Two spherical conductors of capacities \(3 \mu \mathrm{F}\) and \(2 \mu \mathrm{F}\) are charged to same potential having radii \(3 \mathrm{~cm}\) and \(2 \mathrm{~cm}\) respectively. If ' \(\sigma_1\) ' and ' \(\sigma_2\) ' represent surface density of charge on respective conductors then \(\frac{\sigma_1}{\sigma_2}\) is

We know, \(\mathrm{C}=\frac{\mathrm{Q}}{\mathrm{V}}\)
As both the charged spheres are at the same potential, the charge on both spheres is
\(\begin{aligned}
& Q_1=C_1 \mathrm{~V} \\
& Q_2=C_2 \mathrm{~V}
\end{aligned}\)
The charge densities of both spheres are
\(\sigma_1=\frac{\mathrm{Q}_1}{\mathrm{~A}_1}=\frac{\mathrm{C}_1 \mathrm{~V}}{4 \pi \mathrm{r}_1^2}\)
Similiarly,
\(\sigma_2=\frac{\mathrm{Q}_2}{\mathrm{~A}_2}=\frac{\mathrm{C}_2 \mathrm{~V}}{4 \pi \mathrm{r}_2^2}\)
Taking the ratios,
\(\begin{aligned}
\frac{\sigma_2}{\sigma_1} & =\frac{\mathrm{C}_2 \mathrm{r}_1^2}{\mathrm{C}_1 \mathrm{r}_2^2} \\
\frac{\sigma_2}{\sigma_1} & =\frac{2 \times 10^{-6} \times(0.03)^2}{3 \times 10^{-6} \times(0.02)^2}=\frac{3}{2} \\
\therefore \quad \frac{\sigma_1}{\sigma_2} & =\frac{2}{3}
\end{aligned}\)