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. fly to infinity
• D. execute oscillations with frequency $\frac{1}{2\pi }\sqrt{\frac{Q}{4\pi {\epsilon }_{0}M{a}^2}}$
• E. none of the above

Answer 1

The correct option is E none of the above

$F_{net}=-2F\cos \theta =-2\frac{KQ\times 1}{(x^2+a^2)}\times \frac{x}{\sqrt{x^2+a^2}}=\frac{-2KQ}{{(x^2+a^2)}^{3/2}}x$
$\therefore F_{net}\approx -\left(\frac{2KQ}{a^3}\right)x$
Freq $=\frac{1}{2\pi }\sqrt{\frac{2KQ}{m{a}^3}}=\frac{1}{2\pi }\sqrt{\frac{2\times \frac{1}{4\pi \epsilon }Q}{m{a}^3}}=\frac{1}{2\pi }\sqrt{\frac{Q}{2\pi {\epsilon }_{0}m{a}^3}}$