Coherent sources and their importance in interference

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Question

Interference pattern is formed on a screen by Young’s double slits $S_{1}$ and $S_{2}$ which are identical and illuminated by monochromatic light of wavelength $λ$ . Points P and P´ are on the screen closest to and on either side of central maxima, where intensity is one fourth of that at the central maxima. If $P P^{'} = \frac{2 D λ}{d x} .$ Find x.(assume $d > > λ & D > > d$ )

Moderate

Solution

$I_{P} = \frac{4 I_{0}}{4} = I_{0}$
$\Rightarrow I_{0} = I_{0} + I_{0} + 2 I_{0} \cdot \cos ϕ$
$\Rightarrow \cos ϕ = - \frac{1}{2}$
$\Rightarrow ϕ = \frac{2 π}{3} = \frac{2 π}{λ} . \frac{y d}{D}$
$\Rightarrow 2 y = \frac{2 λ D}{3 d}$
$∴ x = 3$

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In an interference arrangement, similar to Young’s double slit experiment, the slits S₁ and S₂ are illuminated with coherent microwave sources, each of frequency 106 Hz. The sources are synchronized to have zero phase difference. The slits are separated by distance d = 150.0 m. The intensity $I_{(θ)}$ is measured as a function of $θ$ , where $θ$ is defined as shown in the figure. If I₀ is maximum intensity, then $I_{(θ)}$ for
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