Question

Express dielectric constant of a medium in terms of capacitance. What is its SI unit?

Answer 1

Capacitance:

1. Capacitance is a property of an electric conductor and is the measure of the amount of electric charge the conductor can hold per unit electric potential.
2. It is expressed by the formula $C=\frac{q}{V}$, where $q$ is amount of charge and $V$ is electric potential.
3. The expression for capacitance of a parallel plate capacitor is $C=\frac{A{\epsilon }_0}{d}$, where $A$ is the area of cross-section of the capacitor, ${\epsilon }_0$ is the permeability of free space and $d$ is the separation between the plates.

Dielectric constant:

1. When a dielectric substance with a dielectric constant $K$ is filled in between the plates, the formula of capacitance of a parallel plate capacitor becomes $C_k=\frac{KA{\epsilon }_0}{d}$.
2. The dielectric constant is the ratio of permeability of material and permeability of free space, $K=\frac{{\epsilon }_r}{{\epsilon }_0}$.
3. To find an expression for dielectric constant take the ratio of the capacitance of a parallel plate capacitor when the dielectric substance is present to the capacitance of a parallel plate capacitor when dielectric substance is absent and solve it further.

$$\begin{gathered} \frac{C_k}{C}=\frac{{\displaystyle \frac{KA{\epsilon }_0}{d}}}{{\displaystyle \frac{A{\epsilon }_0}{d}}} \\ \Rightarrow \frac{C_k}{C}=K \end{gathered}$$

Since it is the ratio of capacitance, it is unitless.

Hence, the dielectric constant is $K=\frac{C_k}{C}$ and it is unitless.