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)

$v\:\propto \:x^{-\frac{1}{2}}$

Answer 2

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