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? $\left(k_{1}>k_{2}\right)$

• A

  $C_{C}>C_{A}>C_{B}$
• B

  $C_{C}>C_{B}>C_{A}$
• Correct answer
  C

  $C_{A}>C_{C}>C_{B}$
• D

  $C_{B}>C_{C}>C_{A}$

Solution

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 $C_C$

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

Hence, the correct answer is option: 'C'.