Question

Calculate the wavelength of the light of frequency $5\times {{10}^{14}}Hz$ in the water of the refractive index $\dfrac{4}{3}$. The velocity of light in free space is $3\times {{10}^{8}}m/s$.

Answer 1

Hint: When light passes from one medium to another medium the lights tend to bend by a certain angle over the edges of the medium. This phenomenon is known as the refraction of light. While refraction the frequency of the light remains constant but the speed of light and the wavelength of the medium changes.

Formula used:
The refractive medium of any medium,
$\mu =\dfrac{{{s}_{i}}}{{{s}_{ref}}}$
Speed of light
$c=\upsilon \lambda $

Complete step by step answer:
As per the given data,
The frequency of the light is $5\times {{10}^{14}}Hz$
The refractive index of the water is $\dfrac{3}{4}$
The speed of light in free space is $3\times {{10}^{8}}m/s$
The speed of the light in the water can be calculated with the help of the formula of the refractive index.
So refractive index is given by
$\mu =\dfrac{{{s}_{i}}}{{{s}_{ref}}}$
So, When the light is passed from air and enters into the water.

By rearranging the formula for the refractive index speed of the light in the water is given can be given by,
$\begin{align}
  & {{s}_{m}}=\mu \times {{s}_{air}} \\
 & \Rightarrow {{s}_{m}}=\dfrac{3}{4}(3\times {{10}^{8}}) \\
 & \Rightarrow {{s}_{m}}=2.25\times {{10}^{8}}m/s \\
\end{align}$
As we know that the speed of light is the product of the wavelength and the frequency of light in the particular medium. While refraction the frequency of the medium is always constant.
$\begin{align}
  & c=\lambda \upsilon \\
 & \Rightarrow \lambda =\dfrac{c}{\upsilon } \\
\end{align}$
By putting the value of the speed of light in the water and the frequency of light. The wavelength of the light will be,
$\begin{align}
  & \lambda =\dfrac{2.25\times {{10}^{8}}}{5\times {{10}^{14}}} \\
 & \Rightarrow \lambda =0.45\times {{10}^{-6}}m \\
\end{align}$

So, the wavelength of the light in water with a refractive index $\dfrac{4}{3}$ is $0.45\times {{10}^{-6}}m$.

Note: Refraction mainly occurs due to the change in the speed of the light in the two mediums. The ratio of this speed of light in two mediums gives the refractive index of a medium concerning another medium. The value of the refractive index of a particular medium can never be less than one.