Two identical charged spheres suspended from a common point by two massless strings of lengths l are initially

Question

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:

Accepted Answer

At equilibrium, the electrostatic force ($F_e$) is balanced by the horizontal component of the tension, while gravity balances the vertical component. For small angles (since $x \ll l$), $	an 	heta \approx \sin 	heta = \frac{x}{2l}$.
1. **Equilibrium Condition:** $F_e = mg 	an 	heta \Rightarrow \frac{kq^2}{x^2} = mg \frac{x}{2l}$.
2. **Relation:** This implies $q^2 \propto x^3$, or $q \propto x^{3/2}$.
3. **Differentiation:** Since charge leaks at a constant rate ($\frac{dq}{dt} = C$), differentiating with respect to time gives $\frac{dq}{dt} \propto \frac{d}{dt}(x^{3/2}) \Rightarrow C \propto x^{1/2} \frac{dx}{dt}$.
4. **Velocity:** Substituting $v = \frac{dx}{dt}$, we get $C \propto x^{1/2} v$, which leads to $v \propto x^{-1/2}$.

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