Question

Write the working of cyclotron in brief. Draw a schematic sketch of the cyclotron showing path of accelerated charged particles (ions) in both Dees.
Derive the following parameters of cyclotron:
(a) Cyclotron frequency and
(b) Kinetic energy of ions of cyclotron.

Answer 1

Cyclotron is a device used to accelerate charged particles to a specified velocity. In a cyclotron, a particle is projected in a region of orthogonal electric and magnetic fields. The particle hence experiences a net force consisting of electric and magnetic fields.

Electric force increases the speed of the particle in the direction (or opposite) of electric field. Magnetic force changes the direction of velocity and acts perpendicular to its direction and that of magnetic field. Hence, the particle follows a helical path as shown in the attached figure. Speed of the particle is fixed depending on the distance from the center $P$. Accordingly a window $W$ is given so that the particles coming out have a predefined speed.

Let the velocity of the particle be $v$ making an angle of $\theta$ with the electric field.

(a)

Magnetic force acting on the particle is:

$F_B=qvB$

This provides the centripetal force for the particle. Hence,

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

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

Time period can be found using:

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

$T=\frac{mv}{qB}$

(b)

Kinetic energy is given by:

$K=\frac{1}{2}m{v}^2$

$=\frac{1}{2}m{\left(\frac{qBr}{m}\right)}^2$

$=\frac{q^2{B}^2{r}^2}{2m}$