Question

Refractive index of a glass with respect to water is (9/8). Refractive index of a glass with respect to air is (3/2). Find the refractive index of water with respect to air.

Answer 1

Hint:The refractive index of a glass with respect to water and that with respect to air is already given in the problem. The refractive index of water with respect to air can be found out by using the formula of refractive index. The formula for the refractive index of medium 1 with respect to medium 2 is given by the division of the refractive index of medium 1 by the refractive index of medium 2.

Formula Used:The refractive index of medium 1 with respect to medium 2 is: $${\mu }_2^1={\displaystyle \frac{{\mu }_1}{{\mu }_2}}$$

where, $${\mu }_1$$ is the refractive index of medium 1 and
$${\mu }_2$$is the refractive index of medium 2

Complete step by step solution:The bending of ray of light on passing from one medium to another is called refraction. Refractive index is defined as the ratio of the velocity of light in the medium of incidence to that in the medium of refraction. The refractive index of medium 1 with respect to medium 2 is given as $${\mu }_2^1={\displaystyle \frac{{\mu }_1}{{\mu }_2}}$$

Let the refractive indices of glass, water and air be $${\mu }_g$$, $${\mu }_w$$and $${\mu }_a$$respectively. It is given that the refractive index of a glass with respect to water is (9/8). Then,
$${\mu }_w^g={\displaystyle \frac{{\mu }_g}{{\mu }_w}}={\displaystyle \frac{9}{8}}$$ $$\to (1)$$

Therefore, $${\mu }_w={\displaystyle \frac{8}{9}}{\mu }_g$$ $$\to (2)$$
Also, the refractive index of a glass with respect to air is (3/2).
$${\mu }_a^g={\displaystyle \frac{{\mu }_g}{{\mu }_a}}={\displaystyle \frac{3}{2}}$$ $$\to (3)$$

Therefore, $${\mu }_a={\displaystyle \frac{2}{3}}{\mu }_g$$ $$\to (4)$$
The refractive index of water with respect to air is given as
$${\mu }_a^w={\displaystyle \frac{{\mu }_w}{{\mu }_a}}$$
Substituting the values of $${\mu }_w$$ and $${\mu }_a$$

Therefore, $${\mu }_a^w={\displaystyle \frac{{\mu }_w}{{\mu }_a}}={\displaystyle \frac{{\displaystyle \frac{8}{9}}{\mu }_g}{{\displaystyle \frac{2}{3}}{\mu }_g}}={\displaystyle \frac{8}{9}}\times {\displaystyle \frac{3}{2}}={\displaystyle \frac{4}{3}}$$

Thus, the refractive index of water with respect to air is $${\displaystyle \frac{4}{3}}$$.

Note: The problem is given with the particular values of refractive indices of a glass with respect to water and glass with respect to air. But the value of refractive index depends on the following factors:
● Nature of the media of incidence and refraction.
● Color of the light or the wavelength of light
● Temperature of the medium
Refractive index decreases with increase in temperature. Also, the refractive index is independent of the angle of incidence.