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 R1 and R2
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 the 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
`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
