Final answer:
The frequency of revolution of a charged particle in a magnetic field is independent of its velocity and energy. It can be shown that the frequency is given by f = qB/(2πm), determined by the charge q, magnetic field B, and mass m of the particle.
Step-by-step explanation:
The frequency of revolution of a charged particle in a magnetic field can be deduced by considering the magnetic force acting as a centripetal force that keeps the particle in a circular path. When a charged particle, with charge q, moves with a velocity v perpendicular to the magnetic field B, it experiences a magnetic force F given by F = qvB. This force serves as the centripetal force Fc = mv2/r, where m is the particle's mass and r is the radius of the circular path.
Equating the magnetic force to the centripetal force and rearranging, qvB = mv2/r, we can find that the velocity v = qBr/m. The period T of the circular orbit is the time taken for one complete revolution, which is T = 2πr/v. Substituting for v, the period T becomes T = 2πm/(qB). The frequency f, which is the inverse of the period, is thus f = 1/T = qB/(2πm).
This frequency is independent of the velocity and radius of the orbit and, therefore, also of the kinetic energy of the particle. Therefore, in devices like cyclotrons, the frequency of the alternating voltage source used to accelerate particles can be kept constant, since it does not depend on the particle's energy.