Under the influence of a uniform magnetic field, a charged particle moves with constant speed v in a circle of radius R. The time period of rotation of the particle

The magnetic force provides the centripetal force: $qvB = \frac{mv^2}{R}$, so $R = \frac{mv}{qB}$
Time period: $T = \frac{2\pi R}{v} = \frac{2\pi}{v}\times\frac{mv}{qB}$
$T = \frac{2\pi m}{qB}$
T depends only on the mass, the charge and the field; v and R cancel out (a faster particle moves in a proportionately bigger circle).
Hence the time period is independent of both v and R.