question_answer

A slit of width 'a' is illuminated by light of wavelength 700 nm, what will be the value of slit width 'a' when

(i) First minimum falls at an angle of diffraction \[30{}^\circ ?\]

(ii) First maximum falls at an angle of diffraction \[30{}^\circ ?\]

(iii) Find the angular spread of central maximum if the slit width a = 4.0 cm. (Assume incidence normal to the plane of the slip)

Answer:

(i) \[1.336\times {{10}^{-\,6}}m\]

(ii) \[2.005\times {{10}^{-\,6}}m\]

(iii) \[3.5\times {{10}^{-5}}rad\]

\[\lambda =700nm\] (i) \[\theta =30{}^\circ =\frac{\pi }{6}\,\,\text{rad}\] For first minimum, \[\theta =\frac{n\lambda }{a}\] \[\frac{\pi }{6}=\frac{1\times 700\times {{10}^{-9}}}{a}\]\[\Rightarrow \] \[a=1.336\times {{10}^{-\,6}}\,m\]             (ii) For first maximum, \[\theta =\left( n+\frac{1}{2} \right)\,\,\frac{\lambda }{a}\]                         \[\frac{\pi }{6}=\left( 1+\frac{1}{2} \right)\frac{700\times {{10}^{-\,9}}}{a}\]             \[\Rightarrow \]   \[a=2.005\times {{10}^{-\,6}}m\] (iii) Angular spread of central maximum is \[2\theta =\frac{2\lambda }{a}=\frac{2\times 700\times {{10}^{-\,9}}}{4\times {{10}^{-2}}}=3.5\times {{10}^{-\,5}}\] rad