(a) Deduce an expression for the frequency of revolution of a charged particle in a magnetic field and show that it is independent of velocity or energy of the particle.

(b) Draw a schematic sketch of a cyclotron. Explain, giving the essential details of its construction, how it is used to accelerate the charged particles.

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

Then, 

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.
b) 

Schematic sketch of a cyclotron is as shown below:
 

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

Construction: The cyclotron is made up of two hollow semi-circular disc like metal containers, D1 and D2, called dees.

It uses crossed electric and magnetic fields. The electric field is provided by an oscillator of adjustable frequency.

Working: A Dee which is at a negative potential accelerates the positive ions which are produced from the source S. Magnetic field which is perpendicular will move the ions in a circular motion inside the Dees. The magnetic field and the frequency of the applied voltages are so chosen that as the ion comes out of a Dee, the Dees change its polarity (positive becoming negative and vice-versa) and the ion is further accelerated. 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.