Question

A parallel plate capacitor of capacitance C has spacing d between two plates having area A. The region between the plates is filled with $N$ dielectric layers, parallel to its plates, each $\delta=\frac{d}{N},$ The dielectric constant of the $m^{th}$ layer is $K_{m}=K\left(1+\frac{m}{N}\right)$ For a very large $N (> 10^3)$, the capacitance $C$ is $\alpha\left(\frac{K\in_{0}A}{d\,In\,2}\right)$. The value of $\alpha$ will be_______.
[$\in_{0}$ is the permittivity of free space with thickness]

Answer 1

$dC=\frac{K\in_{0}A}{dx}$
All are connected in series
$\frac{1}{C}=\int\frac{1}{dc}$
$\frac{1}{C}=\int\frac{dx}{K\in_{0}A\left(1+m/N\right)}$
$x=\frac{md}{N}$
$\frac{1}{C}=\int \frac{dx}{K\in_{0}A\left(1+x/d\right)}$
$\frac{1}{C}=\frac{d}{K\in_{0}A}$ $\int\limits^{{x}}_{{0}}$$\frac{dx}{d+x}$
$\frac{1}{C}=\frac{d\,In\left(2\right)}{K\in_{0}A}$
$C=\frac{K\in_{0}A}{d\,In\left(2\right)}$