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A charged particle having some mass is resting in equilibrium at a height $H$ above the centre of a uniformly charged non-conducting horizontal ring of radius $R$. The force of gravity acts downwards. The equilibrium of the particle will be stable $R$
for all values of $H$
only if $H >$ $\frac{R}{{\sqrt 2 }}$
only if $H <$ $\frac{R}{{\sqrt 2 }}$
only if $H =$ $\frac{R}{{\sqrt 2 }}$
Solution
$F=\frac{k Q x(+q)}{\left(R^{2}+x^{2}\right)^{3 / 2}}-m g$
if $\frac{\mathrm{dF}}{\mathrm{dx}}<0$
Then the particle is in stable equilibrium
$\Rightarrow \frac{\left(R^{2}+x^{2}\right)^{3 / 2}-\frac{3}{2}\left(R^{2}+x^{2}\right)^{1 / 2}\left(2 x^{2}\right)}{\left(R^{2}+x^{2}\right)^{3}}<0$
$\Rightarrow \mathrm{R}^{2}+\mathrm{x}^{2}-3 \mathrm{x}^{2}<0$
$\Rightarrow x>\frac{R}{\sqrt{2}}$ or $x<-\frac{R}{\sqrt{2}}$
$\therefore$ Only if $x>\frac{R}{\sqrt{2}}$
The equilibrium will be stable.
