a
Let \(m\) be mass of each ball and \(q\) be charge on each ball. Force of repulsion

\(F=\frac{1}{4 \pi \varepsilon_{0}} \frac{q^{2}}{r^{2}}\)

In equilibrium

\(T \cos \theta=m g\)    \(...(i)\)

\(T \sin \theta=F\)     \(...(ii)\)

Divide \((ii)\) by \((i),\) we get,

\(\tan \theta=\frac{F}{m g}=\frac{\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{r^{2}}}{m g}\)

From figure \((a),\)

\(\frac{r / 2}{y}=\frac{\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{r^{2}}}{m g}\)    \(....(iii)\)

\(\tan \theta^{\prime}=\frac{\frac{1}{4 \pi \varepsilon_{0}} \frac{q^{2}}{r^{\prime 2}}}{m g}\)

From figure \((b)\)

\(\frac{r^{\prime} / 2}{y / 2}=\frac{\frac{1}{4 \pi \varepsilon_{0}} \frac{q^{2}}{r^{\prime 2}}}{m g} \ldots(\mathrm{iv})\)

Divide \((iv)\) by \((iii),\) we get

\(\frac{2 r^{\prime}}{r}=\frac{r^{2}}{r^{\prime 2}} \Rightarrow r^{\prime 3}=\frac{r^{3}}{2}\)

\(\Rightarrow r^{\prime}=\frac{r}{\sqrt[3]{2}}\)