(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 cyclotron. Explain, giving the essential details of its construction, how it is used to accelerate the charged particles.

OR

(a) Draw a labelled diagram of a moving coil galvanometer. Describe briefly its principle and working.

(b) Answer the following:

(i) Why is it necessary to introduce a cylindrical soft iron core inside the coil of a galvanometer?

(ii) Increasing the current sensitivity of a galvanometer may not necessarily increase its voltage sensitivity. Explain, giving reason

a) Let the charge on the particle be q and its mass be m,

And its velocity be v,

The particle is rotating in the circle of radius r

The force acting on the particle is given as,

F = qvB

Which will account for the centripetal force required,

So the time taken to complete one revolution is,

The frequency of the particle is 1/T,

So, clearly it can be seen that the frequency is independent of velocity or energy of the particle

a) The cyclotron is used to accelerate charged particles to very high energies. The cyclotron exploits both electric and magnetic fields in combination to increase the energy of particles. The frequency of revolution of a charged particle in a magnetic field is independent of its energy. The particles travel inside two semicircular disc-like metal containers, D₁ and D₂, which are called Dees as appear as the letter D. the diagram below shows a schematic view of the cyclotron.

Inside the Dees the particle is not affected by the electric field. The magnetic field acts on the particle and makes it go around in a circular path inside a dee. Every time the particle passes from one dee to another it is gets accelerated by the electric field. The sign of the electric field is changed alternately in tune with the circular motion of the particle. Each time the acceleration increases the energy of the particle. The particles move in a semi-circular path in one of the Dees and arrive in the gap between the Dees in a time interval T/2; where T, the period of revolution, is given by,

This frequency is called the cyclotron frequency.

OR

a) the Moving coil galvanometer consists of a coil having large number of turns, which is allowed to rotate about a fixed axis, in a uniform radial magnetic field. There is a cylindrical soft iron core which not only makes the field radial but also increases the strength of the magnetic field. When a current flows through the coil, a torque acts on it.

b)

This torque is given by.

=NIAB

N is the number of turns, I is the current, A is area of c.s., B is the strength of the magnetic field.

The magnetic torque NIAB tends to rotate the coil. A spring S_P provides a counter torque k that balances the magnetic torque NIAB, resulting in a steady angular deflection . In equilibrium,

k= NI AB

where k is the torsional constant of the spring, i.e. the restoring torque per unit twist. The deflection is indicated on the scale by a pointer attached to the spring. We have.

The term NAB/k is constant for a given galvanometer.

b)

i) When a soft iron core is inserted is used the magnetic field lines tends to crowd through the core. It is because, soft iron core is ferromagnetic in nature. As a result, the strength of the magnetic field increases, which in turn increases the sensitivity of the galvanometer.

ii) Current sensitivity of a galvanometer is defined as the deflection produced when a unit current flows through it.

Voltage sensitivity of a galvanometer is defined as the deflection produced in the galvanometer when a unit voltage is applied across two terminals.

In case of current sensitivity if we increase the no. of turns n its current sensitivity increases. Since resistance of galvanometer R also increase, Voltage sensitivity remain the same