Question

The refractive index of water is 4/3 and the refractive index of glass is 3/2. What is the refractive index of glass with respect to water?

Answer 1

Hint: Refractive index can be calculated by dividing the refractive of glass by the refractive index of water. It is also equal to the velocity of light in water divided by the velocity of light in water. The formulae used to calculate are in the following section.

Formula used:
Refractive index of a medium with respect to other medium:
\[n=\dfrac{{{n}_{i}}}{{{n}_{o}}}=\dfrac{{{v}_{o}}}{{{v}_{i}}}\]
Where, ${{n}_{o}}$is the medium from which light with velocity ${{v}_{o}}$ is leaving and ${{n}_{i}}$ is the medium in which light is entering with velocity ${{v}_{i}}$.

Complete step by step answer:
The refractive index of a material is defined by the following equation:
$n=\dfrac{c}{v}$…(1)
Where, $c$ is the velocity of light in vacuum and $v$ is the velocity of light in the material.
So, we can write the refractive index of water as:
${{n}_{w}}=\dfrac{c}{{{v}_{w}}}$
Now, we can rearrange this equation to find the value of ${{v}_{w}}$. Which is:
${{v}_{w}}=\dfrac{c}{{{n}_{w}}}$
We can now put value of ${{n}_{w}}=\dfrac{4}{3}$
${{v}_{w}}=\dfrac{3c}{4}$…(2)
Similarly, we can find value of velocity of light in glass ${{v}_{g}}$
${{v}_{g}}=\dfrac{c}{{{n}_{g}}}$
We can now put ${{n}_{g}}=\dfrac{3}{2}$
So, we get,
${{v}_{g}}=\dfrac{2c}{3}$…(3)
Now, to find the refractive index of glass with respect to water ${{n}_{gw}}$. We must divide ${{v}_{w}}$by ${{v}_{g}}$. Which will give,
${{n}_{gw}}=\dfrac{{{v}_{w}}}{{{v}_{g}}}$
$=\dfrac{\left( \dfrac{3}{4} \right)}{\left( \dfrac{2}{3} \right)}=\dfrac{\left( \dfrac{3}{2} \right)}{\left( \dfrac{4}{3} \right)}=\dfrac{{{n}_{g}}}{{{n}_{w}}}$
$=\dfrac{9}{8}$
Thus, we get the refractive index of glass with respect to water.

Additional Information
(i)The Refractive index is a dimensionless quantity.
(ii)The refractive index of vacuum is unity and the refractive index of air can also be approximated to 1.
(iii)The frequency of a material is independent of the refractive index but the refractive index of a material is dependent on the wavelength given by the relation $\lambda =\dfrac{{{\lambda }_{o}}}{n}$ where ${{\lambda }_{o}}$denotes the velocity of radiation in a vacuum.
(iv)Speed of anything can’t exceed c = $3\times {{10}^{8}}m{{s}^{-1}}$ but the refractive index can have values below 1, as in the example of X-Rays.

Note:
Students can confuse which value to be put in the numerator and which value to be put in the denominator in the formula. So, remembering one of the two formulas ($n=\dfrac{{{n}_{i}}}{{{n}_{o}}}$ or $n=\dfrac{{{v}_{o}}}{{{v}_{i}}}$) along with $n=\dfrac{c}{v}$ will be helpful to avoid confusion. The value of the refractive index of glass is question specific, different types of glasses can have different refractive indices, so must use only the value mentioned in the question.