Question

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 $k=\lambda sec\left(\frac{\pi y}{2d}\right)$, where $\lambda$ is a dimensionless constant. The capacitance of this capacitor is

• A. $\frac{\pi {ϵ}_0\lambda A}{2d}$
• B. $\frac{\pi {ϵ}_0\lambda A}{d}$
• C. $\frac{2\pi {ϵ}_0\lambda A}{d}$
• D. $\frac{\pi {ϵ}_0\lambda A}{3d}$

Answer 1

The correct option is A $\frac{\pi {ϵ}_0\lambda A}{2d}$


Capacitance of the element is
$dc=\frac{k{ϵ}_{0}A}{dy}$
$=\frac{\lambda {ϵ}_{0}Asec\left(\frac{\pi y}{2d}\right)}{dy}$
$\frac{1}{dc}=\frac{dy}{\lambda {ϵ}_{0}A}cos\left(\frac{\pi y}{2d}\right)$
Each element is in series with the neighboring element.
$\frac{1}{C_{eff}}=\int \frac{1}{dc}$
$=\frac{1}{\lambda {ϵ}_{0}A}{\int }_0^{d}cos\left(\frac{\pi y}{2d}\right)dy$
$=\frac{2d}{\lambda \pi {ϵ}_{0}A}sin{\left(\frac{\pi y}{2d}\right)}_0^d$
$\frac{1}{C_{eff}}=\frac{2d}{\pi \lambda {ϵ}_{0}A}\Rightarrow C_{eff}=\frac{\pi \lambda {ϵ}_{0}A}{2d}$