Discuss the principle and construction of an $A.C.$ generator (dynamo). Derive the formula for the induced $emf$ in an $A.C.$ generator.

(N/A) Principle: The principle of an $A.C.$ generator is based on electromagnetic induction. When a coil is rotated in a uniform magnetic field,the magnetic flux linked with the coil changes continuously,which induces an alternating $emf$ in the coil.
Construction: An $A.C.$ generator consists of a rectangular coil (armature) placed between the pole pieces of a strong permanent magnet. The coil is mounted on a rotor shaft and can be rotated. The ends of the coil are connected to two slip rings,which rotate with the coil. Two stationary carbon brushes press against these slip rings to conduct the induced current to the external circuit.
Derivation of induced $emf$:
Let the area of the coil be $A$,the number of turns be $N$,and the magnetic field be $B$. If the coil rotates with an angular velocity $\omega$,the angle between the area vector $\vec{A}$ and the magnetic field $\vec{B}$ at any time $t$ is $\theta = \omega t$.
The magnetic flux $\phi$ linked with the coil is given by:
$\phi = N B A \cos(\theta) = N B A \cos(\omega t)$
According to Faraday's law of electromagnetic induction,the induced $emf$ $(\varepsilon)$ is:
$\varepsilon = -\frac{d\phi}{dt}$
$\varepsilon = -\frac{d}{dt} (N B A \cos(\omega t))$
$\varepsilon = -N B A (-\sin(\omega t)) \cdot \omega$
$\varepsilon = N B A \omega \sin(\omega t)$
Let $\varepsilon_0 = N B A \omega$ be the peak value of the $emf$. Then:
$\varepsilon = \varepsilon_0 \sin(\omega t)$