Derive an Expression for instantaneous induced emf in an A.C generator

Explanation

In the above figure
N and S are poles of a magnet, `theta` is the angle between the direction of magnetic field B and area vector A. When the coil is rotated in the magnetic field, the flux linked with the coil varies. At any instant of time .t., `A cos theta` is the component of area vector along the direction B. The magnetic flux linked iwth the coil at any instant of time .t. is given by `phi_(B)= B xx` component of area vector along the field direction.
For 1 turn `phi_(B)= BA cos theta`
For n turns `phi_(B)= nAB cos theta`
`phi_(B)= nAB cos omega t` ....(1) `[because theta= omega t]`
where `.omega.` is the angular velocity of the coil at time t
From the Faraday.s second law,
`e= - (d phi)/(dt)`
`e= - (d)/(dt)[ n AB cos omega t]` [From (1) `phi= nAB cos omega t`]
`e= (-nAB)[-sin omega t] xx omega`
`e= nA B omega sin omega t`
`e= e_(0) sin (omega t)`
where `e_(0)` is the peak value of emf `=nAB omega`