Question

A glass slab of thickness \[4\,{\text{cm}}\] contains the same numbers of waves as \[5\,{\text{cm}}\] of water when both are traversed by the same monochromatic ray of light. The refractive index of water is \[\dfrac{4}{3}\], what is that for glass?
A. \[\dfrac{5}{3}\]
B. \[\dfrac{5}{4}\]
C. \[\dfrac{{16}}{{15}}\]
D. \[\dfrac{3}{2}\]

Answer 1

Hint: For this solution, look for optical path which is travelled by both medium through wave, if the number of waves is same then the expression will be \[{n_{\text{w}}}{t_{\text{w}}} = {n_{\text{g}}}{t_{\text{g}}}\] .

Complete step by step answer:
Given,
Glass slab thickness is \[{t_{\text{g}}} = 4\,{\text{cm}}\]
Water medium thickness is \[{t_{\text{w}}} = 5\,{\text{cm}}\]
Refractive index of water \[{n_{\text{w}}} = \dfrac{4}{3}\,{\text{cm}}\]
Optical path travelled by the wave must be the same in both medium as it is provided that the number of waves is the same in both the medium.
\[{n_{\text{w}}}{t_{\text{w}}} = {n_{\text{g}}}{t_{\text{g}}}\]
So,
$\dfrac{4}{3} \times 5 = {n_{\text{g}}} \times 4 \\
{n_{\text{g}}} = \dfrac{5}{3} \\$
Hence, the required answer is \[{n_{\text{g}}} = \dfrac{5}{3}\] .

So, the correct answer is “Option A”.

Additional Information:
Monochromatic light: A monochromatic light is light, where only one optical frequency is in the optical spectrum. In certain spaces, for example, the related electric field intensity shows a strictly sinusoidal oscillation, with an instantaneously continuous frequency and a bandwidth of zero. If they emit monochromatic light, the light sources may also be called monochromatic. The monochromatic antonym is multifarious. Life produced as a thermal emission, for example, is a typical example of polychromatic light in an incandescent lamp.
Refractive index: The refractive index, also called a refraction index, calculates the bending of the ray of light between mediums. If the angle\[i\] is the angle of incidence of the ray (angle of the sun's ray in the vacuum) and the angle of the refractive (angle of the ray in the medium and normal) is \[r\], the index \[n\] is defined as the ratio of the sine of the incidence angle to the sine of the angle of the refraction. That is to say, \[n = \sin i/\sin r\]. The refractive index is equal to even the speed of light \[c\] in the empty space of the given wavelength divided into a material by its speed \[v\] or \[n = c/v\] .

Note:
In this solution, the required expression for the finding the answer is \[{n_w}{t_{\text{w}}} = {n_{\text{g}}}{t_{\text{g}}}\], here glass slab thickness is \[{t_{\text{g}}}\], water medium thickness is \[{t_{\text{w}}}\], refractive index of water \[{{\text{n}}_{\text{w}}}\]. This expression is applicable if the number of waves is the same for both the mediums.