Question

The gap between the plates of a parallel plate capacitor of area A and distance between plates $d$, is filled with a dielectric whose permittivity varies linearly from ${ϵ}_1$ at one plate to ${ϵ}_2$ at the other. The capacitance of capacitor is :

• A. ${ϵ}_0\left({ϵ}_1+{ϵ}_2\right)A/d$
• B. ${ϵ}_0\left({ϵ}_1+{ϵ}_2\right)A/2d$
• C. ${ϵ}_{0}A/\left[dln\left({ϵ}_2/{ϵ}_1\right)\right]$
• D. ${ϵ}_0\left({ϵ}_2-{ϵ}_1\right)A/\left[d\ell n\left({\in }_2/{ϵ}_1\right)\right]$

Answer 1

Answer: (D)

$\epsilon =\left(\frac{{\epsilon }_2-{\epsilon }_1}{d}\right)x+{\epsilon }_1$
$dC=\frac{{\epsilon }_0\epsilon A}{dx}$
$C_{\text{eq }}=\int dc$