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: ?

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: ?

(a) \( v \propto x^{1/2} \)

(b) \( v \propto x \)

(c) \( v \propto x^{-1/2} \)

(d) \( v \propto x^{-1} \)

Answer :  (c) \( v \propto x^{-1/2} \)

\( T \sin\theta = \frac{k q^2}{d^2} \)

\( T \cos\theta = mg \)

\( \tan\theta = \frac{k q^2}{mg \, d^2} \)

\( \frac{d}{2l} = \frac{k q^2}{mg \, d^2} \)

\( x^3 = \frac{2 l k q^2}{mg} \)

\( 3x^2 \frac{dx}{dt} = \text{const.} \cdot 2l \frac{dq}{dt} \)

\( x^2 \cdot V = q \, (\text{const.}) \)

\( \frac{dq}{dt} = \text{const.} \)

\( \frac{dx}{dt} = V \)

V as a function of x: \( x^1, \, x^{1/2}, \, x^3, \, x^{-1} \)

\( x^2 V = x^{3/2} (\text{const.}) \)

\( V = \frac{\text{const.} \cdot x^{3/2}}{x^2} \)

\( V = \text{const.} \cdot x^{3/2 - 2} \)

\( V = \text{const.} \cdot x^{-1/2} \)