Question

A capacitor is formed by two square metal-plates of edge a, separated by a distance d. Dielectrics of dielectric constants $K_1$ and $K_2$ are filled in the gap as shown in figure (31-E25). Find the capacitance.

Answer 1

Consider an elemental capacitor of width dx at a distance 'x' from one end. It is constituted of two capacitance elements of dielectric constants $K_1$ and $K_2$ with plate separation x $tan\theta$ and $d-xtan\theta$ respectively in series.

Effective capacitance

$K=\frac{K_1{K}_2}{K_1-K_2}ln\frac{K_1}{K_2}$

So, $C=\frac{{\in }_{0}KA}{d}$

$\frac{{\in }_0{K}_1{K}_2{a}^2}{(K_1-K_2)d}ln\left(\frac{K_1}{K_2}\right)$ [where, $A=a^2]$