^2}{R^2}$$
D.
$$B = \frac{{m\upsilon }}{e}\,{\text{and}}\,K = {m^2}\pi \upsilon {R^2}$$
Answer :
$$B = \frac{{2\pi m\upsilon }}{e}\,{\text{and}}\,K = 2m{\pi ^2}{\upsilon ^2}{R^2}$$
Solution :
Time period of cyclotron is
$$\eqalign{
& T = \frac{1}{\upsilon } = \frac{{2\pi m}}{{eB}};B = \frac{{2\pi m}}{e}\upsilon ;R = \frac{{m\upsilon }}{{eB}} = \frac{p}{{eB}} \cr
& \Rightarrow P = eBR = e \times \frac{{2\pi m\upsilon }}{e}R = 2\pi m\upsilon R \cr
& K.E.{\text{ }} = \frac{{{p^2}}}{{2m}} = \frac{{{{(2\pi m\upsilon R)}^2}}}{{2m}} = 2{\pi ^2}m{\upsilon ^2}{R^2} \cr} $$