a
(a) In equilibrium \(F_e = T sin\theta \) ....... \((i)\)
\(mg = T cos\theta \) ....... \((ii)\)
\(\tan \theta = \frac{{{F_e}}}{{mg}} = \frac{{{q^2}}}{{4\pi {\varepsilon _o}{x^2} \times mg}}\) also \(\tan \theta \approx \sin \theta = \frac{{x/2}}{L}\)
Hence \(\frac{x}{{2L}} = \frac{{{q^2}}}{{4\pi {\varepsilon _o}{x^2} \times mg}}\)
\(==>\) \({x^3} = \frac{{2{q^2}L}}{{4\pi {\varepsilon _o}mg}}\) \(==>\) \(x = {\left( {\frac{{{q^2}L}}{{2\pi {\varepsilon _o}mg}}} \right)^{1/3}}\)