First slide

Coherent sources and their importance in interference

Question

The ratio of the intensity at the center of a bright fringe to the intensity at a point one-quarter of the distance between two fringe from the center is

Moderate

A

2
B

½
C

4
D

16

Solution

$R e s u l \tan t i n t e n s i t y a t a n y p o i n t o n s c r e e n i s g i v e n b y I_{r} = I_{1} + I_{2} + 2 \sqrt{I_{1} I_{2}} \cos ϕ ......................................... (i) w h e r e ϕ = p h a s e d i f f e r e n c e b e t w e e n t h e w a v e s A t t h e c e n t e r b r i g h t f r i n g e, P u t I_{1} = I_{2} = I a n d ϕ = 0^{o} \Rightarrow I_{r} = 4 I ......................................... (i i) D i s \tan c e o f s e c o n d p o i n t f r o m C B = (\frac{1}{4}) \frac{λ D}{d} ∴ ∆ x = (\frac{1}{4}) λ \Rightarrow \frac{λ}{2 π} ∆ ϕ = (\frac{1}{4}) λ \Rightarrow ∆ ϕ = \frac{π}{2} T o f i n d r e s u l \tan t i n t e n s i t y a t s e c o n d p o i n t P u t I_{1} = I_{2} = I a n d ϕ = \frac{π}{2} i n e q u a t i o n (1) I_{r}' = 2 I ......................................... (i i i) N o w t a k e r a t i o o f \frac{I_{r}}{I_{r}'} = \frac{4 I}{2 I} = 2$

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