A conducting loop of resistance R and radius ‘r’ has its center at the origin of the co-ordinate system in a magnetic field of induction B. When it is rotated about Y-axis through ${90}^0,$ net charge flown in the coil is directly proportional to

 

Induced emf loop when the variation of flux $d\varphi$ during time dt is given as $E=\frac{d\varphi }{dt}$

$\Rightarrow {\int }_{{\varphi }_1}^{{\varphi }_2}d\varphi =\Delta \varphi =\int Edt\mathrm{..........}\left(1\right)$

$\Rightarrow$The total charge flown in the loop $=q=\int idt$

$q=\int \frac{E}{R}dt\mathrm{...........}\left(2\right)$

Using (1) and (2) $q=\frac{\Delta \varphi }{R}$

Where $\Delta \varphi =$change in flux given as $\Delta \varphi ={\varphi }_2-{\varphi }_1=B.\left(\pi r^2\right)$ because initially to flux is linked with the coil and it has maximum flux linkage ${\varphi }_2=B\pi r^2$ when turned through ${90}^0,$ the flux is reduced to zero.

$\Rightarrow q=\frac{\pi B{r}^2}{R}\Rightarrow i.eq\alpha B$

$q \alpha r^2$

$q \alpha \left(\frac{1}{R}\right)$