First slide

Coherent sources and their importance in interference

Question

Monochromatic light waves of wavelength $λ$ from two coherent sources fall on a large plane screen. One of the waves is emanated from a point source S located at distance L from the screen and other one is a broad plane wave as shown in the figure.
Mark the correct options

Moderate

A

Central point of screen is dark spot
B

The shape of fringe on screen is elliptical
C

The distance between nth and (n – 1)th bright fringe is $\sqrt{2 λ L} (\sqrt{n} - \sqrt{n - 1})$
D

None of the above

Solution

$L e t s a y n^{t h} b r i g h t f r i n g e i s f o r m i n g a t p o i n t P P a t h d i f f e r e n c e, Δ x = S P - R P = n λ \Rightarrow Δ x = \sqrt{L^{2} + r^{2}} - L = n λ \Rightarrow \sqrt{L^{2} + r^{2}} = n λ + L \Rightarrow L^{2} + r^{2} = n^{2} λ^{2} + L^{2} + 2 n λ L \Rightarrow r = \sqrt{2 n λ L} [∵ n^{2} λ^{2} m a y b e n e g l e c t e d] ∴ r_{n} = \sqrt{2 n λ L} a n d r_{n - 1} = \sqrt{2 (n - 1) λ L} S p a c i n g b e t w e e n t h e m = r_{n} - r_{n - 1} = \sqrt{2 n λ L} - \sqrt{2 (n - 1) λ L} = \sqrt{2 λ L} (\sqrt{n} - \sqrt{n - 1})$

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