Question

A glass slab of thickness 4cm contains the same number of waves as X cm of water column when both are traversed by the same monochromatic light. If the refractive indices of glass and water (for that light) are 5/3 and 4/3 respectively, the value of x will be:
\[
A. \dfrac{9}{{20}}cm\\
B. \dfrac{{20}}{9}cm \\
C. \dfrac{5}{4}cm \\
D. 5cm \\
 \]

Answer 1

Hint:Since the same number of waves are passing through both the water column and the glass slab, we can say that time taken by light to travel the thicknesses will be the same. Using the formula of refractive index in terms of velocities, use time as a constant and derive relationship between refractive index and thickness of the material.

Complete step by step answer:
Here, the thickness of the glass slab is given as 4cm, let it be shown as \[{t_g}\]. The thickness of the water column is X cm. The refractive index of glass slab \[{n_g}\]is 5/3 and the refractive index of water is given as
\[{n_w}\] is 4/3 for that particular monochromatic light.
Now, the speed of light in vacuum is $c$. Thus the Refractive index n is the ratio of speed of light in vacuum to the speed of light in a given medium. Thus,
\[n = \dfrac{c}{v}\]
Here \[v\] is the velocity of light in the particular medium.
Thus, \[v = \dfrac{c}{n}\].
Time taken by the light to travel a particular thickness would be,
\[T = \dfrac{t}{v} \\
\Rightarrow T= \dfrac{t}{{\dfrac{c}{n}}} \\
\therefore T= \dfrac{{nt}}{c}\].
Now, the same numbers of waves are travelling in the water column as well as the glass slab. Thus $T$ for both the refractive medium is the same.

Hence, option D is the correct option.

Note: Here, actually the same number of waves means the frequency of light is the same, but as frequency and time are related to each other, we can use the above formula. Thus, here we find out that for a constant time or frequency, the refractive index is inversely proportional to the thickness of the material.