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Two identical charged spheres suspended from a common point by two massless strings of lengths l, are initially at a distance d (d << l) apart because of their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity v. Then v varies as a function of the distance x between the spheres, as :

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$V \propto x$

$V \propto x^{\frac{- 1}{2}}$

$V \propto x^{\frac{1}{2}}$

$V \propto x^{- 1}$

answer is C.

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Detailed Solution

$\tan θ = \frac{F_{e}}{mg} ≃ θ$

$\frac{{Kq}^{2}}{x^{2} mg} = \frac{x}{2 ℓ}$

$or x^{3} \propto q^{2} . . . . (1)$

$or x^{3 / 2} \propto q \dots (2)$

$differentiate eq.(i) w.r.t. time$

$3 x^{2} \frac{dx}{dt} \propto 2 q \frac{dq}{dt} but \frac{dq}{dt} is constant$

$so x^{2} (v) \propto q$

$x^{2} (v) \propto x^{3 / 2} or v \propto x^{- 1 / 2}$

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