Two coherent monochromatic point sources and of wavelength are placed symmetrically on either side of the centre of the circle as shown. The sources are separated by a distance . This arrangement produces interference fringes visible as alternate brigh

Question

Two coherent monochromatic point sources $S_{1}$ and $S_{2}$ of wavelength $λ = 600 nm$ are placed symmetrically on either side of the centre 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 [2017]

Smart Achievers · Accepted Answer

(2, 3)

At $P_{2}$ ,

$Δ x = 0 .$ So we will have maxima there. It will be very much like the central maxima in YDSE with $n = 0$ . So (1) is incorrect.

At $P_{1}$ ,

$Δ x = S_{1} P - S_{2} P = d = 1.8 mm$

For maxima, $Δ x = n λ$

$n = \frac{Δ x}{λ} = \frac{1.8 \times 10^{- 3}}{600 \times 10^{- 9}} = \frac{1.8}{600} \times 10^{6} = 3000$

So, the number of fringes between $P_{1}$ and $P_{2}$ will be 3000.

So, (3) is correct. And it will also be the highest order fringe.

So, (2) is correct.

As, for bright fringe, $d \cos θ = n λ$

$\Rightarrow - d \sin θ Δ θ = Δ n λ$

$\Rightarrow Δ θ = - \frac{(Δ n) λ}{d \sin θ}$

As we move from $P_{1}$ to $P_{2}$ , $θ ↓ ↓$ . So $\sin θ$ $↓ ↓$ . Therefore, $Δ θ$ $↑ ↑$ .

So, (4) is incorrect.

Solution

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

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

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

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

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Solution

1 A dark spot will be formed at the point P2

2 At P2 the order of the fringe will be maximum

3 The total number of fringes produced between P1 and P2 in the first quadrant is close to 3000

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

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

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

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

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