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  4. 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 to the initial value. The ratio of the initial charges of the balls is

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Q. 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 to the initial value. The ratio of the initial charges of the balls is

Electric Charges and Fields

Solution:

Suppose the balls have charges $Q_{1}$ and $Q_{2}$, respectively.
Initially:
image
Finally:
image
$F'=\frac{k\left(\frac{Q_{1}+Q_{2}}{2}\right)^{2}}{\left(\frac{r}{2}\right)^{2}}=\frac{k\left(Q_{1}+Q_{2}\right)^{2}}{r^{2}}$
It is given that $F=4.5 F$
so $\frac{k\left(Q_{1}+Q_{2}\right)^{2}}{r^{2}}=4.5 k \frac{Q_{1} Q_{2}}{r^{2}}$
$\Rightarrow \left(Q_{1}+Q_{2}\right)^{2}=4.5 Q_{1} Q_{2}$
On solving, it gives $\frac{Q_{1}}{Q_{2}}=\frac{2}{1}$