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:
\[\lambda =700nm\] (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 (i) \[1.336\times {{10}^{-\,6}}m\] (ii) \[2.005\times {{10}^{-\,6}}m\] (iii) \[3.5\times {{10}^{-5}}rad\]
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