Answer :

**Principle- **Cyclotron is a device that works on the principle that a charged particle moving with a certain velocity in a direction perpendicular to the magnetic field experience Lorentz force, as a result, the particle describes a circular motion.

A charged particle can be accelerated to very high energy by making it pass through a moderate electric field a number of times. This can be done with the help of perpendicular magnetic field which throws the charged particle into the circular motion, the frequency of which does not depend on the speed of the particle and radius of the circular orbit.

The combination of the crossed; electric and magnetic field is used to increase the energy of charged particle.Cyclotron uses the fact that the frequency of revolution of the charged particle in a magnetic field is independent of its energy.

Inside the Dees, the particle is shielded from the electric field and magnetic field acts on the particle and makes it move in circular path.

Every time the particle moves from one Dee to another Dee, it comes under the influence of electric field which ensures increase of energy of the particle as the sign of electric field changes alternately.

The increased energy increases due to radius of the circular path. So accelerated particles move in spiral path.

**Working-**When a positive charge q of mass m is emitted from the source it is accelerated towards the Dee having a negative potential .Due to normal magnetic field,the ions experiences magnetic force and moves in a circular path .When the charged particle arrives at the gap between the Dee the polarities of the Dee changes and the in the gap the charged particles gains extra velocity and is then transferred to the another Dee making a circular path with a greater radius. In this way the charged particles keeps moving in the two Dees each time increasing in velocity and radius of path. When it reaches its maximum radius the charged particle comes out of the Dee with high velocity and hits the deflection plate T.

Suppose the charge on the positive ion is +q ,its mass ‘m’ and strength of magnetic field is B.

Let, at any time, the radius of circular path is ‘r’ and speed of positive ion is ‘v’.

Then, equating the Lorentz’s force and centripetal force we get,

time taken to complete circle of radius ‘r’ is,

Substituting the value of r we get,

We know that, T=. So substituting the value in above equation we get,

This is the required expression.

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