The flux linked with coil (made of wire of radius $r$) of area $A$ and magnetic field $B$, is given by

$\varphi =BA \cos \theta =B \left(\frac{\pi r^2}{2}\right) \cos \omega t$

$\varphi =\frac{\pi B{r}^2}{2} \cos \omega t$   [$\because A=\frac{\pi r^2}{2}$ and $\theta =\omega t$]

By Faraday's law of electromagnetic induction, induced emf, $e=-\frac{d\varphi }{dt}$

Hence, after differentiating,

$e=-\frac{d}{dt}\left(\frac{\pi B{r}^2}{2} \cos \omega t\right)=\frac{\pi B{r}^2\omega }{2} \sin \omega t$  (where, $R=$ resistance in the circuit)

$\therefore$ Power $P=\frac{e^2}{R}=\frac{{\pi }^2{B}^2{r}^4{\omega }^2 {\sin }^2\omega t}{4 \mathrm{R}}$

$\therefore$ Mean power $\overline{P}=\frac{{\pi }^2{B}^2{r}^4{\omega }^2}{4 \mathrm{R}}·\overline{{\sin }^2\omega t}$

$=\frac{{\pi }^2{B}^2{r}^4{\omega }^2}{4 \mathrm{R}}·\frac{1}{2}$   $\left[\because \overline{{\sin }^2\omega t}=\frac{1}{2}\right]$

$=\frac{{\left(\pi B{r}^2\omega \right)}^2}{8 \mathrm{R}}$