With the help of a labelled diagram, state the underlying principle of a cyclotron. Explain clearly how it works to accelerate the charged particles.

Show that cyclotron frequency is independent of energy of the particle. Is there an upper limit on the energy acquired by the particle? Give reason.   

Underlying principle of cyclotron: When a charged particle is kept in a magnetic field it experiences a force and the perpendicular magnetic field causes the particle to rotate.

Working:
Consider the figure which is shown. 

Now, the ions move with higher velocity along a circular path of greater radius. The phenomenon is continued till the ion reaches at the periphery of the Dees where an auxiliary negative electrode (deflecting plate) deflects the accelerated ion on the target to be bombarded.

Expression for cyclotron frequency:

Let the velocity of the positive ion having charge ‘q’ be ‘v’.

Then, 
                                    
 ;m is the mass of ion, r the radius of the path of ion in the dee and B is the strength of the magnetic field.

Angular velocity of the ion is given by, 

Now, time taken by the ion in describing a semi-circle i.e., turning through an angle  is given by, 

The applied alternating potential should also have the same semi-periodic time (T/2) as that taken by the ion to cross either Dee.

That is, 
 
                          
, is the expression for period of revolution. 
So, Frequency of revolution of particle is given by, 

This frequency is called the cyclotron frequency which is independent of the speed of the particle.