Two identical charged spheres suspended from a common point by two massless strings of length I are…
Two identical charged spheres suspended from a common point by two massless strings of length I are initially a distance $d(d< < 1)$ 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,
$v \propto x^{-1}$
$v \propto x^{1 / 2}$
$v \propto x$
$v \propto x^{-1 / 2}$
Solution
At any instant of separation between charges is $x$.
$
\begin{aligned}
& \text { equilibrium condition }=\mathrm{K} \frac{\mathrm{Q}^2}{\mathrm{x}^2}=\omega \frac{\mathrm{x}}{2 \ell} \\
& \Rightarrow \mathrm{Q}^2=\mathrm{Cx}^3 \\
& \Rightarrow 2 \mathrm{Q} \frac{\mathrm{dQ}}{\mathrm{dt}}=\mathrm{C} 3 \mathrm{x}^2 \frac{\mathrm{dx}}{\mathrm{dt}} \\
& \Rightarrow \frac{\mathrm{dx}}{\mathrm{dt}} \propto \frac{\mathrm{x}^{3 / 2}}{\mathrm{x}^2} \propto \mathrm{x}^{-1 / 2}
\end{aligned}
$