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A unit negative charge with mass $M$ resides at the mid-point of the straight line of length $2 a$ adjoining two fixed charges of magnitude $+Q$ each. If it is given a very small displacement $x(x< < a)$ in a direction perpendicular to the straight line, it will
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None of the above
From figure the net force
$\begin{aligned} F_{\text {bet }} &=-F \cos \theta+(-F \cos \theta) \\ &=-2 F \cos \theta \\ &=-2 \times \frac{k Q \times 1}{\left(x^{2}+a^{2}\right)} \times \frac{x}{\sqrt{x^{2}+a^{2}}} \\ &=-\frac{2 k Q}{\left(x^{2}+a^{2}\right)^{3 / 2}} \cdot x \\ F_{\text {net }} &=-\left(\frac{2 k Q}{a^{3}}\right) \cdot x \end{aligned}$
Frequency of oscillation $=\frac{1}{2 \pi} \sqrt{\frac{2 k Q}{M a^{3}}}$
$=\frac{1}{2 \pi} \sqrt{\frac{2 \times \frac{1}{4 \pi \varepsilon_{0}} \cdot Q}{M a^{3}}}$
$=\frac{1}{2 \pi} \sqrt{\frac{Q}{2 \pi e_{0} M a^{3}}}$
$\Rightarrow\left(^{*}\right.$ None of the option matches)
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