For given positive ions in a cyclotron, obtain an expression for the cyclotron frequency.

Explanation

Consider positive ions of charge q and mass m injected in a cyclotron. In the electric-field-free region inside a dee, the ions are acted upon only by the uniform magnetic field. Hence, inside a dee the ions travel in a semicircular path with a constant speed v, in a plane normal to the magnetic field. If B is the induction of the magnetic field, the magnetic force of magnitude qvB is the centripetal force.
`therefore (mv^(2))/(r) =qvB`
`therefore r = (mv)/(qB) " " `...(1)
If t be the time spent in a dee by the ion to describe a semicircular path of radius r,
`t=(pi r)/(v)=(pi)/(v)xx(mv)/(qB)`
`therefore t=(pi m)/(qB) " " ` ...(2)
Thus, t is independent of r and v, i.e., it takes exactly the same time t to travel the semicircular paths inside the dees irrespective of the radius of the path and the speed of the particle.
The period of revolution of an ion in its circular path is
`T=2t =(2pi m)/(qB) " " ` ...(3)
The frequency of revolution,
`f=(1)/(T)=(qB)/(2pi m) " " ` ...(4)
is called the cyclotron frequency or the magnetic resonance frequency.