Two balls of same mass and carrying equal charge are hung from a fixed support of length \(l\). At electrostatic equilibrium, assuming that angles made by each thread is small, the separation, \(x\) between the balls is proportional to

\(\mathrm{mg}=\mathrm{T} \cos \theta\)

\(\tan \theta  = \frac{{{F_e}}}{{mg}} = \frac{{{q^2}}}{{4\pi {_0}\,{x^2}}} \times mg\)

\(\text { also } \tan \theta \approx \sin =\frac{x / 2}{\ell}\)

Hence, \(\frac{x}{2 \ell}=\frac{q^{2}}{4 \pi \epsilon_{0} x^{2} \times m g}\)

\(\Rightarrow x^{3}=\frac{2 q^{2} \ell}{4 \pi \epsilon_{0} m g}\)

\(\therefore x=\left(\frac{q^{2} \ell}{2 \pi \epsilon_{0} m g}\right)^{1 / 3}\)

Therefore \(x \propto \ell^{1 / 3}\)