Question

The absolute refractive indices of glass and water are $\frac{4}{3}$ and $\frac{3}{2}$ respectively. If the speed of light in the glass is $2\times {10}^8\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$, calculate the speed of light in

b. water.

c. air.

Answer 1

Step -1 Given data.

The absolute refractive index of glass, $u_g=\frac{4}{3}$

The absolute refractive index of water $u_w=\frac{3}{2}$

The speed of light in the glass medium $v_g=2\times {10}^{8}m{s}^{-1}$

Step 2- Finding the speed of light in a vacuum

The refractive index of the glass medium is given as

$u_g=\frac{c}{v_g}$ where $c$ is the speed of light in a vacuum,

Substitute the values of the variables to find $c$.

$$\begin{gathered} \frac{4}{3}=\frac{c}{2\times {10}^{8}m/s} \\ c=\frac{4\times 2\times {10}^8}{3} \\ c=2.67\times {10}^{8}m/s \end{gathered}$$

Step 3 - Finding the speed of light in water.

Assume the speed of light is water as $v_w$.

The refractive index of water in terms of speed of light is given by,

${\mu }_w=\frac{c}{v_w}$

Substitute the values into the formula.

$$\begin{gathered} \frac{3}{2}=\frac{2.67\times {10}^8\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}{v_w} \\ {v}_w=\frac{2.67\times {10}^8\times 2}{3} \\ =1.78\times {10}^8\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$

Step 4 - Finding the speed of light in air.

Assume the speed of light in air to be $v$.

The refractive index of air in terms of speed of light is given by,

${\mu }_a=\frac{c}{v_{}}$

Substitute the values into the formula.

$$\begin{gathered} 1.0003=\frac{2.67\times {10}^8\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}{v} \\ v=\frac{2.67\times {10}^8}{1.0003} \\ v=2.6691\times {10}^8\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$

Thus,

1. The speed of light in water is $1.78\times {10}^8\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$.
2. The speed of light in air is $2.6691\times {10}^{8}m/s$.