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 ${}^{\prime }{y}^{\prime }$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. None

Answer 1

Answer: (A)

$E_y=\frac{\sigma }{ky{ϵ}_0}$

$dV=\frac{\sigma }{k{ϵ}_0}dy$

$dV=\frac{\sigma }{\lambda {ϵ}_0}\cos \left(\frac{\pi y}{2d}\right)dy$

Integrating to dy and other side;

${\int }_0^{V}dV={\int }_0^d\frac{\sigma }{\lambda {ϵ}_0}\cos \left(\frac{\pi y}{2d}\right)dy$

$V=\frac{\sigma }{\lambda {ϵ}_0}.\frac{2d}{\pi }$

$C=\frac{Q}{\sigma }\left(\frac{\lambda {ϵ}_0\pi }{2d}\right)$

$C=\frac{A\lambda {ϵ}_0\pi }{2d}$