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

let $$C_0=\frac{\varepsilon_0A}{d}$$

case (A)

top layer (d/2, full area, $$k_1$$​):

$$C_t=\frac{k_1\varepsilon_0A}{d/2}=\frac{2k_1\varepsilon_0A}{d}$$

bottom layer (d/2 split):

$$C_{b1}=\frac{k_1\varepsilon_0(A/2)}{d/2}=\frac{k_1\varepsilon_0A}{d}$$

$$C_{b2}=\frac{k_2\varepsilon_0(A/2)}{d/2}=\frac{k_2\varepsilon_0A}{d}$$

parallel:

$$C_b=\frac{(k_1+k_2)\varepsilon_0A}{d}$$

series:

$$C_A=\frac{\varepsilon_0A}{d}\cdot\frac{2k_1(k_1+k_2)}{3k_1+k_2}$$

case (B)

top: $$k_2$$

$$C_t=\frac{2k_2\varepsilon_0A}{d}$$

bottom same as before:

$$C_b=\frac{(k_1+k_2)\varepsilon_0A}{d}$$

$$C_B=\frac{\varepsilon_0A}{d}\cdot\frac{2k_2(k_1+k_2)}{k_1+3k_2}$$

case (C)

top split:

$$C_t=\frac{(k_1+k_2)\varepsilon_0A}{d}$$

bottom split:

$$C_b=\frac{(k_1+k_2)\varepsilon_0A}{d}$$

series of equal:

$$C_C=\frac{1}{2}\cdot\frac{(k_1+k_2)\varepsilon_0A}{d}$$

now compare (given $$k_1>k_2$$):

$$C_A>C_C>C_B$$