question_answer

Let a beam of wavelength \[\lambda \] falls on parallel reflecting planes with separation \[d,\]then the angle \[\theta \] that the beam should make with the planes so that reflected beams from successive planes may interfere constructively should be (where,\[n=1,2,...\])

A) \[{{\cos }^{-1}}\left( \frac{n\lambda }{2d} \right)\]

B)  \[{{\sin }^{-1}}\left( \frac{n\lambda }{2d} \right)\]

C)  \[{{\sin }^{-1}}\left( \frac{n\lambda }{d} \right)\]

D) \[{{\tan }^{-1}}\left( \frac{n\lambda }{d} \right)\]

Correct Answer: B

Solution :

 From the adjoining figure path difference between the beams- reflected from successive planes \[=AM+AN=2d\,\sin \theta \] For constructive interference \[2d\,\sin \theta =n\lambda \] \[\sin \theta =\frac{n\lambda }{2d}\] \[\theta ={{\sin }^{-1}}\left( \frac{n\lambda }{2d} \right)\]