Question

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

Answer 1

Capacitance is described as the property to hold charge. Capacitance is dependent on the geometry of the capacitor.
The mathematical expression for a parallel plate capacitor is ${C_0} = \dfrac{{A{\varepsilon _0}}}{d}$ where C is the capacitance, A is the area of cross section of the capacitor, d is the separation between the plates and ${\varepsilon _0}$ is the permeability of free space. This is the expression for the capacitor with no dielectric in between.
When a dielectric of dielectric constant K is present between the plates, the capacitance is given by ${C_k} = \dfrac{{KA{\varepsilon _0}}}{d}$.
The dielectric constant is the ratio of permeability of a material to the permeability of free space.

Complete step by step solution:
We know that the dielectric constant is the ratio of permeability of a material to the permeability of free space. The expression is given as $K = \dfrac{{{\varepsilon _r}}}{{{\varepsilon _0}}}$ .
Let the capacitance in a dielectric medium be ${C_r}$ . Its value is given by ${C_r} = \dfrac{{A{\varepsilon _r}}}{d}$ which simplifies to ${C_r} = \dfrac{{KA{\varepsilon _0}}}{d}$ .
But we know that the value of capacitance with the same geometry but in free space will be ${C_0} = \dfrac{{A{\varepsilon _0}}}{d}$ .
Taking the ratio of ${C_r}$ and ${C_0}$ ,
$\dfrac{{{C_r}}}{{{C_0}}} = \dfrac{{\dfrac{{KA{\varepsilon _0}}}{d}}}{{\dfrac{{A{\varepsilon _0}}}{d}}}$
Further solving this equation, we get
$\dfrac{{{C_r}}}{{{C_0}}} = K$ where K is the dielectric constant of the medium.
This is the required ratio.
Since the dielectric constant is the ratio of permeability of a material to the permeability of free space, it is a unitless quantity.

Note: Capacitance is dependent on its geometry and this formula is applicable only for parallel plate capacitors. The units of any physical quantity can be easily found by substituting proper units in place of the physical quantities in a relation satisfied by the physical quantity whose units are desirable.