Show that the time period of revolution of particles in a cyclotron is independent of their speeds. Why is this property necessary for the operation of a cyclotron?

#### Solution

Let a particle of charge q and mass m enter a region of magnetic field B with a velocity v normal to the field. The particle follows a circular path inside the cyclotron, and the necessary centripetal force is provided by the magnetic field.

Therefore, we have

`qvB=(mv^2)/r`

`:.r=(mv^2)/(qvB)=(mv)/(qB)`

Now, the time period of revolution will be

`T=d/v=(2pir)/v=(2pimv)/(qBv)=(2pim)/(qB)`

Therefore, from the above expression, we see that the time period is independent of the speed of the particle.

The time period should be independent of speed so that the frequency of revolution of the particle remains equal to the frequency of the ac source applied to the cyclotron.