A thin rectangular metallic plate of dimensions 15 cm × 20 cm is...

Total charge given to the metallic plate, $Q = 12 \mu\mathrm{C} = 12 \times 10^{-6} \mathrm{C}$.

Area of one surface of the plate, $A = 15 \text{ cm} \times 20 \text{ cm} = 300 \text{ cm}^2 = 3 \times 10^{-2} \text{ m}^2$.

Since it is a thin metallic plate, the charge distributes uniformly over both of its outer surfaces.

Charge on each surface, $q = \dfrac{Q}{2} = 6 \times 10^{-6} \mathrm{C}$.

Surface charge density, $\sigma = \dfrac{q}{A} = \dfrac{6 \times 10^{-6}}{3 \times 10^{-2}} = 2 \times 10^{-4} \mathrm{C} \mathrm{m}^{-2}$.

Answer: $2 \times 10^{-4} \mathrm{C} \mathrm{m}^{-2}$