A double slit setup is shown in the figure. One of the slits is in medium 2 of refractive index . The other slit is at the interface of this medium with another medium 1 of refractive index . The line joining the slits is perpendicular to the interface an

Question

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 $n_{1} (n_{1} \neq n_{2})$ . 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 [2022]

Which of the following statement(s) is(are) correct? [2022]

Smart Achievers · Accepted Answer

(1, 2)

We have, $n_{1} (n_{1} \neq n_{2})$

and $B C = A B \sin α = d \tan θ \sin α$

Also, $A D = A B \sin θ = d \tan θ \sin θ$

So, optical path difference

$= n_{1} B C - n_{2} A D$

$= n_{1} (d \tan θ \sin α) - n_{2} (d \tan θ \sin θ)$

$= d \tan θ (n_{1} \sin α - n_{2} \sin θ)$ $= d \tan θ \times 0 = 0$

So, (1) and (2) are correct, and (3) and (4) are incorrect.

Solution

1 The phase difference between the two rays is independent of $d .$

2 The two rays interfere constructively at the detector.

3 The phase difference between the two rays depends on $n_{1}$ but is independent of $n_{2}$ .

4 The phase difference between the two rays vanishes only for certain values of $d$ and the angle of incidence of the beam, with $θ$ being the corresponding angle of refraction.

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Solution

1 The phase difference between the two rays is independent of d.

2 The two rays interfere constructively at the detector.

3 The phase difference between the two rays depends on n1 but is independent of n2.

4 The phase difference between the two rays vanishes only for certain values of d and the angle of incidence of the beam, with θ being the corresponding angle of refraction.

Ans. (1, 2) We have, AB=d tan θ and BC=ABsinα=dtanθsinα Also, AD=ABsinθ=dtanθsinθ So, optical path difference =n1BC-n2AD =n1(dtanθsinα)-n2(dtanθsinθ) =dtanθ(n1sinα-n2sinθ)=dtanθ×0=0 So, (1) and (2) are correct, and (3) and (4) are incorrect.

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1 The phase difference between the two rays is independent of $d .$

3 The phase difference between the two rays depends on $n_{1}$ but is independent of $n_{2}$ .

4 The phase difference between the two rays vanishes only for certain values of $d$ and the angle of incidence of the beam, with $θ$ being the corresponding angle of refraction.