Coulomb's Law: Practice Problem & Solution
3. Two identical charged spheres suspended from a common point by two massless strings of lengths $\ell$, are initially at a distance $d$ ($d \ll \ell$) 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: (2016 - I)
Solution Explained:
To solve this problem, we apply the core principles of Coulomb's Law. Understanding the underlying formula is key to arriving at the correct answer below:
For equilibrium, $\tan\theta = \frac{F_e}{mg} \implies \frac{x}{2\ell} = \frac{k q^2}{x^2 mg} \implies q \propto x^{3/2}$. Differentiating w.r.t time, $\frac{dq}{dt} \propto x^{1/2} V$. Since $\frac{dq}{dt}$ is constant, $V \propto x^{-1/2}$.
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