A parallel plate capacitor with square plates is filled with four dielectrics of dielectric constants $\mathrm{K}_1, \mathrm{~K}_2, \mathrm{~K}_3, \mathrm{~K}_4$ arranged as shown in the figure. The effective dielectric constant K will be:
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Solution
Ans. (No option is correct) [Bonus]
Sol.
$$
C_{e q}=\frac{C_1 C_2}{C_1+C_2}+\frac{C_3 C_4}{C_3+C_4}, C_1=K_1 C, C=\frac{\epsilon_0 A / 2}{d / 2}
$$
Similarly
$$
\begin{aligned}
& C_2=K_2 C \\
& C_3=K_3 C \\
& C_4=K_4 C \\
& K_{e q}\left(\frac{\epsilon_0 A}{d}\right)=\left(\frac{K_1 K_2}{K_1+K_2}+\frac{K_3 K_4}{K_3+K_4}\right) \frac{\epsilon_0 A / 2}{d / 2} \\
& \Rightarrow K_{e q}=\frac{K_1 K_2}{K_1+K_2}+\frac{K_3 K_4}{K_3+K_4}
\end{aligned}
$$
No option is correct.
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