A ring of radius R has charge Q distributed uniformly on it. At a distance x $\left(>0\right)$ from the centre on its axis, (consider axial points only)

A) $\text{V}=\frac{\text{KQ}}{\text{R}}$is maximum at x=0 or  ${\text{V}}_{\max }=\frac{\text{KQ}}{\text{R}}$

Option (A) is correct.
B) $\text{E}=\frac{\text{KQx}}{{\left({\text{R}}^{\text{2}}{\text{+x}}^{\text{2}}\right)}^{\text{3/2}}}$for ${\text{E}}_{\max },\frac{\text{dE}}{\text{dx}}=0\Rightarrow \text{x}=\frac{\text{R}}{\sqrt{\text{2}}}$

${\text{E}}_{\text{max}}=\frac{\text{KQ}\left(\frac{\text{R}}{\sqrt{\text{2}}}\right)}{{\left({\text{R}}^{\text{2}}\text{+}{\left(\frac{\text{R}}{\sqrt{\text{2}}}\right)}^{\text{2}}\right)}^{\text{3/2}}}=\frac{\text{KQR}}{\sqrt{\text{2}}}\frac{1}{{\text{R}}^{\text{3}}\left(\frac{3\sqrt{3}}{2\sqrt{2}}\right)}=\frac{\text{2KQ}}{3\sqrt{3}{\text{R}}^2}=\frac{\text{Q}}{6\sqrt{3}{\text{πε}}_{\text{0}}{\text{R}}^2}$

Option (B) is correct.
C)  $\text{F}=\text{ma}=\text{qE}=\frac{\text{KqQx}}{{\left({\text{R}}^{\text{2}}{\text{+x}}^{\text{2}}\right)}^{\text{3/2}}}\approx \frac{1}{4{\text{πε}}_{\text{0}}}\frac{\text{Qqx}}{{\text{R}}^{\text{3}}}\propto \text{x}\Rightarrow \text{SHM}$

Time period, $\text{T}=\text{2π}\sqrt{\frac{\text{x}}{\text{a}}}=\text{2π}\sqrt{\frac{{\text{4πε}}_{\text{0}}{\text{mR}}^{\text{3}}}{\text{qQ}}}$

Option (C) is correct.
D) for just escape, total mechanical energy=0

$-\frac{\text{KQq}}{\text{R}}\text{+}\frac{\text{1}}{\text{2}}{\text{mv}}^2=0\Rightarrow \text{v}=\sqrt{\frac{\text{2KQq}}{\text{mR}}}=\sqrt{\frac{\text{Qq}}{{\text{2πε}}_{\text{0}}\text{mR}}}$

Option (D) is correct.