Question

A parallel plate capacitor is made of two square plates of side ${}^{\prime }{a}^{\prime }$, separated by a distance $d(d<<a)$. The lower triangular portion is filled with a dielectric of dielectric constant $K$. The capacitance of this capacitor is :

• A. $\frac{1}{2}K\frac{{ϵ}_0{a}^2}{d}$
• B. $K\frac{{ϵ}_0{a}^2}{d}\ln K$
• C. $K\frac{{ϵ}_0{a}^2}{d(K-1)}\ln K$
• D. $K\frac{{ϵ}_0{a}^2}{2d(K+1)}$

Answer 1

The correct option is C $K\frac{{ϵ}_0{a}^2}{d(K-1)}\ln K$

Suppose the dielectric was not present then the value of $K$ must have been $1$.
This means that the capacitance in that case would be $C=\frac{{ϵ}_0{a}^2}{d}$-------(1)
Now, let us to check if any of our option matches (1) on putting $K=1$ .
In option (a) it comes out to be $\frac{{ϵ}_0{a}^2}{2}$.
In option (b) it will be 0.
In option (c) it will be $\frac{0}{0}$ indeterminate form. This means it can take any value between 0 to infinity.
In option (d) it will be $\frac{{ϵ}_0{a}^2}{4d}$
So, the only option correct will be (c).