The area of the plates of a parallel plate capacitor is $A$ and the gap between them is $d$. The gap is filled with a non-homogeneous dielectric whose dielectric constant varies with the distance $‘y’$ from one plate as : $K = \lambda \ sec(\pi y/2d)$, where $\lambda $ is a dimensionless constant. The capacitance of this capacitor is
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$E=\frac{\sigma}{K \epsilon_{o}}$
$\frac{d v}{d y}=\frac{\sigma}{K \epsilon_{o}}$
$\int_{0}^{v} d v=\frac{\sigma}{\lambda \epsilon_{o}} \int_{0}^{d} \cos \left(\frac{\pi y}{2 d}\right) d y V$
$=\frac{\sigma}{\lambda \epsilon_{o}} \times \frac{2 d}{\pi}\left[\sin \frac{\pi y}{2 d}\right]_{0}^{d}$
$=\frac{\sigma}{\lambda \epsilon_{o}} \times \frac{2 d}{\pi}$
$C=\frac{Q}{V}=\frac{A \lambda \epsilon_{o} \pi}{2 d}$
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