Coulumbs law in vector form

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

Moderate

Solution

$\text{From figure,}T\mathrm{cos}\theta =mg$…..(i)

$T\mathrm{sin}\theta =\frac{k{q}^{2}}{{x}^{2}}$……..(ii)

$\text{From eqns. (i) and (ii),}\mathrm{tan}\theta =\frac{k{q}^{2}}{{x}^{2}mg}$

$\text{Since}\theta \text{is small,}\therefore \mathrm{tan}\theta =\mathrm{sin}\theta =\frac{x}{2l}$

$\begin{array}{l}\\ \therefore \frac{x}{2l}=\frac{k{q}^{2}}{{x}^{2}mg}\Rightarrow {q}^{2}={x}^{3}\frac{mg}{2lk}\text{or}q\propto {x}^{3/2}\\ \Rightarrow \frac{dq}{dt}\propto \frac{3}{2}\sqrt{x}\frac{dx}{dt}=\frac{3}{2}\sqrt{x}v.\\ \text{Since,}\frac{dq}{dt}=\text{constant}\\ \therefore v\propto \frac{1}{\sqrt{x}}\end{array}$

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