Give the principle of a cyclotron. Draw a labelled diagram of cyclotron and describe how a positively charged particle is accelerated in it?

Principle of cyclotron : 
The working of the cyclotron is that a positively charged particle can be accelerated to sufficiently high energy using smaller oscillating electric field by making the cyclotron cross the same elctric field time and again by magnetic field.  


 
Working of cyclotron: 
1.The positive ion which is to be accelerated is bought at a point in between D₁ and D_2. Suppose, D₁ is at a negative potential and D₂ is at a positive potential. Hence, the ion will be accelerated towards D₁.
2. The ion will be in a free space once it enters D₁ and therefore, it moves with a constant speed say, v.
3. A perpendicular magnetic field B is applied. So, the ion will undergo a circular path of radius r in D₁ .
The circular path is given by,
                           $qvB = \frac{m{v}^2}{r}$ 
where, m is the mass of ion and,
q is the charge on ion.

                       $\therefore \mathrm{r} = \frac{\mathrm{mv}}{\mathrm{qB}}$ 
4. Now, time taken by an ion to describe a semicircular path is given by,  

       t =$\frac{\mathrm{\pi r}}{\mathrm{v}} = \frac{\pi m}{qB} = \frac{\pi }{B{\displaystyle \raisebox{1ex}{$(q$}\!\left/ \!\raisebox{-1ex}{$m)$}\right.}} = cons\tan t$  

5. If, this time during which the positive ion describes a semi-circular path is the time during which half-cycle of electric oscillator is completed, then as the ion arrives in the gap between the two dees, polarity of the two dees gets reversed. Consequently, positive ion is accelerated towards D₂ and the ion enters D₂ with greater speed which will remain constant in D₂. 
6. Ion will describe a  semicircular path of greater radius  due to perpendicular magnetic field and will again arrive in a gap between the two dees exactly at that instant when, the polarity of two dees is reversed. 
7. Thus, positive ion will keep on accelerating everytime it comes into the gap between the dees and will describe a circular path of greater and greater radius henceforth, acquiring sufficiently high kinetic energy.