Question

The refractive indices of water for red and violet colours are 1.325 and 1.334 respectively. Find the difference between the velocities of rays for these two colours in water. ($c=3\times {{10}^{8}}m{{s}^{-1}}$)

Answer 1

Hint: Refractive index is defined as the ratio of speed of light in vacuum to the speed of light in a given medium, i.e. $\mu =\dfrac{c}{v}$. The value of c is the same for all the colours of light. With this formula find the velocities of red and violet in water and then calculate the difference.

Formula Used:
$\mu =\dfrac{c}{v}$

Complete step-by-step answer:
Speed of light is different in different mediums. If a ray of light passes from one medium to another, the speed of light changes. The medium in which the speed of light is faster is called the rarer medium and the medium in which the speed of light is slower is called denser medium. When the speed of light changes, it also changes its direction and the light appears to be bending at the interface of the mediums. This bending of light is called refraction.
To understand the speed of light in a medium, we have something known as a refractive index. Refractive index ($\mu $) of a medium is that characteristic which decides the speed of light. It is defined as the ratio of the speed of light in vacuum (c) to the speed of light (v) in the given medium i.e. $\mu =\dfrac{c}{v}$.
Light consists of electromagnetic waves of different wavelengths. Each wave with a particular wavelength has a particular colour. Understand these colours with the help of the light spectrum.
In vacuum, all the colours have the same speed i.e. c. When they pass from vacuum into another medium, their velocities differ. Hence, the colours have different refractive indices.
It is given that the refractive index of water for red is 1.325. Let the velocity of red light in water be ${{v}_{1}}$.
This means that $1.325=\dfrac{3\times {{10}^{8}}}{{{v}_{1}}}$.
$\Rightarrow {{v}_{1}}=\dfrac{3\times {{10}^{8}}}{1.325}=2.26\times {{10}^{8}}m{{s}^{-1}}$.
It is given that the refractive index of water for violet is 1.325. Let the velocity of violet light in water be ${{v}_{2}}$.
This means that $1.334=\dfrac{3\times {{10}^{8}}}{{{v}_{2}}}$.
$\Rightarrow {{v}_{2}}=\dfrac{3\times {{10}^{8}}}{1.334}=2.24\times {{10}^{8}}m{{s}^{-1}}$.
$\Rightarrow {{v}_{1}}-{{v}_{2}}=(2.26\times {{10}^{8}}-2.24\times {{10}^{8}})m{{s}^{-1}}=0.02\times {{10}^{8}}m{{s}^{-1}}$.
Therefore, the difference between the velocities of rays for these two colours in water is $0.02\times {{10}^{8}}m{{s}^{-1}}$.

Note: When white light passes through different mediums, the frequency of light remains constant. However, the wavelengths of the different colours change and change is different for different colours.
The wavelength of red light is maximum in a given medium. Whereas the violet light has the least wavelength.