Question

$\ln$ the adjacent diagram, CP represents a wavefront and AO a BP, the corresponding two rays. Find the condition on $\theta$ for constructive interference at $P$ between the ray BP and reflected ray OP.

• A. $\cos \theta =3\lambda /2d$
• B. $\cos \theta =\lambda /4d$
• C. $\sec \theta -\cos \theta =\lambda /d$
• D. $\sec \theta -\cos \theta =4\lambda /d$

Answer 1

The correct option is B $\cos \theta =\lambda /4d$

From geometry,

$OP=\frac{d}{\cos \theta }$

Path difference between the rays=$CO+OP=\frac{d}{\cos \theta }\cos 2\theta +\frac{d}{\cos \theta }=2d\cos \theta$

Since there is a phase change by $\pi$ due to reflection from optically denser medium, condition for constructive interference is

Path difference=$\frac{\lambda }{2}$

$⟹\cos \theta =\frac{\lambda }{4d}$