Question

The space between the plates of a parallel plate capacitor is filled with a 'dielectric' whose 'dielectric constant' varies with distance as per the relation :
$K\left(x\right)=K_0+\lambda x\left(\lambda =a\text{constant}\right)$
The capacitance $C$, of this capacitor, would be related to its 'vacuum' capacitance $C_o$ as per the relation :

• A. $C=\frac{\lambda d}{\ell n\left(1+K_o\lambda d\right)}{C}_o$
• B. $C=\frac{\lambda }{d.\ell n\left(1+K_o\lambda d\right)}{C}_o$
• C. $C=\frac{\lambda d}{\ell n\left(1+\lambda d/K_o\right)}{C}_o$
• D. $C=\frac{\lambda d}{d.\ell n\left(1+K_o/\lambda d\right)}{C}_o$

Answer 1

Answer: (C)

$d_C=\frac{\left(K_0+{\lambda }_x\right)A}{dx}$
$\int \frac{1}{d_C}=\underset{0}{\overset{\varphi }{\int }}\frac{dx}{A\cdot \left(K_0+\lambda x\right)}$
$C_0=\frac{{\epsilon }_{0}A}{d}\frac{1}{\lambda }\underset{K_0}{\overset{K_0+\lambda d}{\int }}\frac{dt}{At}$
$K_0+\lambda x=t\lambda dx=dt$
$C=\frac{1}{\lambda A}\ln \left[\frac{K_0+\lambda d}{K_0}\right]$