Question

(a) Deduce an expression for the frequency of revolution of a charged particle in a magnetic field and show that it is independent of velocity or energy of the particle.

(b) Draw a shematic sketch of a cyclotron. Explain, giving the essential details of its construction, how it is used to accelerate the charged particles.

Answer 1

(a) When a charged particle having charge q having velocity v moves inside a magnetic field $\overrightarrow{B}$ , it experiences a force, which is given by :

$\overrightarrow{F}=q(\overrightarrow{v}\times \overrightarrow{B})$

Here,

$\overrightarrow{v}$ is perpendicular to $\overrightarrow{B}$,

$\overrightarrow{F}$ is the force on the charged particle which behaves as the centripetal force and make it move in a circular path.


Let m be the mass of the charged particle and r be the radius of the circular path.

$\therefore q(\overrightarrow{v}\times \overrightarrow{B})=\frac{m{v}^2}{r}$

v and B are at right angles :

$\therefore qvB=\frac{m{v}^2}{r}$

$r=\frac{mv}{Bq}$

Time period of circular motion of the charged particle can be calculated by,

$T=\frac{2\pi r}{v}$

= $\frac{2\pi }{v}\frac{mv}{Bq}$

$T=\frac{2\pi m}{Bq}$

$\therefore$ Angular frequency is

$\omega =\frac{2\pi }{T}$

$\omega =\frac{Bq}{m}$

Therefore, the frequency of the revolution of the charged particle is independent of the velocity or the energy of the particle.

(b) The working principle of a cyclotron is that a charge particle can be accelerated to high velocity and kinetic energy by an oscillating electric field. A cyclotron uses an electric field to accelerate charge particles across the gap between the two D-shaped magnetic field regions. The magnetic field is perpendicular to the paths of the charged particles that makes them follow in circular paths within the two Ds. Each time the charged particles cross the Ds, it is accelerated by an alternating voltage. As its speed increases the radius of path of each particle also increases. So, the accelerated particles move in a spiral path to the outer wall of the cyclotron.


Square wave electric fields are used to accelerate the charged particles in a cyclotron.


At the time the charge particle finishes its half circle, the accelerating electric field reverse so that it gets accelerated across the gap between the Ds.

The particle gets accelerated again and again, and its velocity increases. Therefore, a high kinetic energy is achieved.