State the principle of an ac generators and explain its working with the help of a labelled diagram. Obtain the expression for the emf induced in a coil havin N turns each of cross-sectional area, rotating with a constant angular speed 'ω' in a magnetic field $\overrightarrow{\mathrm{B}}$, directed perpendicular to the axis of rotation.

Principle − AC generator based on the phenomenon of electromagnetic induction.

Construction:

Main parts of an ac generator:

1. Armature − Rectangular coil ABCD

2. Filed Magnets − Two pole pieces of a strong electromagnet

3. Slip Rings − The ends of coil ABCD are connected to two hollow metallic rings R₁ and R₂.

4. Brushes − B₁ and B₂ are two flexible metal plates or carbon rods. They are fixed and are kept in tight contact with R₁ and R₂ respectively.

Theory and Working − As the armature coil is rotated in the magnetic field, angle θ between the field and normal to the coil changes continuously. Therefore, magnetic flux linked with the coil changes. An emf is induced in the coil. According to Fleming’s right-hand rule, current induced in AB is from A to B and it is from C to D in CD. In the external circuit, current flows from B₂ to B₁.

To calculate the magnitude of emf induced:

Suppose

A → Area of each turn of the coil

N → Number of turns in the coil

$\overrightarrow{\mathrm{B}}$→ Strength of magnetic field

θ → Angle which normal to the coil makes with at any instant t

∴ Magnetic flux linked with the coil in this position:

$\mathrm{\Phi } = \mathrm{N} (\overrightarrow{\mathrm{B}}.\overrightarrow{\mathrm{A}})$= NBA cosθ= NBA cosωt …(i)

Where, ‘ω’ is angular velocity of the coil

As the coil rotates, angle θchanges. Therefore, magnetic flux Φ linked with the coil changes and hence, an emf is induced in the coil. At this instant t, if e is the emf induced in the coil, then

$$\begin{gathered} \mathrm{e} = - \frac{\mathrm{d\theta }}{\mathrm{dt}} =-\frac{\mathrm{d}}{\mathrm{dt}} (\mathrm{NAB} \cos \mathrm{\omega t}) \\ = -\mathrm{NAB}\frac{\mathrm{d}}{\mathrm{dt}}(\cos \mathrm{\omega t}) \\ =-\mathrm{NAB} (-\sin \mathrm{\omega t}) \mathrm{\omega } \end{gathered}$$