First slide

Coherent sources and their importance in interference

Question

Two coherent point sources $S_{1} {and S}_{2}$ vibrating in phase emit light of wavelength $λ$ . The separation between the sources is $2 λ$ . The smallest distance from $S_{2}$ on a line passing through $S_{2}$ and perpendicular to $S_{1} {,S}_{2}$ , where a minimum of intensity occurs is:

Moderate

A

$\frac{7 λ}{12}$
B

$\frac{15 λ}{4}$
C

$\frac{λ}{2}$
D

$\frac{3 λ}{4}$

Solution

[Concept]: Path difference at S₂ is $2 λ$ . Therefore for minimum intensity(dark band) at P.
[Step 1]:
$\begin{array}{l} F o r D a r k b a n d : p a t h d i f f e r e n c e Δ x = (\frac{2 n - 1}{2}) λ \\ S_{1} P - S_{2} P = \frac{3 λ}{2} .......... (i) \end{array}$
[Step 2]: $\sqrt{4 λ^{2} + x^{2}} - x = \frac{3 λ}{2}$
[Step 3]:
$\begin{array}{l} \Rightarrow \sqrt{4 λ^{2} + x^{2}} = x + \frac{3 λ}{2} \\ s q u a r i n g o n b o t h s i d e s \\ \Rightarrow {(\sqrt{4 λ^{2} + x^{2}})}^{2} = {(x + \frac{3 λ}{2})}^{2} \\ \Rightarrow 4 λ^{2} + x^{2} = x^{2} + {(\frac{3 λ}{2})}^{2} + 2 x \frac{3 λ}{2} \\ \Rightarrow 4 λ^{2} - \frac{9 λ^{2}}{4} = 3 x λ \\ \Rightarrow (16 - 9) \frac{λ^{2}}{4} = 3 x λ \end{array}$
Solving equation, $x = \frac{7 λ}{12}$

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