Droplets of mercury \(=64\)

Density of small droplet \(=\sigma_{\text {small }}\)

Density of big droplet \(=\sigma_{\text {biz }}\)

We know that,

\(\frac{\sigma_{\text {small }}}{\sigma_{\text {big }}}=\frac{ q }{ Q } \times \frac{ R ^2}{ r ^2}\)

\(\frac{\sigma_{\text {small }}}{\sigma_{\text {big }}}=\frac{q}{(n q)} \times \frac{\left( n ^{\frac{1}{3}} r \right)}{ r ^2}\)

\(\frac{\sigma_{\text {small }}}{\sigma_{\text {big }}}= n ^{-\frac{1}{3}}\)

Now, put the value of \(n\)

\(\frac{\sigma_{\text {small }}}{\sigma_{\text {big }}}=(64)^{-\frac{1}{3}}\)

\(\frac{\sigma_{\text {small }}}{\sigma_{\text {big }}}=\frac{1}{4}\)

Hence, the ratio is \(\frac{1}{4}\)