Q. An alternating electric field, of frequency $v$, is applied across the dees (radius $= R$) of a cyclotron that is being used to accelerated 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
A) $B = \frac{mv}{e}$ and $K = 2m\pi^{2}v^{2}R^{2}$
B) $B = \frac{2\pi mv}{e}$ and $K = m^{2}\pi vR^{2}$
C) $B = \frac{2\pi mv}{e}$ and $K = 2m\pi^{2}v^{2}R^{2}$
D) $B = \frac{mv}{e}$ and $K = m^{2}\pi vR^{2}$
Explanation
(c) Time period of cyclotron is
$T = \frac{1}{v} = \frac{2\pi m}{eB}$
$B = \frac{2\pi m}{e}v$
$R = \frac{mv}{eB} = \frac{p}{eB} \Rightarrow p = eBR = e \times \frac{2\pi mv}{e}R = 2\pi mvR$
$K.E. = \frac{p^{2}}{2m} = \frac{(2\pi m\ v\ R)^{2}}{2m} = 2\pi^{2}\ mv^{2}R^{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.