Discuss the characteristics of the induced $emf$ in an $AC$ generator.

(N/A) In an $AC$ generator,a coil rotates in a uniform magnetic field $B$. The magnetic flux $\phi$ linked with the coil at any time $t$ is given by $\phi = NBA \cos(\omega t)$,where $N$ is the number of turns,$A$ is the area of the coil,and $\omega$ is the angular velocity.
According to Faraday's law of induction,the induced $emf$ is $\varepsilon = -\frac{d\phi}{dt} = NBA\omega \sin(\omega t)$.
Let $\varepsilon_0 = NBA\omega$ be the peak value of the $emf$. Then $\varepsilon = \varepsilon_0 \sin(\omega t)$.
Characteristics:
$1$. The induced $emf$ varies sinusoidally with time.
$2$. At $\omega t = 0, 180^\circ, 360^\circ$,the $emf$ is zero because the plane of the coil is perpendicular to the magnetic field.
$3$. At $\omega t = 90^\circ$,the $emf$ is maximum positive $(\varepsilon_0)$ because the rate of change of flux is maximum.
$4$. At $\omega t = 270^\circ$,the $emf$ is maximum negative $(-\varepsilon_0)$ as the coil moves in the opposite direction relative to the field lines.