The magnetic field at all points within the cylindrical region whose cross-section is indicated in the accompanying figure start increasing at a constant rate $\alpha T/s$. Find the magnitude of electric field as a function of , the distance from the geometric centre of the region.

For $r\le R$

Using $El=S\left|\frac{dB}{dt}\right|$ or $E\left(2\mathrm{\pi r}\right)=\left({\mathrm{\pi r}}^2\right)\alpha$

$E=\frac{r\alpha }{2}$

$E\propto r$, i.e. E-r graph is a straight line passing through origin.

At $r=R$, $E=\frac{R\alpha }{2}$

For $r\ge R$

Using $El=S\left|\frac{dB}{dt}\right|$,

$$\begin{gathered} E\left(2\mathrm{\pi r}\right)=\left(\pi R^2\right)\alpha \\ E=\frac{\alpha R^2}{2r} \end{gathered}$$

$E\propto \frac{1}{r}$, i.e. E-r graph is a rectangular hyperbola.

The E-r graph is as shown in figure.

The direction of electric field is shown in above figure.