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Two identical balls having like charges and placed at a certain distance apart repel each other with a certain force. They are brought in contact and then moved apart to a distance equal to half their initial separation. The force of repulsion between them increases $4.5$ times in comparison with the initial value. The ratio of the initial charges of the balls is
$2$
$3$
$4$
$6$
Solution
(a) Suppose the balls having charges $Q_1$ and $Q_2$ respectively.
Initially : $(i)$
Finally : $(ii)$
$F' = \frac{{k\,{{\left( {\frac{{{Q_1} + {Q_2}}}{2}} \right)}^2}}}{{{{\left( {\frac{r}{2}} \right)}^2}}} = \frac{{k{{({Q_1} + {Q_2})}^2}}}{{{r^2}}}$
It is given that $F' = 4.5\,F$ so $\frac{{k\,{{({Q_1} + {Q_2})}^2}}}{{{r^2}}} = 4.5\,k.\frac{{{Q_1}{Q_2}}}{{{r^2}}}$
$==>$ ${({Q_1} + {Q_2})^2} = 4.5\,{Q_1}{Q_2}$. On solving it gives $\frac{{{Q_1}}}{{{Q_2}}} = \frac{2}{1}.$
