Question

The refractive index of glass with respect to air is $\dfrac{3}{2}$ and the refractive index of water with respect to air is $\dfrac{4}{3}$. What will be the refractive index of water with respect to glass?

Answer 1

Hint: Here, we will proceed by finding the ratio of refractive index of water to the refractive index of air and the ratio of refractive index of glass to the refractive index of air. Then, we will finally divide the first ratio by the second ratio.
Formula Used- Refractive index of medium A with respect to medium B = $\dfrac{{{\text{Refractive index of medium A}}}}{{{\text{Refractive index of medium B}}}}$.

Complete Step-by-Step solution:
Given, Refractive index of glass with respect to air ${\mu _{{\text{ga}}}} = \dfrac{3}{2}$
Refractive index of water with respect to air ${\mu _{{\text{wa}}}} = \dfrac{4}{3}$
According to the formula,
Refractive index of medium A with respect to medium B = $\dfrac{{{\text{Refractive index of medium A}}}}{{{\text{Refractive index of medium B}}}}$
As we know that the refractive index of glass with respect to air will be equal to the ratio of the refractive index of glass to the refractive index of air
i.e., Refractive index of glass with respect to air = $\dfrac{{{\text{Refractive index of glass}}}}{{{\text{Refractive index of air}}}}$
\[
   \Rightarrow {\mu _{{\text{ga}}}} = \dfrac{{{\mu _{\text{g}}}}}{{{\mu _{\text{a}}}}} \\
   \Rightarrow \dfrac{3}{2} = \dfrac{{{\mu _{\text{g}}}}}{{{\mu _{\text{a}}}}} \\
   \Rightarrow \dfrac{{{\mu _{\text{g}}}}}{{{\mu _{\text{a}}}}} = \dfrac{3}{2}{\text{ }} \to {\text{(1)}} \\
 \]

Also we know that the refractive index of water with respect to air will be equal to the ratio of the refractive index of water to the refractive index of air
i.e., Refractive index of water with respect to air = $\dfrac{{{\text{Refractive index of water}}}}{{{\text{Refractive index of air}}}}$
\[
   \Rightarrow {\mu _{{\text{wa}}}} = \dfrac{{{\mu _{\text{w}}}}}{{{\mu _{\text{a}}}}} \\
   \Rightarrow \dfrac{4}{3} = \dfrac{{{\mu _{\text{w}}}}}{{{\mu _{\text{a}}}}} \\
   \Rightarrow \dfrac{{{\mu _{\text{w}}}}}{{{\mu _{\text{a}}}}} = \dfrac{4}{3}{\text{ }} \to {\text{(2)}} \\
 \]
By dividing equation (2) by equation (1), we get
\[
   \Rightarrow \dfrac{{\left( {\dfrac{{{\mu _{\text{w}}}}}{{{\mu _{\text{a}}}}}} \right)}}{{\left( {\dfrac{{{\mu _{\text{g}}}}}{{{\mu _{\text{a}}}}}} \right)}} = \dfrac{{\left( {\dfrac{4}{3}} \right)}}{{\left( {\dfrac{3}{2}} \right)}} \\
   \Rightarrow \dfrac{{{\mu _{\text{w}}}}}{{{\mu _{\text{a}}}}} \times \dfrac{{{\mu _{\text{a}}}}}{{{\mu _{\text{g}}}}} = \dfrac{4}{3} \times \dfrac{2}{3} \\
   \Rightarrow \dfrac{{{\mu _{\text{w}}}}}{{{\mu _{\text{g}}}}} = \dfrac{8}{9} \\
 \]
Also, the refractive index of water with respect to glass will be equal to the ratio of the refractive index of water to the refractive index of glass
i.e., Refractive index of water with respect to glass = $\dfrac{{{\text{Refractive index of water}}}}{{{\text{Refractive index of air}}}} = \dfrac{{{\mu _{\text{w}}}}}{{{\mu _{\text{g}}}}} = \dfrac{8}{9}$
Therefore, the required refractive index of water with respect to glass is $\dfrac{8}{9}$.

Note- To find the answer to this problem one has to know what refractive index means. Refractive index (also called refractive index) is defined as the measure of a ray of light bending while passing from one medium to another. It can also be defined as measuring the reduction of light velocity in a medium.