Two spherical conductors of radii $4 \mathrm{~cm}$ and $5 \mathrm{~cm}$ are charged to the same potential.…
Two spherical conductors of radii $4 \mathrm{~cm}$ and $5 \mathrm{~cm}$ are charged to the same potential. If ' $\sigma_1$ ' and ' $\sigma_2$ ' be the respective value of the surface density of charge on the two conductors then the ratio $\sigma_1: \sigma_2$ is
5:4
3:2
4:3
2:1
Solution
Spheres are at the same potential
$\begin{aligned}
& \therefore \mathrm{V}=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{\mathrm{q}_1}{\mathrm{r}_1}=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{\mathrm{q}_2}{\mathrm{r}_2} \\
& \therefore \frac{\mathrm{q}_1}{\mathrm{q}_2}=\frac{\mathrm{r}_1}{\mathrm{r}_2} \\
& \sigma_1=\frac{\mathrm{q}_1}{4 \pi \mathrm{r}_1^2}, \sigma_2=\frac{\mathrm{q}_2}{4 \pi \mathrm{r}_2^2} \\
& \therefore \frac{\sigma_1}{\sigma_2}=\frac{\mathrm{q}_1}{\mathrm{q}_2} \cdot \frac{\mathrm{r}_2^2}{\mathrm{r}_1^2}=\frac{\mathrm{r}_1}{\mathrm{r}_2} \cdot \frac{\mathrm{r}_2^2}{\mathrm{r}_1^2}=\frac{\mathrm{r}_2}{\mathrm{r}_1}=\frac{5}{4}
\end{aligned}$
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