Question

The area of the plates of a parallel plante 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. $\frac{\pi {\epsilon }_0\lambda A}{2d}$
• B. $\frac{\pi {\epsilon }_0\lambda A}{d}$
• C. $\frac{2\pi {\epsilon }_0\lambda A}{d}$
• D. None

Answer 1

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

$E=\frac{\sigma }{K{ϵ}_o}$

$\frac{dv}{dy}=\frac{\sigma }{K{ϵ}_o}$

${\int }_0^{v}dv=\frac{\sigma }{\lambda {ϵ}_o}{\int }_0^d\cos \left(\frac{\pi y}{2d}\right)dyV=\frac{\sigma }{\lambda {ϵ}_o}\times \frac{2d}{\pi }{\left[\sin \frac{\pi y}{2d}\right]}_0^d=\frac{\sigma }{\lambda {ϵ}_o}\times \frac{2d}{\pi }C=\frac{Q}{V}=\frac{A\lambda {ϵ}_o\pi }{2d}$