Two coherent monochromatic point sources $S_{1}$ and $S_{2}$ of wavelength $λ = 600 n m$ are placed symmetrically on either side of the center of the circle as shown. The sources are separated by a distance d = 1.8 mm. This arrangement produces interference fringes visible as alternate bright and dark spots on the circumference of the circle. The angular separation between two consecutive bright spots is $Δ θ$ . Which of the following options is/are correct?

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A dark spot will be formed at the point $P_{2}$

At $P_{2}$ the order of the fringe will be maximum

The angular separation between two consecutive bright spots decreases as we move from $P_{1}$ and $P_{2}$ along the first quadrant

The total number of bright fringes produced between $P_{1}$ and $P_{2}$ in the first quadrant is close to 3000

answer is A, C.

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Detailed Solution

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A) at point $P_{1}$ path difference $Δ x = 0$ (central maxima) at point $P_{2}$ path difference $Δ x = d$
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$n = \frac{Δ x}{λ} = \frac{1.8 \times 10^{- 3}}{600 \times 10^{- 9}}$
$n = 3000$ (maxima)
C) As the path difference is maximum at $P_{2}$ the order of the fringe will be maximum.
D) Angular fringe width increases as we more from $P_{1}$ and $P_{2}$ as shown below
path difference = $x = d \sin θ$
$∴ Δ x = d \cos θ . Δ θ ∴ λ = d \cos θ . Δ θ$
$Δ θ = \frac{λ}{d \cos θ}$ as $θ$ increases $\cos θ$ decreases and $Δ θ$ increases.