Two coherent monochromatic point sources S₁ and S₂ of wavelength 𝜆 = 800 𝑛𝑚 are placed symmetrically on either side of centre of the circle as shown. The sources are separated by a distance d = 2 mm. The 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 P₂

The angular separation between two consecutive bright spots decreases as we move from P₁ to P₂ along the first quadrant

At P₂ the order of the fringe will be maximum

The total number of fringes produced between P₁ and P₂ in the first quadrant is close to 2500

answer is C, D.

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Detailed Solution

Path difference at point P on the circle when angle is $θ$ (see the picture) is given by
Question Image
$\begin{array}{l} Δx = 2 X \frac{d}{2} \sin θ = dsin θ \\ At P_{2}, θ = 90^{\circ} => Δx = dsin 90^{\circ} = d \\ d = nλ \Rightarrow n = \frac{d}{λ} = \frac{2 X 10^{- 3}}{800 \times 10^{- 9}} = 2500 \end{array}$
=>At P₂ order of the maximum is 2500 and this is maximum order of the fringe.
=>Total number of fringes in 1st quadrant is 2500
$dsin θ = nλ \Rightarrow dcos θdθ = dnλ$
For successive fringes dn =1
$\Rightarrow dθ = \frac{λ}{dcos θ}$
=> d $θ$ (angular width of successive maximum) increases as we move from P1 to P₂