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Question: The absolute refractive index of glass and water is \[\dfrac{3}{2}\]and \[\dfrac{4}{3}\], respective...

The absolute refractive index of glass and water is 32\dfrac{3}{2}and 43\dfrac{4}{3}, respectively. If the speed of light in glass is 2×1082\times {{10}^{8}}, calculate the speed of light in: a) vacuum b) water

Explanation

Solution

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:- μ=cv\mu =\dfrac{c}{v}
Where,
μ\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 32\dfrac{3}{2}and 43\dfrac{4}{3}, respectively.
Let, absolute refractive index of glass and water be μg{{\mu }_{g}} and μw{{\mu }_{w}}
μg=32{{\mu }_{g}}=\dfrac{3}{2} and μw=43{{\mu }_{w}}=\dfrac{4}{3}
It is given that the speed of light in glass vg{{v}_{g}} is 2×1082\times {{10}^{8}}. Therefore, the absolute refractive index of the glass can be given by,
μg=cvg{{\mu }_{g}}=\dfrac{c}{{{v}_{g}}}…………. (1)
After substituting the values,
32=c2×108\dfrac{3}{2}=\dfrac{c}{2\times {{10}^{8}}}
On solving
We get,
c=3×108c=3\times {{10}^{8}}
Therefore, the speed of light in vacuum is 3×1083\times {{10}^{8}}.
Now, for the speed of light in water.
It is given the refractive index of water μw=43{{\mu }_{w}}=\dfrac{4}{3}.
From (1) we can say that refractive index of water is given by,
μw=cvw{{\mu }_{w}}=\dfrac{c}{{{v}_{w}}}
After substituting the values,
43=3×108vw\dfrac{4}{3}=\dfrac{3\times {{10}^{8}}}{{{v}_{w}}}
On solving
We get,
vw=2.25×108{{v}_{w}}=2.25\times {{10}^{8}}
Therefore, the speed of light in water is 2.25×1082.25\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.