Physics - Capacitor Question with Solution | TestHub

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 $\begin{array}{rl}& Q_1=C_1\mathrm{V}\\ & Q_2=C_2\mathrm{V}\end{array}$ 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, $\begin{array}{rl}\frac{{\sigma }_2}{{\sigma }_1}& =\frac{{\mathrm{C}}_2{\mathrm{r}}_1^2}{{\mathrm{C}}_1{\mathrm{r}}_2^2}\\ \frac{{\sigma }_2}{{\sigma }_1}& =\frac{2\times 10^{-6}\times (0.03{)}^2}{3\times 10^{-6}\times (0.02{)}^2}=\frac{3}{2}\\ \therefore \frac{{\sigma }_1}{{\sigma }_2}& =\frac{2}{3}\end{array}$