question_answer

Two identical charged spheres suspended from a common point by two massless strings of length l are initially a distance \[d(d<<l)\] apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate. As a result the charges approach each other with a velocity v. Then as a function of distance \[x\] between them                              [AIEEE 2011]

A) \[\upsilon \propto x\]

B) \[\upsilon \propto {{x}^{-\frac{1}{2}}}\]

C) \[\upsilon \propto {{x}^{-1}}\]

D) \[\upsilon \propto {{x}^{\frac{1}{2}}}\]

Correct Answer: B

Solution :

[b] \[\tan \theta =\frac{F}{mg}\]

or \[\frac{x}{2l}=\frac{k{{q}^{2}}}{mg{{x}^{2}}}\]

\[\frac{{{x}^{3}}}{2l}=\frac{k{{q}^{2}}}{mg}\]

\[\frac{3{{x}^{2}}\frac{dx}{dt}}{2l}=\frac{2kq\frac{dq}{dt}}{mg}\]

Also, \[q\propto {{x}^{3/2}}\]

\[\Rightarrow \,\,\frac{dx}{dt}\propto \frac{{{x}^{3/2}}}{{{x}^{2}}},i.e.,\,\upsilon \propto {{x}^{-1/2}}\]