• # question_answer 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)   $v\propto {{x}^{-1}}$

Solution :

$\tan \theta =\frac{{{F}_{e}}}{mg}\simeq \theta$                 $\frac{K{{q}^{2}}}{{{x}^{2}}mg}=\frac{x}{2\ell }$                 or                                                     ?..(1)                 or            ${{x}^{3/2}}\propto q$                                               ?..(2) differentiate eq.(i) w. r .t. time $3{{x}^{2}}\frac{dx}{dt}\propto 2q\frac{dq}{dt}$but $\frac{dq}{dt}$ is constant so x${{x}^{2}}(v)\propto q$      replace q from eq. (2) ${{x}^{2}}(v)\propto {{x}^{3/2}}$ or 

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