Question

(a) State the laws of refraction of light. Give an expression to relate the absolute refractive index of a medium with the speed of light in vacuum.
(b) The refractive index of water and glass with respect to air are 4/3 and 3/2 respectively. If the speed of light in glass is \[2\times {{10}^{8}}\,m/s\], find the speed of air in (i) water (ii) air.

Answer 1

Hint:Refraction is concerned with the bending of light at the interface of transparent media such as water, glass, air etc. This phenomenon is solely a result of different wavelengths in different media which arise due to different speeds in the media.

Complete step by step answer:
(a) The incident ray, refracted ray and the normal to the interface lie in the same plane.The product of the absolute refractive index of first medium(incident medium) and the sine of angle of incidence is equal to the product of the absolute refractive index and the sine of angle of refraction, i.e,
μisin(i)= μrsin(r)
The above law is also known as Snell’s law.

The absolute refractive index of a medium is actually the ratio of speed of light in vacuum/air to the speed of light in that medium.
\[\therefore \mu =\dfrac{c}{v}\];
where ‘$c$’ is the speed of light in vacuum, i.e \[3\times {{10}^{8}}\,m/s\], ‘$v$’ is the speed of light in the medium.

(b) The refractive index of a medium 1; w.r.t medium 2 is the ratio of speed of light in medium 2 to the speed of light in medium 1. From the above statement:
\[~{{\mu }_{air}}=\dfrac{c}{{{V}_{glass}}}\];
\[ \Rightarrow c={{v}_{glass}}\times \mathbf{\mu } \\
\Rightarrow c=2\times {{10}^{8}}m/s\times \dfrac{3}{2} \\
\therefore c=3\times {{10}^{8}}\,m/s \]
${{\mathbf{\mu }}_{\mathbf{glass}/\mathbf{water}}}=\dfrac{{{v}_{water}}}{{{v}_{glass}}}=\dfrac{{{\mathbf{\mu }}_{\mathbf{glass}}}}{{{\mathbf{\mu }}_{\mathbf{water}}}} \\
\Rightarrow \dfrac{{{\mathbf{\mu }}_{\mathbf{glass}}}}{{{\mathbf{\mu }}_{\mathbf{water}}}}=\dfrac{(3/2)}{(4/3)}=9/8 \\
\Rightarrow {{v}_{water}}={{v}_{glass}}\times {{\mathbf{\mu }}_{\mathbf{glass}/\mathbf{water}}} \\
\Rightarrow {{v}_{water}}=2\times {{10}^{8}}m/s\times 9/8 \\
\therefore {{v}_{water}}=9/4\times {{10}^{8}}m/s$

Note: The speed of light in a medium can be directly calculated by dividing the speed of light in vacuum by the absolute refractive index of the medium. The vice versa of this can also be done to obtain the speed of light in vacuum.