At equilibrium, the electrostatic repulsion force ($F_e$) is balanced by the component of tension and gravity ($mg$). For a small angle $\theta$, $\tan \theta \approx \frac{r/2}{y}$, where $y$ is the vertical height from the suspension point to the line joining the balls. The equilibrium condition is $F_e = mg \tan \theta$. Substituting Coulomb's law: $\frac{kq^2}{r^2} = mg \frac{r}{2y}$. Rearranging gives $r^3 = \frac{2kq^2 y}{mg}$, which implies $r^3 \propto y$. When the strings are clamped at half the height, the new height $y' = y/2$. Let the new separation be $r'$. Then $(r')^3 \propto y'$. Taking the ratio: $\frac{(r')^3}{r^3} = \frac{y'}{y} = \frac{y/2}{y} = \frac{1}{2}$. Therefore, $r' = \frac{r}{\sqrt{2}}$.