A thin spherical shell of radius R has charge Q spread uniformly over its surface. Which of the following graphs most closely represents the electric field E(r) produced by the shell in the range $0\le r<\mathrm{\infty }$ where r is the distance from the center of the shell?

From the Gauss's Law we have,

$\text{∮}{\mathrm{E}}_{\mathrm{r}}\cdot \mathrm{dS}=\frac{{\text{ q}}_{\mathrm{enclosed}}}{{ϵ}_0}$

Since, charge lies on surface of shell, so no charge enclosed from Gaussian surface lying inside shell.

Therefore, from Gauss's law, total electric field inside shell =0 

$\Rightarrow E_{\mathrm{r}}=0$ for $r<R$

The Outside of the shell : $r\ge R$

Now, the Charge which is enclosed by Gaussian surface 

${\text{ q}}_{\mathrm{enclosed}}=Q$

Again from the Gauss's law, 

${\mathrm{E}}_{\mathrm{r}}\text{∮}\mathrm{dS}=\frac{{\text{ q}}_{\mathrm{enclosed}}}{{ϵ}_0}$

We obtained,

${\mathrm{E}}_{\mathrm{r}}\times 4\pi {\mathrm{r}}^2=\frac{\mathrm{Q}}{{ϵ}_0}$

Now, the Electric field at the point outside of the shell 

$E_{\mathrm{r}}=\frac{Q}{4\pi {ϵ}_0{r}^2}$ for $r\ge R$

So, electric field decreases with $\frac{1}{{\mathrm{r}}^2}$  outside of the shell and it is represented by the graph of option (A).