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(C) The capacitance of a parallel plate capacitor with air as the medium is given by $C_1 = \frac{\epsilon_0 A}{d}$.When a metal sheet of thickness $t

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$3$

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(C) The capacitance of a parallel plate capacitor with air as the medium is given by $C_1 = \frac{\epsilon_0 A}{d}$.When a metal sheet of thickness $t = \frac{2d}{3}$ is introduced between the plates,the effective distance between the plates decreases.The capacitance $C_2$ of a capacitor with a dielectric slab (or metal sheet) of thickness $t$ is given by $C_2 = \frac{\epsilon_0 A}{d - t + \frac{t}{K}}$.For a metal sheet,the dielectric constant $K = \infty$.Substituting the values,we get $C_2 = \frac{\epsilon_0 A}{d - \frac{2d}{3} + \frac{2d/3}{\infty}} = \frac{\epsilon_0 A}{d - \frac{2d}{3} + 0} = \frac{\epsilon_0 A}{d/3} = 3 \frac{\epsilon_0 A}{d}$.Since $C_1 = \frac{\epsilon_0 A}{d}$,we have $C_2 = 3C_1$.Therefore,the ratio $\frac{C_2}{C_1} = 3:1$.

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