Question

Two similar balls, each of mass m and charge q, are hung from a common point by two silk threads, each of length $l$. Prove that separation between the balls is $x={\left[\frac{q^{2}l}{2\pi {\epsilon }_{0}mg}\right]}^{1/3}$, if $\theta$ is small.
Find the rate dq/dt with which the charge should leak off each sphere if the velocity of approach varies as $v=a/\sqrt{x}$, where a is a constant

Answer 1

$tan\theta =\frac{F}{mg}=\frac{k{q}^2}{x^{2}mg}$

$\frac{x}{2l}=\frac{q^2}{4\pi {\epsilon }_{0}mg}$

$x={\left[\frac{q^{2}l}{2\pi {\epsilon }_{0}mg}\right]}^{1/3}$. Hence proved.

Now $q^2=\frac{2\pi {\epsilon }_{0}mg{x}^3}{l}$

or $q=\sqrt{\frac{2\pi {\epsilon }_{0}mg}{l}}{x}^{3/2}$

or $\frac{dq}{dt}=\sqrt{\frac{2\pi {\epsilon }_{0}mg}{l}}\frac{3}{2}\sqrt{x}\frac{dx}{dt}$

or $(\sqrt{x}\frac{dx}{dt}=\sqrt{x}(-v)=-\sqrt{x}a/\sqrt{x}=-a)$

or $\frac{dq}{dt}=-\frac{3}{2}a\sqrt{\frac{2\pi {\epsilon }_{0}mg}{l}}$