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 $\frac{1}{2\pi }\sqrt{\frac{Q}{4\pi {\epsilon }_{0}M{a}^3}}$
• C. execute oscillations with frequency $\frac{1}{2\pi }\sqrt{\frac{Q}{2\pi {\epsilon }_{0}m{a}^3}}$
• D. None of the above

Answer 1

Answer: (D)

From figure the net force
$F_{\text{net }}=-F\cos \theta +(-F\cos \theta )$
$=-2F\cos \theta$
$=-2\times \frac{kQ\times 1}{\left(x^2+a^2\right)}\times \frac{x}{\sqrt{x^2+a^2}}$
$=-\frac{2kQ}{{\left(x^2+a^2\right)}^{3/2}}\cdot x$
$\therefore F_{\text{net }}=-\left(\frac{2kQ}{a^3}\right)\cdot x$
Frequency of oscillation
$=\frac{1}{2\pi }\sqrt{\frac{2kQ}{M{a}^3}}$
$=\frac{1}{2\pi }\sqrt{\frac{2\times \frac{1}{4\pi {\epsilon }_0}\cdot Q}{M{a}^3}}$
$=\frac{1}{2\pi }\sqrt{\frac{Q}{2\pi {\epsilon }_{0}M{a}^3}}$