For a uniformly charged ring of radius $$R,$$ the electric field on its axis has the largest magnitude at a distance $$h$$ from its centre. Then value of $$h$$ is:

Solution :
Electric field on the axis of a ring of radius $$R$$ at a distance $$h$$ from the centre,
$$E = \frac{{kQh}}{{{{\left( {{h^2} + {R^2}} \right)}^{\frac{3}{2}}}}}$$
Condition for maximum electric field
We have $$\frac{{dE}}{{dh}} = 0$$
$$ \Rightarrow \frac{d}{{dh}}\left[ {\frac{{kQh}}{{{{\left( {{R^2} + {h^2}} \right)}^{\frac{3}{2}}}}}} \right] = 0$$
On solving we get, $$h = \frac{R}{{\sqrt 2 }}$$