Question

If \[{\mu _r}\] be the relative permeability and \[{\varepsilon _r}\] is the relative dielectric constant of a medium, its refractive index is
(A) \[\dfrac{1}{{\sqrt {{\mu _r}{\varepsilon _r}} }}\]
(B) \[\dfrac{1}{{{\mu _r}{\varepsilon _r}}}\]
(C) \[\sqrt {{\mu _r}{\varepsilon _r}} \]
(D) \[{\mu _r}{\varepsilon _r}\]

Answer 1

Hint:The refractive index of a medium is the ratio of velocity of light in free space to the velocity of light in that medium. Express the speed of light in a medium of permeability \[\mu \] and dielectric constant \[\varepsilon \] and use the relations \[\mu = {\mu _r}{\mu _0}\] and \[\varepsilon = {\varepsilon _r}{\varepsilon _0}\] to rewrite it.

Complete step by step answer:
We know the refractive index of a medium is the ratio of velocity of light in free space to the velocity of light in that medium. Therefore,
\[\mu = \dfrac{c}{v}\] …… (1)
Here, c is the speed of light and v is the speed of light in a medium of refractive index \[\mu \].We know the speed of light in free space is expressed as,
\[c = \dfrac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }}\] …… (2)
Here, \[{\mu _0}\] is the permeability of free space and \[{\varepsilon _0}\] is the dielectric constant of the free space.
Also, the speed of light in the medium of permeability \[\mu \] and dielectric constant \[\varepsilon \] is given as,
\[v = \dfrac{1}{{\sqrt {\mu \varepsilon } }}\]
We know that permeability of the medium is the product of relative permeability and permeability of free space. Therefore, we can write,
\[\mu = {\mu _r}{\mu _0}\]
Also, the dielectric constant of a medium is the product of relative dielectric constant and dielectric constant of free space. Therefore, we can write,
\[\varepsilon = {\varepsilon _r}{\varepsilon _0}\].
Therefore, we can express the speed of light as follows,
\[v = \dfrac{1}{{\sqrt {{\mu _r}{\mu _0}{\varepsilon _r}{\varepsilon _0}} }}\] … (3)
Using equations (2) and (3) in equation (1), we get,
\[\mu = \dfrac{{\dfrac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }}}}{{\dfrac{1}{{\sqrt {{\mu _r}{\mu _0}{\varepsilon _r}{\varepsilon _0}} }}}}\]
\[ \therefore\mu = \sqrt {{\mu _r}{\varepsilon _r}} \]

So, the correct answer is option (C).

Note:The refractive index of any medium is greater than 1, since the numerator, speed of light in the free space is always greater than speed of light in the given medium. If the relative permeability is equal to the permeability of free space and relative dielectric constant is equal to the relative dielectric constant of free space then the refractive index of the medium is usually written as, \[\mu = \sqrt {{\mu _o}{\varepsilon _o}} \].