Question

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 the value of $h$ is:

• A. $R$
• B. $\frac{R}{\sqrt{5}}$
• C. $\frac{R}{\sqrt{2}}$
• D. $R\sqrt{2}$

Answer 1

The correct option is C $\frac{R}{\sqrt{2}}$

Electric field on the axis of a ring of radius $R$ at a distance $h$ from the centre,

$E=\frac{kQh}{(h^2+R^2{)}^{3/2}}$

Condition for maximum electric field $\frac{dE}{dh}=0$

$\Rightarrow \frac{d}{dh}\left[\frac{kQh}{(R^2+h^2{)}^{3/2}}\right]=0$

$\Rightarrow \frac{kQ}{(R^2+h^2{)}^{3/2}}\left(\right(R^2+h^2{)}^{3/2}\left(1\right)-\left(h\right)\frac{3}{2}(R^2+h^2{)}^{1/2}(2h\left)\right)$

$\Rightarrow h=\frac{R}{\sqrt{2}}$

Hence, option $\left(B\right)$ is correct.