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Two identical charged spheres suspended from a common point by two massless strings of length $l$ are initially a distance $d(d << I) $ apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate. As a result charges approach each other with a velocity $v$. Then as a function of distance $x$ between them,
$v$ $ \propto \;{x^{ - \frac{1}{2}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$v $ $ \propto \;{x^{ - 1}}$
$v $ $ \propto \;{x^{\frac{1}{2}}}$
$v $ $ \propto \;x$
Solution
At any instant
$T \cos \theta=m g………(i)$
$T \sin \theta=F_{e}………(ii)$
$\Rightarrow \quad \frac{\sin \theta}{\cos \theta}=\frac{F_{e}}{m g} \Rightarrow \mathrm{F}_{\mathrm{e}}=\mathrm{mg} \tan \theta$
$\Rightarrow \quad \frac{k q^{2}}{x^{2}}=m g \tan \theta \Rightarrow q^{2} \propto x^{2} \tan \theta$
$\sin \theta =\frac{x}{2 l} $
For small $\theta$, $\sin \theta \approx \tan \theta $
$\therefore q^{2} \propto x^{3}$
$\Rightarrow \quad q \frac{d q}{d t} \propto x^{2} \frac{d x}{d t}$
$\therefore \quad \frac{d q}{d t}=$ const.
$\therefore \quad q \propto x^{2}, v \Rightarrow x^{3 / 2} \alpha x^{2}, v\left[\because q^{2} \propto x^{3}\right]$
$\Rightarrow v \propto x^{-1 / 2}$
