Question

Show that the equilibrium is unstable.

Answer 1

Let a charge $Q$ be placed at a distance x from $q$. $Q$ has to be a negative charge for the system to be in equilibrium.
As the system is in equilibrium, net force on each charge is zero. Therefore,
$\frac{1}{4\pi ϵ}\frac{qQ}{x^2}=\frac{1}{4\pi ϵ}\frac{q4q}{L^2}$ -(1)
$\frac{1}{4\pi ϵ}\frac{qQ}{x^2}=\frac{1}{4\pi ϵ}\frac{4qQ}{(L-x{)}^2}$ -(2)
$\frac{1}{4\pi ϵ}\frac{4qQ}{(L-x{)}^2}=\frac{1}{4\pi ϵ}\frac{q4q}{L^2}$ -(3)
From 1 & 3 ,
$\frac{1}{x^2}=\frac{4}{(L-x{)}^2}\Rightarrow x=L/3$
And $Q=-4q/9$