First slide

Coherent sources and their importance in interference

Question

In a Young's double slit experiment, the slits are 2 mm apart and are illuminated with a mixture of two wavelength $λ_{0} = 750 mm$ and $λ = 900 mm$ . The minimum distance from the common central bright fringe on a screen 2m from the slits where a bright fringe from one interference pattern coincides with a bright fringe from the other is

Moderate

A

1.5 mm
B

3 mm
C

4.5 mm
D

6 mm

Solution

From the given data, note that the fringe width ( $β_{1}$ ) for $λ_{1} = 900 mm$ is greater than fringe width ( $(β_{2})$ ) for $λ_{2} = 750 mm$ . This means that at though the central maxima of the two coincide, but first maximum for $λ_{1} = 900 mm$ will be further away from the first maxima for $λ_{2} = 750 mm$ and so on. A stage may come when this mismatch equals $β_{2}$ , then again maxima of $λ_{1} = 900 mm$ will coincide with a maxima of $λ_{2} = 750 mm$ , let this correspond to n^thorder fringe for $λ_{1}$ . Then it will correspond to $(n + 1)^{th}$ order fringe for $λ_{2}$ .
$Therefore \frac{{nλ}_{1} D}{d} = \frac{(n + 1) λ_{2} D}{d} \Rightarrow n \times 900 \times 10^{- 9} = (n + 1) 750 \times 10^{- 9} \Rightarrow n = 5$
Minimum distance from
$Central maxima = \frac{{nλ}_{1} D}{d} = \frac{5 \times 900 \times 10^{- 9} \times 2}{2 \times 10^{- 3}} = 45 \times 10^{- 4} m = 4 .5 mm$

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