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

Hint: 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.