Question

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 \([\frac{(n^2_2-n^2_1)d}{2h}]\).
• B. It will deflect up by an angle tan⁻¹ \([\frac{(n_2-n_1)d}{h}]\)
• C. It will not deflect
• D. The deflection angle depends only on (n₂-n₁) and not on the individual values of n₁ an_{d n}2.

Answer 1

The Correct Option is B, D

The incidence of a monochromatic light wave on a glass slab with a refractive index that increases linearly from \(n_1\) to \(n_2\) over a height \(h\) requires understanding how the light will behave as it exits the slab. When the refractive index varies, the exit angle of the light wave from the slab changes based on this variation. Let's analyze the situation: 

1. Overall Concept: When the refractive index of a medium changes, the speed of light within that medium varies, resulting in refraction. For a linearly varying refractive index, this can cause the light to bend gradually.

2. Deflection Calculation: The light wave deflection angle, due to the refractive index gradient across the height of the slab, can be calculated using the expression for a linear refractive index gradient: \[ \theta = \tan^{-1}\left(\frac{(n_2-n_1)d}{h}\right) \]

Here, \(\theta\) is the angle of deflection, \(d\) is the thickness of the slab, and \(h\) is the height over which the refractive index changes from \(n_1\) to \(n_2\).

3. Key Observations:

• The angle of deflection depends on the difference \( (n_2 - n_1) \), not the absolute values of \( n_1 \) and \( n_2 \). Hence, it's evident that the change in refractive index across the height, rather than specific values, dictates the deflection.
• The relationship \( \tan^{-1}\left(\frac{(n_2-n_1)d}{h}\right) \) combines known parameters to determine the deviation angle.

Based on these calculations and observations, the correct statements are:

It will deflect up by an angle \( \tan^{-1}\left(\frac{(n_2-n_1)d}{h}\right) \).

The deflection angle depends only on \( (n_2-n_1) \) and not on the individual values of \( n_1 \) and \( n_2 \).