Two coherent narrow slits emitting light of wavelength $λ$ in the same phase are placed parallel to each other at a small separation of 2 $λ$ . The light is collected on a screen S which is placed at a distance $(D >> λ)$ from the slit S₁ as shown in figure. The distance $x = \sqrt{n} D$ then the value of n is _____(such that the intensity at P equal to intensity O)

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answer is 3.

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Detailed Solution

The path difference between the waves arriving at P
$Δx = 2 λcos θ$
The minimum path difference $Δx = 0, when θ = 90^{\circ}$ , and maximum path difference, $Δx = 2 λ, when θ = 0$ . Thus there must be one more maximum between these two.

For this, $Δx = λ \Rightarrow 2 λcos θ = λ$

$∴ \cos θ = \frac{1}{2}$
From the geometry,
$\begin{array}{ll} \cos θ = \frac{D}{\sqrt{D^{2} + x^{2}}} \\ ∴ \frac{D}{\sqrt{D^{2} + x^{2}}} = \frac{1}{2} \\ \Rightarrow x = \sqrt{3} D \end{array}$