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,

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$v \propto x^{- 1}$

$v \propto x^{1 / 2}$

$v \propto x^{- 1 / 2}$

$v \propto x$

answer is D.

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Detailed Solution

$\tan θ = \frac{F}{mg} \approx θ$

$\frac{K q^{2}}{x^{2} m g} = \frac{x}{2 L} \Rightarrow x^{3 / 2} \propto q$

Differentiating wrt time, we get

$3 x^{2} \frac{dx}{dt} \propto 2 q \frac{dq}{dt} where, \frac{dq}{dt} is constant \Rightarrow x^{2} (v) \propto q$

As $x^{3 / 2} \propto q$

We get $v \propto x^{- 1 / 2}$

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