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Two spheres A and B of radii $a$ and $b$ respectively are at the same electric potential. The ratio of the surface charge densities of A and B is

1. $\frac{b}{a}$
2. $\frac{a}{b}$
3. $\frac{b^2}{a^2}$
4. $\frac{a^2}{b^2}$

Correct answer: $\frac{a}{b}$

Solution

For a conducting sphere, $V=\frac{1}{4\pi\varepsilon_0}\frac{Q}{R}$. If two spheres are at the same potential, then $Q\propto R$. Surface charge density is $\sigma=\frac{Q}{4\pi R^2}\propto \frac{1}{R}$. Hence, $\sigma_A/\sigma_B=b/a$? Wait, comparing carefully gives $\sigma_A/\sigma_B=\frac{b}{a}$ only if $Q$ is same; for equal potential, $Q\propto R$, so $\sigma\propto 1/R$, giving $\sigma_A/\sigma_B=\frac{b}{a}$.

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