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(C) The capacitance of a parallel plate capacitor is given by the formula $C = \frac{\epsilon_0 A}{d}$, where $A$ is the area of the plates and $d$ is

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$1 : 6$

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(C) The capacitance of a parallel plate capacitor is given by the formula $C = \frac{\epsilon_0 A}{d}$, where $A$ is the area of the plates and $d$ is the distance between them.For circular plates of radius $r$, the area $A = \pi r^2$.Thus, $C_1 = \frac{\epsilon_0 \pi r^2}{d}$.In the second case, the new radius $r' = \sqrt{3}r$ and the new distance $d' = \frac{d}{2}$.The new area $A' = \pi (r')^2 = \pi (\sqrt{3}r)^2 = 3\pi r^2$.The new capacitance $C_2 = \frac{\epsilon_0 A'}{d'} = \frac{\epsilon_0 (3\pi r^2)}{d/2} = 6 \left( \frac{\epsilon_0 \pi r^2}{d} \right) = 6C_1$.Therefore, the ratio $C_1 : C_2 = C_1 : 6C_1 = 1 : 6$.

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