Show that for a uniformly charged hollow sphere, the electric field has a maximum value at the surface of the sphere.

Consider a thin spherical shell of radius 'R' with centre O. Let, a charge +q be distributed uniformly on the surface of the shell. 
According to gaussian theorem, 

           
Surface area of the sphere = 4$\mathrm{\pi }$r² 

$\therefore \mathrm{E}. 4{\mathrm{\pi r}}^2 = \frac{\mathrm{q}}{{\mathrm{\epsilon }}_{\mathrm{o}}}$

Now, at a point on the surface area of the shell 
Let, $\mathrm{\sigma }$ be the surface charge density on the shell, 
then, 
             q = 4$\mathrm{\pi }$R² . $\mathrm{\sigma }$ 

$\Rightarrow$  $\mathrm{E} = \frac{4{\mathrm{\pi R}}^2 \mathrm{\sigma }}{4{\mathrm{\pi \epsilon }}_{\mathrm{o}}{\mathrm{R}}^2} = \frac{\mathrm{\sigma }}{{\mathrm{\epsilon }}_{\circ }}$ 

Therefore, 
                   $\mathrm{E} = \frac{\mathrm{\sigma }}{{\mathrm{\epsilon }}_{\mathrm{o}}}$ 
which is the required electric field at the surface of the shell.