Question

The absolute refractive index of glass and water is $\dfrac{3}{2}$and $\dfrac{4}{3}$, respectively. If the speed of light in glass is $2\times {{10}^{8}}$, calculate the speed of light in: a) vacuum b) water

Answer 1

In this question we have been asked to calculate the speed of light in vacuum and in water. It is given that the refractive index of glass and water is 3/2 and 4/3 respectively. We know that the refractive index is the refractive index in a vacuum. Therefore, using this definition, we shall calculate the speed of light in vacuum and water.
Formula Used:- $\mu =\dfrac{c}{v}$
Where,
$\mu$ is the absolute refractive index of the medium
C is the speed of light in a vacuum
V is the speed of light in a medium

Complete step by step solution:
It is given that the absolute refractive index of glass and water is $\dfrac{3}{2}$and $\dfrac{4}{3}$, respectively.
Let, absolute refractive index of glass and water be ${{\mu }_{g}}$ and ${{\mu }_{w}}$
${{\mu }_{g}}=\dfrac{3}{2}$ and ${{\mu }_{w}}=\dfrac{4}{3}$
It is given that the speed of light in glass ${{v}_{g}}$ is $2\times {{10}^{8}}$. Therefore, the absolute refractive index of the glass can be given by,
${{\mu }_{g}}=\dfrac{c}{{{v}_{g}}}$…………. (1)
After substituting the values,
$\dfrac{3}{2}=\dfrac{c}{2\times {{10}^{8}}}$
On solving
We get,
$c=3\times {{10}^{8}}$
Therefore, the speed of light in vacuum is $3\times {{10}^{8}}$.
Now, for the speed of light in water.
It is given the refractive index of water ${{\mu }_{w}}=\dfrac{4}{3}$.
From (1) we can say that refractive index of water is given by,
${{\mu }_{w}}=\dfrac{c}{{{v}_{w}}}$
After substituting the values,
$\dfrac{4}{3}=\dfrac{3\times {{10}^{8}}}{{{v}_{w}}}$
On solving
We get,
${{v}_{w}}=2.25\times {{10}^{8}}$
Therefore, the speed of light in water is $2.25\times {{10}^{8}}$.

Note: The absolute refractive index of a medium is defined as the ratio of the speed of light in the vacuum over the speed of light in the medium. The refractive index of a medium depends on the wavelength of light, optical density, temperature, and refractive index of the surrounding. Impurities present in a medium will increase the refractive index of the medium. As the refractive index of a medium increases its optical density increases. Therefore, the speed of light in that medium is slower.