Question

In the adjacent diagram, CP represents a wave front and AO and BP, the corresponding two rays. Find the INCORRECT 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

Answer: (A), (C), (D)

The correct options are
A $\cos \theta =3\lambda /2d$
C $\sec \theta -\cos \theta =\lambda /d$
D $\sec \theta -\cos \theta =4\lambda /d$


$$\therefore PR=d\Rightarrow PO=d\sec \theta$$and $$CO=PO\cos 2\theta =d\sec \theta \cos 2\theta$$ is Path difference between the two rays $$\Delta =CO+PO$$ $$=(d\sec \theta +d\sec \theta \cos 2\theta )$$ Phase difference between the two rays is $$\varphi =\pi$$(One is reflected, while another is direct) Therefore condition for constructive interference should be $$\Delta =\frac{\lambda }{2},\frac{3\lambda }{2},...$$ Or $$d\sec \theta (1+cos2\theta )=\frac{\lambda }{2}$$ Or $$\frac{d}{\cos \theta }\left(2co{s}^2\theta \right)=\frac{\lambda }{2}\Rightarrow \cos \theta =\frac{\lambda }{4d}$$