1. The cyclotron
The cyclotron was first designed to avoid the need for a long linear accelerator to obtain high energy particles. It consists of two semicircular hollow conductors or “D’s” .Perpendicular to the D’s is set a large electromagnet.
A source of ions is placed at the centre of the D’s. These are bent into a circular path by the perpendicular magnetic field.

2. The cyclotron D2 a c D1
A high frequency alternating potential difference is applied to the D’s which must change over just as the ions are crossing the gap between the D’s. A positive ion emerging at position a is attracted to the negative potental of D1. When the ion gets to C, D1 is positive and D2 is now negative and again the ion is attracted across the gap.

4. This part of the motion is where the acceleration takes place due to the electric field- +

5. This part of the motion is where the acceleration takes place due to the electric field- +

6. This part of the motion is where the acceleration takes place due to the electric field- +

7. This part of the motion is where the acceleration takes place due to the electric field- +

8. Within the D the perpendicular magnetic field forces the particle into a circular path

9. Inside a D the path of the ion is circular and the force towards the centre is given by
F is the centripetal force applied to the ion to make its path circular and so :
The length of path in a D is
The time an ion spends in each D is
Hence:
Where T is the time taken for 1 rotation

10. The cyclotron Circular motion theory gives us From and So 1
Substituting this value into 1
Gives:
This formula tells us that the period of the motion does not depend on the radius of the ion path or on the speed of the particles. It is the same for all ions of the same type.
So once the frequency is adjusted correctly the energy of the ions is increased every time they cross the gap between Ds