First slide

Coherent sources and their importance in interference

Question

In Young’s double slit experiment intensity at a point is (1/4) of the maximum intensity. Angular position of this point is

Moderate

A

$\sin^{- 1} (λ / d)$
B

$\sin^{- 1} (λ / 2 d)$
C

$\sin^{- 1} (λ / 3 d)$
D

$\sin^{- 1} (λ / 4 d)$

Solution

$L e t P b e t h e p o i n t o n t h e c e n t r a l m a x i m a w h o s e i n t e n s i t y i s o n e f o u r t h o f t h e m a x i m u m i n t e n s i t y . F o r i n t e r f e r e n c e w e k n o w t h a t I = I_{1} + I_{2} + 2 \sqrt{I_{1} I_{2}} \cos ϕ . W h e r e I i s t h e i n t e n s i t y a t P a n d I_{1}, I_{2} a r e t h e i n t e n s i t y i s o n e f o u r t h o f l i g h t o r i g i n a t i n g f r o m A a n d B r e s p e c t i v e l y a n d ϕ i s t h e p h a s e d i f f e r e n c e a t P . I n Y D S E, I_{1} = I_{2} a n d I_{\max} = 4 I W e a r e c o n c e n t r a t i n g a t a p o i n t w h e r e t h e i n t e n s i t y i s o n e f o u r t h o f t h e m a x i n t e n s i t y . ∴ I = I + I + 2 I \cos ϕ \Rightarrow \cos ϕ = - \frac{1}{2} \Rightarrow ϕ = \frac{2 π}{3} \Rightarrow \frac{2 π}{λ} ∆ x = \frac{2 π}{3} \Rightarrow ∆ x = \frac{λ}{3} \Rightarrow d \sin θ = \frac{λ}{3} \Rightarrow \sin θ = \frac{λ}{3 d} \Rightarrow θ = \sin^{- 1} (\frac{λ}{3 d})$

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