Question

Draw a labelled diagram of an ac generator. Obtain the expression for the emf induced in the rotating coil of N turns each of cross-sectional area A, in the presence of a magnetic field $\overrightarrow{B}$.

Answer 1

Let say at the instant $t=0$, the plane of the coil is perpendicular to the direction of magnetic field i.e. area vector of coil points in same direction as that of magnetic field. So maximum flux passes through the coil.

Flux passing through the coil at $t=0$, ${\varphi }_{max}=NBA$

The coil rotates about an axis with constant angular velocity $w$ as shown in the figure.

Angle rotated by coil in time $t$ is $wt$.

Flux passing through the coil at that instant, $\varphi =NBA\cos wt$

Rate of change of flux $\frac{d\varphi }{dt}=NBA\frac{d}{dt}\left(\cos wt\right)$

$\therefore$ $\frac{d\varphi }{dt}=NBA\times (-w\sin wt)$

We get $\frac{d\varphi }{dt}=-NBAw\sin wt$

Induced emf in the coil $E=-\frac{d\varphi }{dt}$

$⟹$ $E=NBAw\sin wt$