The phase difference between the flux linkage and the induced e.m.f. in a rotating coil in a uniform magnetic field is

Flux: $\mathrm{\varphi }=\mathrm{BA} \mathrm{cos\theta }$ 

where $\mathrm{\theta }$ is the angle between area and field

$\mathrm{e}=\frac{-\mathrm{d\varphi }}{\mathrm{df}}=-\mathrm{BA} \mathrm{sin\theta }·\left(\frac{\mathrm{d\theta }}{\mathrm{dt}}\right)$

$\Rightarrow \mathrm{e}=\mathrm{BA}\left(\frac{\mathrm{d\theta }}{\mathrm{dt}}\right) \cos ({90}^0+\mathrm{\theta })$

Here phase difference $=90°=\frac{\mathrm{\pi }}{2}$