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 `vecB` directed perpendicular to the axis of rotation.

#### Solution

**Principle - **Based on the phenomenon of electromagnetic induction

**Construction**

Main parts of an ac generator:

Armature − Rectangular coil ABCD

Filed Magnets − Two pole pieces of a strong electromagnet

Slip Rings − The ends of coil ABCD are connected to two hollow metallic rings R_{1} and R_{2}

Brushes − B_{1} and B_{2} are two flexible metal plates or carbon rods. They are fixed and are kept in tight contact with R_{1} and R_{2} 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 the CD. In the external circuit, current flows from B_{2} to B_{1}

To calculate the magnitude of emf induced:

Suppose

A → Area of each turn of the coil

N → Number of turns in the coil

`vecB`→ Strength of magnetic field

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

∴ Magnetic flux linked with the coil in this position:

`phi = N(vecB.vecA)` = NBA cosθ= NBA cosωt …(i)

Where ‘ω’ is angular velocity of the coil

As the coil rotates, angle θ changes. 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

`e = -(dtheta)/(dt) = - d/(dt) (NAB cos omegat)`

`= -NAB d/(dt) (cos omegat)`

`=-NAB(-sin omegat)omega`

∴ e = NAB ω sinωt