JEE Advanced 2022 — Paper 2 — PHY-II solution

JEE Advanced2022Paper 2PHY-II

A double slit setup is shown in the figure. One of the slits is in medium 2 of refractive index $n_{2}$ . The other slit is at the interface of this medium with another medium 1 of refractive index $n_{1} (\neq = n_{2})$ . The line joining the slits is perpendicular to the interface and the distance between the slits is $d$ . The slit widths are much smaller than $d$ . A monochromatic parallel beam of light is incident on the slits from medium 1. A detector is placed in medium 2 at a large distance from the slits, and at an angle $θ$ from the line joining them, so that $θ$ equals the angle of refraction of the beam. Consider two approximately parallel rays from the slits received by the detector.

Which of the following statement(s) is(are) correct?

A
The phase difference between the two rays is independent of d.
B
The two rays interfere constructively at the detector.
C
The phase difference between the two rays depends on n $_{1}$ but is independent of n $_{2}$ .
D
The phase difference between the two rays vanishes only for certain values of d and the angle of incidence of the beam, with q being the corresponding angle of refraction.

Solution

Let $i$ be the angle of incidence in medium 1 and $θ$ the angle of refraction in medium 2, with Snell's law $n_{1} sin i = n_{2} sin θ$ .

The extra path in medium 1 before reaching the lower slit (which sits in medium 2 a distance $d$ below the interface slit) is $d sin i$ (geometric path along the incident direction projected onto the slit line). After both slits radiate, the extra geometric path in medium 2 to the detector along the refracted direction is $d sin θ$ .

The optical path difference is therefore $Δ = n_{1} (d sin i) - n_{2} (d sin θ) = d (n_{1} sin i - n_{2} sin θ) = 0$ by Snell's law.

Hence the phase difference is identically zero, independent of $d$ , of $n_{1}$ , and of $n_{2}$ . The waves interfere constructively at the detector.

(A) Correct: phase difference does not depend on $d$ (it is always 0).
(B) Correct: constructive interference.
(C) Incorrect: the result is independent of both $n_{1}$ and $n_{2}$ .
(D) Incorrect: the phase difference vanishes for every $d$ and every incidence angle (as long as $θ$ is the corresponding refraction angle).

$∴$ Correct options: (A), (B).

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