Question

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

• A. $v\propto x$
• B. $v\propto x^{-\frac{1}{2}}$
• C. $v\propto x^{-1}$
• D. $v\propto x^{\frac{1}{2}}$

Answer 1

Answer: (B)

$\tan \theta =\frac{F}{mg}$
or $\frac{x}{2l}=\frac{k{q}^2}{mg{x}^2}$
$\frac{x^3}{2l}=\frac{k{q}^2}{mg}$
$\frac{3x^2\frac{dx}{dt}}{2l}=\frac{2kq\frac{dq}{dt}}{mg}$
Also, $q\propto x^{3/2}$
$\Rightarrow \frac{dx}{dt}\propto \frac{x^{3/2}}{x^2},$ i.e., $v\propto x^{-1/2}$