A monochromatic light wave is incident normally on a glass slab of thickness 𝑑, as shown in the figure. The refractive index of the slab increases linearly from 𝑛1 to 𝑛2 over the height ℎ. Which of the following statement(s) is(are) true about the light wave emerging out of the slab? (A) It will deflect up by an angle tan−1 [ (𝑛2 2−𝑛1 2 )𝑑 2ℎ ]. (B) It will deflect up by an angle tan−1 [ (𝑛2−𝑛1 )𝑑 ℎ ]. (C) It will not deflect. (D) The deflection angle depends only on (𝑛2 − 𝑛1 ) and not on the individual values of 𝑛1 and 𝑛2.

To determine the correct statement(s) about the light wave emerging out of the glass slab, let's analyze the situation step by step.

Given:

• A monochromatic light wave is incident normally on a glass slab.

• The slab has a thickness $d$.

• The refractive index of the slab increases linearly from $n_1$ at the bottom to $n_2$ at the top over a height $h$.

Key Concepts:

1. Refraction and Snell's Law: When light passes through a medium with a varying refractive index, it bends towards the region of higher refractive index.

2. Linear Variation of Refractive Index: Since the refractive index increases linearly from $n_1$ to $n_2$ over height $h$, the light will experience a continuous change in direction as it propagates through the slab.

Analysis:

• The light enters the slab normally (perpendicular to the surface), so initially, there is no deflection.

• As the light travels through the slab, it encounters regions with increasing refractive index, causing it to bend gradually.

• The deflection angle depends on the rate of change of the refractive index and the thickness of the slab.

Calculation of Deflection Angle:

The deflection angle $\theta$ can be approximated using the small-angle approximation and the relationship between the refractive index gradient and the slab's geometry. The deflection angle is given by:

$$\theta \approx {\tan }^{-1}\left(\frac{(n_2-n_1)d}{h}\right)$$

This formula arises because the deflection is proportional to the difference in refractive indices ($n_2-n_1$) and the thickness of the slab ($d$), and inversely proportional to the height over which the refractive index changes ($h$).

Evaluating the Statements:

• (A): This statement suggests a deflection angle of ${\tan }^{-1}\left(\frac{(n_2^2-n_1^2)d}{2h}\right)$. This is incorrect because the deflection angle depends on $n_2-n_1$, not $n_2^2-n_1^2$.

• (B): This statement suggests a deflection angle of ${\tan }^{-1}\left(\frac{(n_2-n_1)d}{h}\right)$. This is correct as it matches our derived formula.

• (C): This statement claims there will be no deflection. This is incorrect because the varying refractive index will cause the light to bend.

• (D): This statement claims the deflection angle depends only on $n_2-n_1$ and not on the individual values of $n_1$ and $n_2$. This is correct because the deflection angle is determined by the difference in refractive indices, not their individual values.

Conclusion:

The correct statements are (B) and (D). The light wave will deflect up by an angle ${\tan }^{-1}\left(\frac{(n_2-n_1)d}{h}\right)$, and this deflection angle depends only on the difference $n_2-n_1$, not on the individual values of $n_1$ and $n_2$.