Two conducting spheres A and B of radii 'a' and 'b', respectively are…

The correct answer is Option (2) → $\frac{b}{a}$

Let the potentials of both spheres be equal.

Potential of a charged sphere, $V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{R}$

For spheres A and B:

$\frac{Q_A}{a} = \frac{Q_B}{b}$

⇒ $\frac{Q_A}{Q_B} = \frac{a}{b}$

Surface charge density, $\sigma = \frac{Q}{4\pi R^2}$

Hence,

$\frac{\sigma_A}{\sigma_B} = \frac{Q_A / a^2}{Q_B / b^2} = \frac{Q_A b^2}{Q_B a^2}$

Substitute $\frac{Q_A}{Q_B} = \frac{a}{b}$

$\frac{\sigma_A}{\sigma_B} = \frac{a b^2}{b a^2} = \frac{b}{a}$

∴ Ratio of surface charge densities = $\frac{b}{a}$