Cyclotron Frequency

The cyclotron frequency comes from equating the magnetic force to the centripetal force, $qvB=\frac{m{v}^2}{r}$, which yields $r=\frac{mv}{qB}$. The period of one revolution is $T=\frac{2\pi r}{v}=\frac{2\pi m}{qB}$, and the frequency is $f=\frac{qB}{2\pi m}$. Strikingly, the speed $v$ cancels out, so the frequency depends only on the charge-to-mass ratio and the field, not on how fast the ion moves. This is exactly why a cyclotron can use a fixed-frequency oscillator: as the ion gains energy and spirals outward, it still completes each loop in the same time. The option that frequency increases with speed is wrong because the larger radius compensates for the higher speed. The option that it decreases with speed similarly ignores this cancellation. The square-of-speed option has no basis in the derivation. NCERT highlights this constancy as the central design principle of the cyclotron. A check shows $f$ carries units of hertz from tesla times coulomb per kilogram. This speed independence is what makes the classical cyclotron practical, since a single radio-frequency oscillator tuned to the constant cyclotron frequency keeps accelerating the ion turn after turn. The limitation appears only at very high energies, where relativistic mass increase slowly lowers the true frequency and the simple constant-frequency picture breaks down, which is the motivation for the synchrocyclotron that adjusts its frequency to compensate for the growing relativistic mass.