Question 70 of 180

Three parallel plate capacitors each with area $A$ and separation $d$ are filled with two dielectric ($k_{1}$ and $k_{2}$) in the following fashion. Which of the following is true? $\left(k_{1}\gt k_{2}\right)$

A$C_{C}\gt C_{A}\gt C_{B}$B$C_{C}\gt C_{B}\gt C_{A}$C$C_{A}\gt C_{C}\gt C_{B}$D$C_{B}\gt C_{C}\gt C_{A}$

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Answer: (C) $C_{A}\gt C_{C}\gt C_{B}$

For $C_A$

Let $\dfrac{\varepsilon_0 A}{d} = C$
$\therefore C_A = \dfrac{K_1 C}{2} + \dfrac{K_1 K_2 C}{K_1 + K_2}$
$= K_1 C \left[ \dfrac{K_1 + 2K_2}{2(K_1 + K_2)} \right]$

For $C_B$

$C_B = \dfrac{K_2 C}{2} + \dfrac{K_1 K_2 C}{K_1 + K_2}$
$= K_2 C \left[ \dfrac{K_1 + 2K_2}{2(K_1 + K_2)} \right]$

For $C_C$

$\therefore C_C = \dfrac{2K_1 K_2 C}{K_1 + K_2}$
$C_A \gt C_C \gt C_B$