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