Question

A capacitor is half filled with a dielectric (K = 2) as shown in the figure A. If the same capacitor is to be filled with the same dielectric as shown in figure B, what should be the thickness $t$ of the dielectric so that capacitance still has same value?

Answer 1

Initially, $C=C_1+C_2=\frac{{ϵ}_0\frac{A}{2}}{d}+\frac{k{ϵ}_0\frac{A}{2}}{d}=(1+k)\frac{{ϵ}_{0}A}{2d}=\frac{3{ϵ}_{0}A}{2d}$
Finally, $C=\frac{C_1{C}_2}{C_1+C_2}=\frac{\frac{{ϵ}_0.A}{d-t}.\frac{k{ϵ}_{0}A}{t}}{\left(\frac{{ϵ}_{0}A}{d-t}\right)+\left(\frac{{ϵ}_{0}A}{t}\right).k}=\frac{k{ϵ}_0^2{A}^2}{{ϵ}_{0}At+k{ϵ}_{0}A(d-t)}=\frac{2.{ϵ}_{0}A}{t+2.(d-t)}$
Now, $\frac{3{ϵ}_{0}A}{2d}=\frac{2{ϵ}_{0}A}{t+2(d-t)}\Rightarrow \frac{3}{2d}=\frac{2}{2d-t}\Rightarrow 6d-3t=4d$
$\Rightarrow t=\frac{2d}{3}$