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

• A

  come back to its original position and stay there
• B

  execute oscillations with frequency $$ \frac{1}{2 \pi} \sqrt{\frac{Q}{4 \pi \varepsilon_{0} M a^{3}}} $$
• Correct answer
  C

  None of the above
• D

  execute oscillations with frequency $$ \frac{1}{2 \pi} \sqrt{\frac{Q}{4 \pi \varepsilon_{0} M a^{2}}} $$

Solution

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)

Hence, the correct answer is option: 'C'.