Question

Two identical charged spheres suspended from a common point by two massless strings of length $l$ are initially a distance $d(d<<\ell )$ apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate. As a result the charges approach each other with a velocity $v$. Then as a function of distance $x$ between them:

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

Answer 1

The correct option is A $v\infty x^{-1/2}$

$Tsin\theta =\frac{K{q}^2}{x^2}$ ....(i)
$T\cos \theta =mg$ .....(ii)
or
$tan\theta =\frac{K{q}^2}{x^{2}mg}=\frac{x}{l}$(from the FBD); since $d>>l$
or
$x^3=\frac{K{q}^2\ell }{mg}$ ($...iii$)
$x^3\propto q^2$
$3x^2\frac{dx}{dt}\propto 2q\frac{dq}{dt}$
$x^2\cdot v\propto q$

Substituting the value of $q$ from equation (iii).
$v\propto x^{\frac{-1}{2}}$