Question

A unit negative charge with mass $M$ resides at the midpoint of the straight line of length $2a$ 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 $\frac{1}{2\pi}\sqrt{\frac{Q}{4\pi\varepsilon_{0}Ma^{3}}}$
• C. execute oscillations with frequency $\frac{1}{2\pi}\sqrt{\frac{Q}{2\pi\varepsilon_{0}ma^{3}}}$
• D. None of the above

Answer 1

Answer: (D)

From figure the net force
$F_{\text {net }} =-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 $
$\therefore F_{\text {net }} =-\left(\frac{2 k Q}{a^{3}}\right) \cdot x$
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}}} $
$=\frac{1}{2 \pi} \sqrt{\frac{Q}{2 \pi \varepsilon_{0} M a^{3}}}$