Question

State the principle of an ac generator and explain its working with the help of a labelled diagram. Obtain the expression for the emf induced in a coil having N turns each of cross-sectional area A, rotating with a constant angular speed ‘ω’ in a magnetic field  vector B, directed perpendicular to the axis of rotation.

Answer 1

The AC Generator works on the principle of electromagnetic induction. when the magnetic flux through a coil changes, an emf is induced in it.

As the coil rotates in magnetic field the effective area of the loop, (i.e. A cosθ) exposed to the magnetic field keeps on changing, hence magnetic flux changes and an emf is induced.

When a coil is rotated with a constant angular speed 'ω ', the angle 'θ' between the magnetic field vector B  and the area vector A , of the coil at any instant 't' equals ωt; (assuming θ = 0° at t = 0) As a result, the effective area of the coil exposed to the magnetic field changes with time ; The flux at any instant 't' is given by

Detailed Answer : 

(a) AC generator

A.C generator is a device which converts mechanical energy into electric energy.

Principle: It works on the principle of electromagnetic induction. It consists of :

(i) Armature coil of large number of turns of copper wire wound over soft iron core, soft iron core is used to increase magnetic flux.

(ii) Field magnets are used to apply magnetic field in which armature coil is rotated with its axis perpendicular to field lines.

(iii) Slip rings used to provide movable contacts of armature coil with external circuit containing load.

(iv) Brushes are the metallic pieces used to pass an electric current from armature coil to the external circuit containing load.

Theory :

Consider a coil PQRS free to rotate in a uniform magnetic field B. The initial flux through the coil is maximum i.e., φ = BA but as the coil is rotating with angular velocity ω, at any instant ‘t’ the flux is given by

As the coil rotates, the magnetic flux linked with it changes. An induced emf is set up in the coil which is given by,

If the coil has N turns, then the total induced emf will be

ε =  NBAω sin ωt.

Thus, the induced emf varies sinusoidally with time ’t’. The value of induced emf is maximum when sin tω = 1 or ωt = 90° , i.e., when the plane of the coil parallel to the field B . Denoting this maximum value by ε₀ , we have

ε₀ = NBAω

ε = ε₀ sin ωt  = = sin sin 2πft where f is the frequency of rotation of the coil.

We consider the following special cases :

(i) When ωt = 0° , the plane of the coil is perpendicular to B, sin ωt = sin0° so that ε = ₀ 

(ii) When ωt = π/2 ,  the plane of coil is parallel to field B, sin sin ωt =  sin π/2 =1 so that  ε = ε₀ 

(iii) When ωt = π , the plane of the coil is again perpendicular to B, sinωt = sinπ = 0 so that ε = 0

(iv) When ωt = 3π/2 , the plane of the coil is again parallel to B, sinωt-sin 3π/2  = − 1 so that ε = − ε₀

(v) When ω = 2π , the plane of the coil again becomes 1 to B after completing one rotation, sin ωt = sin2π =  0 so that ε = 0.

As the coil continues to rotate in the same sense the same cycle of changes repeats again and again. Such an emf is called sinusoidal or alternating emf. Both the magnitude and direction of this emf changes regularly with time.

1. The fact that an induced emf is set up in a coil when rotated a magnetic field forms the basics principle of a dynamo or a generator.

2. The electric current produced varies sinusoidally with time, so is known as ‘alternating current’ and hence the generator is known as ‘A.C. generator’.