Import Question JSON

$\text{Hint: The two capacitors are in a series combination.}$ $\text{Step 1: Find the capacitance of individual capacitors.}$ $\text{The capacitance of a parallel plate capacitor filled with a dielectric block of thickness } d_1 \text{ and dielectric constant } K_1 \text{ is given by;}$ $C_1 = \frac{K_1 \varepsilon_0 A}{d_1}$ $\text{Similarly, the capacitance of a parallel plate capacitor filled with a dielectric block of thickness } d_2 \text{ and dielectric constant } K_2 \text{ is given by;}$ $C_2 = \frac{K_2 \varepsilon_0 A}{d_2}$ $\text{Step 2: Find the equivalent capacitance.}$ $\text{Since the two capacitors are in a series combination, the equivalent capacitance is given by;}$ $\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$ $C = \frac{C_1 + C_2}{C_1 C_2} = \frac{K_1 \varepsilon_0 A}{d_1} + \frac{K_2 \varepsilon_0 A}{d_2}$ $C = \frac{K_1 \varepsilon_0 A}{d_1} + \frac{K_2 \varepsilon_0 A}{d_2}$ $\Rightarrow C = \frac{K_1 K_2 \varepsilon_0 A}{K_1 d_2 + K_2 d_1}$ $\text{Step 3: Find the equivalent dielectric constant.}$ $\text{But the equivalent capacitances are given by;}$ $C = \frac{K \varepsilon_0 A}{d_1 + d_2}$ $\text{On comparing we have,}$ $\Rightarrow K = \frac{K_1 K_2 (d_1 + d_2)}{K_1 d_2 + K_2 d_1}$ $\text{Hence, option (3) is the correct answer.}$