The maximum electric field intensity on the axis of a uniformly charged ring of charge q and radius R will be :

$\mathrm{E}=\frac{\mathrm{kqx}}{{({\mathrm{R}}^2+{\mathrm{x}}^2)}^{3/2}},$for maximum $\mathrm{E}, \frac{\mathrm{dE}}{\mathrm{dx}}=0$

$\Rightarrow \frac{\mathrm{kq}\left(1\right)}{{({\mathrm{R}}^2+{\mathrm{x}}^2)}^{3/2}}+\left(-\frac{3}{2}\right)\frac{\mathrm{kqx}}{{({\mathrm{R}}^2+{\mathrm{x}}^2)}^{5/2}}\left(2\mathrm{x}\right)=0$

$\Rightarrow {\mathrm{R}}^2+{\mathrm{x}}^2=3{\mathrm{x}}^2\Rightarrow {\mathrm{x}}^2=\frac{{\mathrm{R}}^2}{2}$

$\Rightarrow {\mathrm{E}}_{\max }=\frac{\mathrm{kq}(\mathrm{R}/\sqrt{2})}{{\left({\mathrm{R}}^2+{\displaystyle \frac{{\mathrm{R}}^2}{2}}\right)}^{3/2}}=\frac{2\mathrm{kq}}{3\sqrt{3}{\mathrm{R}}^2}$