CYCLOTRON

 

Scientists E. O. Lawrence and M. S. Livingston constructed the first cyclotron in 1934 A. D., which is used to accelerate charged particles. To understand how a cyclotron works, consider the motion of a positively charged particle moving with velocity v and entering perpendicularly uniform magnetic field of intensity B as shown in the figure.

The force acting on the charged particle is

1. F = q (v * B) = q v B sin q = q v B   therefore Sin q = p / 2

Under the effect of this force, the charged particle performs uniform circular motionin a plane perpendicular to the plane formed by v and B.

Therefore qvB = mv2 / r and r = mv / qB = p/ qB

Where p is the linear momentum of the charged particle.

Now Putting v = r wc, where wc is called the angular frequency of the cyclotron,

r = mrw / qB because w = qB /m and fc= qB / 2pm

The equation shows that the frequency does not depend on the momentum. Hence On increasing momentum of the particle, the radius of its circular path increases but Its frequency does not. This fact is used in the design of the cyclotron. The figure shows side view and top view of a cyclotron. 

Construction: -

Two D-shaped boxes are kept with their diameters facing each other with a small gap as shown in the figure. A uniform magnetic field is developed

in the space enveloped by the two boxes with a strong electromagnet. These two boxes are called Dees as they are D-shaped. An A.C. of high frequency is applied between the two Dees. The device is kept in an evacuated chamber in order

to avoid the collision of charged particles with the air molecules. 
WORKING: -

A positively charged particle is released at the center P of the gap at time t = 0. It gets attracted towards the Dee, which is at a negative potential at that time. It enters the uniform magnetic field between the Dees perpendicularly and performs uniform circular motion in the gap. As there is no electric field inside the Dees, it moves on a circular path of radius depending upon its momentum and comes out of the Dee after completing a half circle. As the frequency of A.C. (fA) is equal to fc, the diameter of the opposite Dee becomes negative when the particle emerges from one Dee and attracts it with a force, which increases its momentum. The particle then enters the other Dee with larger velocity and hence moves on a circular path of larger radius. This process keeps on repeating and the particle gains momentum and hence radius of its circular path goes on increasing but the frequency remains the same. Thus the charged particle goes on gaining energy, which becomes maximum on reaching the circumference of the Dee. When the particle is at the edge, it is deflected with the help of another magnetic field, brought out and allowed to hit the target. Such accelerated particles are used in the study of nuclear reactions, preparation of artificial radioactive substances, treatment of cancer and ion implantation in solids.

Limitations: -

• According to the theory of relativity, as velocity of the particle approaches that of light, its mass goes on increasing. In this situation, the condition of resonance (fA = fc) is not satisfied.
• To accelerate very light particles like electrons, A.C. of very high frequency (of the order of GHz) is required.
• It is difficult to maintain a uniform magnetic field over large sized Dees. Hence accelerators like synchrotron are developed.