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:

• A. $v\propto x^{\frac{1}{2}}$
• B. $v\propto x$
• C. $v\propto x^{-\frac{1}{2}}$
• D. $v\propto x^{-1}$

Answer 1

Answer: (C)

$\Rightarrow \tan \theta =\frac{F}{mg}\approx \theta$
$\Rightarrow \frac{K{q}^2}{x^{2}mg}=\frac{x}{21}$
$\Rightarrow x^3\propto q^2$
$\Rightarrow x^{3/2}\propto q$
Differentiating wrt time, we get
$\Rightarrow 3x^2\frac{dx}{dt}\propto 2q\frac{dq}{dt}$
where, $\frac{dq}{dt}$ is constant
$\Rightarrow x^2(v)\propto q$
On substituting q, we get
$\Rightarrow x^2(v)\propto x^{3/2}$
$\Rightarrow v\propto x^{-1/2}$