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.