Write down the formula for the induced $emf$ in an $AC$ generator and discuss how it varies with time.

(N/A) The formula for the induced $emf$ $(\varepsilon)$ in an $AC$ generator is given by:
$\varepsilon = \varepsilon_{0} \sin(\omega t) = \varepsilon_{0} \sin(2 \pi \nu t) \quad \dots (1)$
where $\varepsilon_{0}$ is the peak value of the $emf$,$\omega$ is the angular frequency,and $\nu$ is the frequency of rotation.
Equation $(1)$ represents the instantaneous value of the $emf$. The $emf$ varies periodically between $+\varepsilon_{0}$ and $-\varepsilon_{0}$ as the coil rotates.
Based on the rotation of the coil in the magnetic field $\vec{B}$,we can analyze different stages:
$1$. Stage $1$ $(\omega t = 0^{\circ})$: The plane of the coil is perpendicular to the magnetic field $\vec{B}$. The magnetic flux is maximum,but the rate of change of flux is zero. Thus,$\varepsilon = \varepsilon_{0} \sin(0^{\circ}) = 0$.
$2$. Stage $2$ $(\omega t = 90^{\circ})$: The plane of the coil is parallel to the magnetic field $\vec{B}$. The rate of change of flux is maximum. Thus,$\varepsilon = \varepsilon_{0} \sin(90^{\circ}) = \varepsilon_{0}$.
$3$. Stage $3$ $(\omega t = 180^{\circ})$: The plane of the coil is again perpendicular to $\vec{B}$. Thus,$\varepsilon = \varepsilon_{0} \sin(180^{\circ}) = 0$.
$4$. Stage $4$ $(\omega t = 270^{\circ})$: The plane of the coil is parallel to $\vec{B}$,but the coil has rotated to the opposite side. Thus,$\varepsilon = \varepsilon_{0} \sin(270^{\circ}) = -\varepsilon_{0}$.
$5$. Stage $5$ $(\omega t = 360^{\circ})$: The coil returns to its initial position. Thus,$\varepsilon = \varepsilon_{0} \sin(360^{\circ}) = 0$.