Question

The frequency of a light wave in a material is \[2\times {{10}^{14}}Hz\] and wavelength is \[5000\overset{o}{\mathop{\text{A}}}\,\]. The refractive index of the material will be
A) 1.50
B) 3.00
C) 1.33
D) 1.40

Answer 1

Hint: We have given the wavelength and frequency of the material and we have to find the refractive index. Now the refractive index of the material can be given as the ratio of the speed of light in a vacuum to the speed of light in that medium in which it is traveling. The velocity can be given as a product of wavelength and frequency which is given here so we can calculate the refractive index.
Formula used:
\[\begin{align}
  & n=\dfrac{c}{v} \\
 & v=\lambda f \\
\end{align}\]

Complete step-by-step solution
The refractive index is the measure of the deviation of a light ray from its path while traveling from one media to another media. Now the refractive index of the material can be given as the ratio of the speed of light in a vacuum to the speed of light in that medium in which it is traveling. Hence, the refractive index can be given as
\[n=\dfrac{c}{v}\]
Where c is the speed of light and v is the velocity of light in a particular medium.
Now velocity is given as a product of wavelength and the frequency of light in that particular medium. Hence we can write v as
\[v=\lambda f\]
Where $\lambda$ is the wavelength and f is the frequency.
By using the above formula of velocity, we can rewrite the formula of refractive index as
\[n=\dfrac{c}{\lambda f}\]
Now we have given the frequency of the light wave in a material as \[2\times {{10}^{14}}Hz\]and wavelength as \[5000\overset{o}{\mathop{\text{A}}}\,\]. And c is the speed of light in vacuum which is a universal constant and given as \[3\times {{10}^{8}}m/s\].
Substituting this value in the above equation we will get the refractive index of the material. So, we can write
\[\begin{align}
  & n=\dfrac{3\times {{10}^{8}}}{\left( 5000\times {{10}^{-10}} \right)\left( 2\times {{10}^{14}} \right)} \\
 & \Rightarrow n=\dfrac{3\times {{10}^{8}}}{\left( 5\times {{10}^{-7}} \right)\left( 2\times {{10}^{14}} \right)} \\
 & \Rightarrow n=\dfrac{3\times {{10}^{8}}}{10\times {{10}^{7}}} \\
 & \Rightarrow n=\dfrac{3\times {{10}^{8}}}{{{10}^{8}}} \\
 & \Rightarrow n=3 \\
\end{align}\]
Hence correct option is B.

Note: While substituting the values we multiplied the wavelength with \[{{10}^{-10}}\] as to convert it from angstrom into meters. As the SI unit of length is meter, in case we don’t change the unit we won’t get the right answer. Note that the speed of light is constant and it can’t exceed its value from \[3\times {{10}^{8}}m/s\] but the speed can slow down while traveling through material and so it can have less than \[3\times {{10}^{8}}m/s\]. Here the speed of light slows down by factor 3.