First slide

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

A

$v \propto x^{- 1 / 2}$
B

$v \propto x^{- 1}$
C

$y \propto x^{1 / 2}$
D

$v \propto x$

Solution

$From figure, T \cos θ = m g$ …..(i)
$T \sin θ = \frac{k q^{2}}{x^{2}}$ ……..(ii)
$From eqns. (i) and (ii), \tan θ = \frac{k q^{2}}{x^{2} m g}$
$Since θ is small, ∴ \tan θ = \sin θ = \frac{x}{2 l}$
$\begin{array}{l} ∴ \frac{x}{2 l} = \frac{k q^{2}}{x^{2} m g} \Rightarrow q^{2} = x^{3} \frac{m g}{2 l k} or q \propto x^{3 / 2} \\ \Rightarrow \frac{d q}{d t} \propto \frac{3}{2} \sqrt{x} \frac{d x}{d t} = \frac{3}{2} \sqrt{x} v . \\ Since, \frac{d q}{d t} = constant \\ ∴ v \propto \frac{1}{\sqrt{x}} \end{array}$

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