Question

If $ {\varepsilon}_0 $ and $ {\mu}_0 $ are the electric permittivity and magnetic permeability in a free space, $ \varepsilon $ and $\mu $ are the corresponding quantities in medium, the index and refraction of the medium is

• A. $\sqrt{ \frac{ {\varepsilon}_0 {\mu}_0}{\varepsilon \mu }}$
• B. $\sqrt{ \frac{\varepsilon \mu}{{\varepsilon}_0 {\mu}_0}}$
• C. $\sqrt{ \frac{{\varepsilon}_0 \mu }{\varepsilon {\mu}_0}}$
• D. $\sqrt{ \frac{\varepsilon}{{\varepsilon}_0}}$

Answer 1

The Correct Option is B

$ c= \frac{ 1}{ \sqrt{{\mu}_0 { \varepsilon}_0 }} $(free space)
$v= \frac{1}{ \sqrt{\mu \varepsilon}}$ (medium)
$ \therefore \, \, \, \, \, \mu = \frac{c}{v} = \sqrt{ \frac{ \mu \varepsilon }{ { \mu}_0 {\varepsilon}_0 }}$

Answer 2

ε is called the electric permittivity of a medium. It is a proportional constant used in the formula for the electric force between two point-charged particles separated by some distance, i.e. F=\(\frac{q_1q_2}{4\pi\epsilon r^2}\)

When the charges are present in the medium of vacuum, the electric permittivity is called as permittivity of free space and is denoted as ε₀.

μ is called the magnetic permeability of a medium. It is a proportional constant used in the formula for the magnetic field produced by a current-carrying wire of length l at a point located at a distance r, i.e. 

  \(\vec{B}\)=\(\frac{\mu(\vec{l} \times \hat{a})}{4\pi r^2}\)

When the medium surrounding the wire is a vacuum, the magnetic permeability is called as permeability of free space and is denoted as μ₀

The relation between the electric permittivity and the magnetic permeability of a medium is given with help of the speed (v) of light traveling in that medium as 

v=\(\frac{1}{\sqrt{\mu \epsilon}}\)…..(i)

When we talk about a vacuum, the speed of light in a vacuum is equal to c. In vacuum, 

ε=ε₀ and μ=μ₀

Therefore,

 c=\(\frac{1}{\mu_0\epsilon_0}\) …….(ii)

The refractive index, or index of refraction, of a medium is defined as the ratio of the speed (c) of light in a vacuum to the speed (v) of light in that medium.

Hence, the refractive index is given as 

\(\frac{c}{v}\)

Substitute the values of c and v from equations (i) and (ii).

Hence, the value of the refractive index can be written as 

\(\frac{c}{v}\) =\(\frac{\frac{1}{\sqrt{\mu_0 \epsilon_0}}}{\frac{1}{\sqrt{\mu \epsilon}}}\) =\(\frac{\sqrt{\mu \epsilon}}{\mu_0 \epsilon_0}\)

 

Hence, the correct option is B.