\(F _{\text {net}}=-\left[2 F _{ q / q } \cos \theta\right]\)

\(F _{\text {net}}=-2 \cdot \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{ q ^{2}}{\left(\sqrt{ d ^{2}+ x ^{2}}\right)^{2}} \cdot \frac{ x }{\sqrt{ d ^{2}+ x ^{2}}}\)

\(=-\frac{q^{2}}{2 \pi \varepsilon_{0}} \frac{x}{\left(d^{2}+x^{2}\right)^{3 / 2}}\)

For \(x ( x << d )\)

\(F_{\text {net }}=-\frac{q^{2}}{2 \pi \varepsilon_{0} d^{3}} x\)

\(\therefore a =-\frac{ q ^{2}}{2 \pi \varepsilon_{0} \cdot md ^{3}} x\)

Comparing with equation of \(SHM \left( a =-\omega^{2} x \right)\)

\(\therefore \omega=\sqrt{\frac{ q ^{2}}{2 \pi \varepsilon_{0} md ^{3}}}\)