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 schematic sketch of a cylotron. Explain, giving the essential details of its construction, how it is used to accelerate the charged particles.

Answer 1

(a)

Magnetic force on charged particle in magnetic field is give by:

$F=qvB$

This is equal to the centripetal force acting on the particle.

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

$\frac{v}{r}=\frac{qB}{m}$

Frequency is given by:

$f=\frac{v}{2\pi r}=\frac{qB}{2\pi m}$

Hence, frequency is independent of velocity of the particle.

(b)

Schematic diagram of cyclotron is attached in the given figure.

It is used to accelerate particles to very high speeds. Protons are injected from a source in a region consisting of orthogonal electric and magnetic fields. The particle feels a centripetal force due to the magnetic field and an accelerating force due to the electric field.

Radius of a particle in the cyclotron is given by:

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

Hence, given the radius of the particle, speed of the particle can be found. This is used to accelerate the particle to the given value.