Question

In a medium of dielectric constant $K$ , the electric field is $\vec E$ . If ${\varepsilon _0}$ is the permittivity of free space, the electric displacement vector is:
(A)\dfrac{{K\vec E}}{{{\varepsilon _0}}} \\\ (B)\dfrac{{\vec E}}{{k{\varepsilon _0}}} \\\ (C)\dfrac{{{\varepsilon _0}\vec E}}{k} \\\ (D)K{\varepsilon _0}\vec E \\\

Answer 1

Hint : In order to solve this question, we are going to first find the permittivity of the given medium from the value of the permittivity of the free space ${\varepsilon _0}$ and the dielectric constant $K$ . After that the displacement current is found from the permittivity so obtained and the electric field vector as given.
Formula used: If the dielectric constant of a medium is $K$ and the permittivity of free space is ${\varepsilon _0}$
Then, the permittivity of the medium is given by
$\varepsilon = K{\varepsilon _0}$
If $\vec E$ is the electric field inside a medium and $\varepsilon$ is the permittivity of the medium, then, the displacement current vector of the medium is given by
$\vec D = \varepsilon \vec E$

Complete Step By Step Answer:
As we are given that the dielectric constant of the medium is $K$ , thus, the permittivity of the given medium can be written as the product of the dielectric constant and the permittivity of the free space.
i.e.
$\varepsilon = K{\varepsilon _0}$
Now the displacement current vector is mathematically the product of the permittivity of the medium and the electric field $\vec E$
So, $\vec D = \varepsilon \vec E$
Now putting the value of the permittivity, $\varepsilon$ in the above relation, we get
$\vec D = K{\varepsilon _0}\vec E$
Thus, the option $(D)K{\varepsilon _0}\vec E$ is the correct answer.

Note :
In electromagnetism, displacement current density is the quantity $\dfrac{{\partial D}}{{\partial t}}$ appearing in Maxwell's equations that is defined in terms of the rate of change of $D$ , the electric displacement field. It depends directly on the value of the electric field applied across the medium and also the permittivity of the medium.