Cyclotron is used to accelerate

$A$ charged particle is subjected to acceleration in a cyclotron as shown. The charged particle undergoes an increase in its speed:

$A$ charged particle moves along a circular path in a uniform magnetic field in a cyclotron. The kinetic energy of the charged particle increases to $4$ times its initial value. What will be the ratio of the new radius to the original radius of the circular path of the charged particle?

In a cyclotron experiment, if we assume that the maximum radius attained by any charged particle is equal to the radius of the dees, then the minimum kinetic energy gained by the particles will be for

$A$ cyclotron's oscillator frequency is $n$ and the radius of the dees is $r$. The operating magnetic field $(B)$ for accelerating protons of charge $q$ and the kinetic energy of the protons produced by the accelerator are respectively ($m$ and $v$ are the mass and velocity of the proton).

Write the resonance condition for a cyclotron.