An alternating electric field, of frequency $f$, is applied across the dees $(\, radius \,\approx\, R)$ of a cyclotron that is being used to accelerate protons $(\,mass \,\approx \, m).$ The operating magnetic field $(B)$ used in the cyclotron and the kinetic energy $(K)$ of the proton beam, produced by it, are given by
AIPMT 2012, Medium
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$\text { Frequency, } v=\frac{e B}{2 \pi m}$
or $\quad B=\frac{2 \pi m v}{e}.........(i)$
As $\frac{m v^{2}}{R}=e v B$
${\text{or }}v = \frac{{eBR}}{m} = \frac{{e2\pi mvR}}{{me}}\,\,\,\,\,(U\sin g\,(i))$
$ = 2\pi vR{\text{ }}.........{\text{(ii)}}$
${\text{Kinetic energy, }}K = \frac{1}{2}m{v^2} = \frac{1}{2}m{(2\pi vR)^2}{\text{ }}$
$=2m\pi ^2v^2R^2$
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