First slide

Coherent sources and their importance in interference

Question

In the adjacent diagram, CP represents a wavefront and AO & BP, the corresponding two rays. Find the condition on q for constructive interference at P between the ray BP and reflected ray OP

Moderate

A

cosq = 3l/2d
B

cosq = l/4d
C

secq – cosq = l/d
D

secq – cosq = 4l/d

Solution

PR = d
Þ PO = d secq and CO = PO cos 2q $= d \sec θ \cos 2 θ$
Path difference between the two rays
D = CO + PO = (d secq + d secq cos 2q)
Phase difference between the two rays is
f = p (One is reflected, while another is direct)
Therefore condition for constructive interference should be
$Δ = \frac{λ}{2}, \frac{3 λ}{2} ...... \Rightarrow d \sec θ (1 + \cos 2 θ) = \frac{λ}{2} \Rightarrow \frac{d}{\cos θ} (2 \cos^{2} θ) = \frac{λ}{2} \Rightarrow \cos θ = \frac{λ}{4 d}$

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