Cyclotron Basic Construction and Working Principle

Before understanding the basic working principle of a cyclotron, it is necessary to understand the force on a moving charged particle in a magnetic field and its motion within that field.

Force on a Moving Charged Particle in a Magnetic Field

When a current-carrying conductor of length L meters with current I amperes is placed perpendicularly in a magnetic field of flux density B Weber per square meter, a magnetic force acts on the conductor.

Now, consider there are N mobile free electrons in the conductor of length L meters, causing the current of I amperes.

Where, e is the electric charge of one electron and it is 1.6 × 10^-19 coulomb.
Now from equation (1) and (2) we get

Here, N number of electrons causing current I ampere, and consider that they travel length L metre in time t, therefore drift velocity of the electrons would be

From equation (3) and (4), we get

It is the force acting on N number of electrons in the magnetic field hence force on a single electron in that magnetic field can be

Motion of Charged Particle in a Magnetic Field

When a charged particle moves in a magnetic field, there are two main conditions: the particle moves either along the direction of the magnetic field or perpendicular to it.
When the particle moves along the axis of the direction of magnetic field, magnetic force acting on it,

Hence there will be no force acting on the particle, hence no change in the velocity of particle and hence it moves in straight line with constant speed.

Now if the charged particle moves perpendicular to the magnetic field then there will be no change in the speed of the particle. This is because the force acting on the particle is perpendicular to the motion of the particle hence the force will not do any work on the particle so there will be no change in the speed of the particle.
But this force acting on the particle perpendicular to its motion and the direction of the motion of the particle will change continuously. As a result the particle will move in a in the field in a circular path of a constant radius with constant speed.
If the radius of the circular motion is R metre then

Now,

Hence radius of the motion depends upon the velocity of the motion.
Angular speed and time period are constant.

Basics Principle of Cyclotron

The cyclotron uses the concept of charged particles moving in a magnetic field. Though simple in design, it has significant applications in engineering, physics, and medicine. It is a device that accelerates charged particles using a perpendicular magnetic field.

Construction of Cyclotron

This device basically has three main constructional parts

1. Large sized electromagnet to create uniform magnetic field in between its two face-to-face placed magnetic opposite poles.
   
2. Two low height hollow half cylinders made of high conductive metals. These components of cyclotron are called Dees.
   
3. A high-frequency alternating high voltage source.

Constructional Details

The Dees are placed face to face in between the electromagnetic poles. The dees are so placed, that the straight edge will be face-to-face with small gap between them. Also the magnetic flux of the electromagnet cut these Dees exact perpendicularly. Now these two Dees are connected to two terminals of an alternating voltage source so that if one Dees is in the positive potential than other will be in exact opposite negative potential at same time. As the source is alternating the potential of the Dees are altered according to the frequency of the source. Now if a charged particle is thrown from a point near to the centre of one Dees with certain velocity V₁. As the movement of the particle now perpendicular to the externally applied magnetic field, there will be no change of velocity but the charged particle will follow a circular path of radius

Where, m gram is the mass and q coulomb is the charge of the thrown particle and B Weber/metre² is the flux density of externally applied perpendicular magnetic field.
After travelling π radians or 180^o with radius R₁ the charged particle comes to the edge of the Dee. Now the time period and frequency of the applied voltage source is so adjusted with the time period of circular motion that is

That the polarity of the other Dee becomes opposite that of the charged of the particle. Hence due to attraction of the Dee ahead the moving particle and also due to repulsion of the Dee in which the particle is now situated, the particle gets extra kinetic energy.

Where ν₁ is the velocity of the particle at previous Dee and ν₂ is the velocity of the particle in next Dee. Now the particle will move with this greater velocity with radius R₂ metre.

Again due to constant perpendicular magnetic field the particle travels another half cycle with this new radius R₂ metre and comes to the edge of present Dee. When it comes to this edge, the ahead Dee again becomes in opposite polarity of the behind and the particle crosses the gap between Dees with gain of kinetic energy qV and hence again there is gain of velocity and radius of the circularly travelling charged particle. In this way the charged particle follows a spiral path of motion with continually increasing velocity. Therefore the charged particle gets sufficiently high required velocity before leaving the cyclotron gun head.
The frequency of voltage source say f.

Here, 2π is constant, m, q and B are known hence T can be calculated and hence frequency of the voltage source would be

Application of Cyclotron

There are mainly two types of application of cyclotron. One is in lavatory of different physics experiments when highly accelerated photons are required. Also highly accelerated photons are used to irradiate tissues.