Question

If $ {v_1} $,$ {v_2} $ and $ c $ represent the speed of light in medium 1, medium 2, and vacuum respectively, write the expressions for
(a) absolute refractive index of medium 1
(b) absolute refractive index of medium 2
Hence show that \[{}_1{n_2} = \dfrac{{{n_2}}}{{{n_1}}}\]

Answer 1

Hint:
- According to Snell’s law, when a light or other waves passing through a boundary between two different isotopic media then the ratio of the sines of the angles of incidence and refraction is constant and value of this constant depends on the types of medium through which the waves propagating and wavelength of the propagating wave.
- According to Snell’s of refraction, the refractive index $ n $ of the second medium with respect to first medium is given by
$n = \dfrac{{\sin (r)}}{{\sin (i)}}$, where $i$ be the incident angle and $r$ be the Refracted angle.
- According to wave theory of light,
${n_1} = \dfrac{{\sin (r)}}{{\sin (i)}} = \dfrac{c}{{{v_1}}}$, where the velocity of light in vacuum is $c$ and the velocity of light in that medium \[ - 1\] is ${v_1}$.

Complete step by step solution:
If a light propagate through vacuum then if it is refracted to another medium then the ratio of sines of angles of incidence and refraction is called the absolute refractive refractive index of that medium i.e., refractive index of a medium with respect to vacuum is called absolute refractive index of that medium.
If ${n_1}$ is the absolute refractive index of medium\[ - 1\] then
${n_1} = \dfrac{{\sin (r)}}{{\sin (i)}}$.....(1)
According to wave theory of light,
${n_1} = \dfrac{c}{{{v_1}}}$.....(2)
Similarly if ${n_2}$is the absolute refractive index of medium\[ - 2\]then
${n_2} = \dfrac{c}{{{v_2}}}$.....(3)
Where $ {v_2} $ is the velocity of light in medium\[ - 2\].

Again according to wave theory of light, the refractive index of medium \[ - 1\] with respect to medium-2
\[{}_1{n_2} = \] velocity of light in medium \[ - 1\] velocity of light in medium\[ - 2\]
\[{}_1{n_2} = \dfrac{{{v_1}}}{{{v_2}}}\]......(4)
From equation\[ - \left( 2 \right){\text{ }}and{\text{ }}\left( 3 \right)\], velocity of light in medium\[ - 1\], ${v_1} = \dfrac{c}{{{n_1}}}$and the velocity of light in medium\[ - 2\], ${v_2} = \dfrac{c}{{{n_2}}}$. Putting these expressions of velocities in equation (4) we have
\[{}_1{n_2} = \dfrac{{{v_1}}}{{{v_2}}} = \dfrac{{\dfrac{c}{{{n_1}}}}}{{\dfrac{c}{{{n_2}}}}} = \dfrac{{{n_2}}}{{{n_1}}}\]

Note:
If medium\[ - 2\] is more denser than medium 1 then \[{v_2} < {\text{ }}{v_1}\] and hence in that case \[{}_1{n_2} > 1.\]
If medium\[ - 1\] is more denser than medium 2 then \[{v_2} > {\text{ }}{v_1}\] and hence in that case \[{}_1{n_2} < 1.\]