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 $\alpha \left(\frac{K{\in }_{0}A}{dIn2}\right)$. The value of $\alpha$ will be_______.
[${\in }_0$ is the permittivity of free space with thickness]

Answer 1

Answer: 1.00

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