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 $\mathrm{n}_{1}$ to $\mathrm{n}_{2}$ over the height h . Which of the following statement(s) is (are) true about the light wave emerging out of the slab ?

(B) It will deflect up by an angle $\tan ^{-1}\left[\frac{\left(n_{2}-n_{1}\right) d}{h}\right]$; (D) The deflection angle depends only on $\left(n_{2}-n_{1}\right)$ and not on the individual values of $n_{1}$ and $n_{2}$.

(A) It will deflect up by an angle $\tan ^{-1}\left[\frac{\left(n_{2}^{2}-n_{1}^{2}\right) d}{2 h}\right]$

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Geometrical Optics

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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

n_{1}

n_{2}

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 $tan^{- 1} [\frac{( n _{2}^{2} - n _{1}^{2} ) d}{2 h}]$
(B)It will deflect up by an angle $tan^{- 1} [\frac{( n _{2} - n _{1} ) d}{h}]$ ✓ Correct
(C)It will not deflect.
(D)The deflection angle depends only on $(n_{2} - n_{1})$ and not on the individual values of $n_{1}$ and $n_{2}$ .✓ Correct

Answer: (B), (D)

Solution

As the light travels through the slab with varying refractive index, it bends toward regions of higher refractive index.

The optical path difference across the slab causes a lateral shift. Using Fermat's principle and the geometry:

n_{1} d + ℓ = n_{2} d

where

ℓ

is the lateral displacement. This gives

ℓ = (n_{2} - n_{1}) d

.

The deflection angle is:

tan θ = \frac{ℓ}{h} = \frac{( n _{2} - n _{1} ) d}{h}

Therefore

θ = tan^{- 1} [\frac{( n _{2} - n _{1} ) d}{h}]

.

The angle depends only on the difference

(n_{2} - n_{1})

, not on individual values.

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JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Geometrical Optics

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