Question

The refractive indices of water and diamond are $\dfrac{4}{3}$ and $2.42$ respectively. Find the speed of light in water and diamond.
$\left( c=3\times {{10}^{8}}m/s \right)$

Answer 1

Hint: This problem can be solved by using the direct formula for the refractive index of a material in terms of the speed of light in vacuum (or air) and the speed of light in that medium. By plugging in the information given in the question in the direct formula, we can get the required answers.
Formula used:
$n=\dfrac{c}{v}$

Complete step by step answer:
We will use the formula for the refractive index of a material in terms of the speed of light in that material for this problem.
The refractive index $n$ of a medium in which the speed of light is $v$ is given by
$n=\dfrac{c}{v}$ --(1)
Where $c=3\times {{10}^{8}}m/s$ is the speed of light in air (or vacuum).
Hence, let us analyze the question.
The refractive index of water is ${{n}_{water}}=\dfrac{4}{3}$
Let the speed of light in water be ${{v}_{water}}$.
The refractive index of diamond is ${{n}_{diamond}}=2.42$
Let the speed of light in diamond be ${{v}_{diamond}}$.
The speed of light in air is given to be $c=3\times {{10}^{8}}m/s$.
Hence, using (1), we get
${{n}_{water}}=\dfrac{c}{{{v}_{water}}}$
$\therefore {{v}_{water}}=\dfrac{c}{{{n}_{water}}}$
\[\therefore {{v}_{water}}=\dfrac{3\times {{10}^{8}}}{\dfrac{4}{3}}=\dfrac{3\times 3\times {{10}^{8}}}{4}=\dfrac{9\times {{10}^{8}}}{4}=2.25\times {{10}^{8}}m/s\]
Hence, the speed of light in water is $2.25\times {{10}^{8}}m/s$.
Also, using (1), we get
${{n}_{diamond}}=\dfrac{c}{{{v}_{diamond}}}$
$\therefore {{v}_{diamond}}=\dfrac{c}{{{n}_{diamond}}}$
\[\therefore {{v}_{diamond}}=\dfrac{3\times {{10}^{8}}}{2.42}\approx 1.24\times {{10}^{8}}m/s\]
Hence, the speed of light in diamond is $1.24\times {{10}^{8}}m/s$.

Note: Students should note that the speed and wavelength of light change when travelling from one medium to another. However, the frequency of the light remains the same. The speed of light can be mathematically written as the product of its wavelength and frequency. If the frequency remains constant, this means that the factor by which the speed of light changes when travelling from one medium to another, will be the same factor by which the wavelength will also change.