Explanation

(d) Restoring force on charge $q$
[image:1_image252.emf]
$\frac{1}{4\pi\varepsilon_{0}} \times \frac{q.q}{(a + x)^{2}} - \frac{1}{4\pi\varepsilon_{0}}\frac{q.q}{(a - x)^{2}}$
$= \frac{1}{4\pi\varepsilon_{0}} \times \frac{q^{2}}{1}\left\lbrack \frac{(a - x)^{2} - (a + x)^{2}}{\left( a^{2} - x^{2} \right)^{2}} \right\rbrack$
$\frac{q^{2}}{4\pi\varepsilon_{0}}\left\lbrack \frac{(a - x + a + x)(a - x - a - x)}{a^{4}\left( 1 - \frac{x^{2}}{a^{2}} \right)^{2}} \right\rbrack$
$\frac{- 4ax.q^{2}}{4\pi\varepsilon_{0}a^{4}} = - \frac{q^{2}x}{\pi\varepsilon_{0}a^{3}} \Rightarrow a = \frac{F}{m} = \frac{- a^{2}x}{\pi\varepsilon_{0}m.a^{3}} \Rightarrow a = - \omega^{2}x$
$\omega^{2} = \frac{a^{2}}{\pi\varepsilon_{0}ma^{3}} \Rightarrow \omega = \sqrt{\frac{a^{2}}{\pi\varepsilon_{0}m.a^{3}}}$