Question

If the absolute refractive indices of glass and water are $\dfrac{3}{2}$ and $\dfrac{4}{3}$ respectively, what is the refractive index of glass with respect to water?

Answer 1

Hint: - Absolute refractive index of a medium is the ratio of the refractive index of the medium and refractive index of vacuum. In other words, the absolute refractive index of a medium is the ratio of the speed of light in a vacuum and the speed of light in a medium. Moreover, the refractive index of medium 1 with respect to medium 2 is found by dividing the absolute refractive index of medium 1 by the absolute refractive index of medium 2. This will give us our final answer.

Formula used:
Absolute refractive index, ${n_a} = \dfrac{{{n_m}}}{{{n_v}}} = \dfrac{{{v_c}}}{{{v_m}}}$ where ${n_m}$ = refractive index of medium, ${n_v}$ = refractive index of vacuum, ${v_C}$= speed of light in vacuum, and ${v_m}$= speed of light in medium.
Refractive index of medium 1 with respect to medium 2 is given by,
${n_{12}}$ = $\dfrac{{{n_1}}}{{{n_2}}}$ where ${n_1}$= refractive index of medium 1, and ${n_2}$ = refractive index of medium 2.

Step by step solution:
Here, we need to find the refractive index of glass with respect to water. Let us consider glass as medium 1 and water as medium 2.
Therefore, ${n_1}$= $\dfrac{3}{2}$, ${n_2}$ = $\dfrac{4}{3}$ (${n_1}$ = refractive index of glass, ${n_2}$ = refractive index of water)
The refractive index of glass with respect to water = $\dfrac{{{n_1}}}{{{n_2}}}$ (dividing the two values)
Which gives, $\dfrac{3}{2} \div \dfrac{4}{3}$
$\Rightarrow \dfrac{3}{2} \times \dfrac{3}{4} = \dfrac{9}{8}$
The refractive index of glass with respect to water is found to be $\dfrac{9}{8}$.

Note: Refractive indices have no units because while finding out the refractive index, we divide two values, i.e. speed of light in medium 1 and speed of light in medium 2, and these two values have the same unit and hence these units get cancelled while dividing.