Light incident normally on a grating of total ruled width of 5.1 × 10 − 3 m with 2520 lines in all. Find out if two sodium lines with wavelengths 5890 A ∘ and 5896 A ∘ can be seen distinctly by the grating.

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Light incident normally on a grating of total ruled width of   5.1  ×      10          −      3         m with 2520 lines in all. Find out if two sodium lines with wavelengths   5890       A    ∘   and   5896       A    ∘   can be seen distinctly by the grating.

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Given:The rule width of diffraction grating is   L  =  5.1  ×      10          −      3         m.The number of lines in grating is   N  =  2520.The wavelength of first line of sodium is       λ    1    =  5890               A      ˙      .The wavelength of second line of sodium is       λ    2    =  5896               A      ˙      .The formula to calculate the distance between two lines of grating is,   d  =      L    N  Here, d is the distance between two lines of grating, L is the rule width and N is the number of lines.Substitute the known values in the formula to calculate the distance between two lines of grating.  d  =            5.1      ×              10                  −          3                     m        2520      =  2.02  ×      10          −      6           mThe formula to calculate the angle of diffraction for first line is,  sin  ⁡      θ    1    =            n              λ        1              d  Here,       θ    1   is the angle of diffraction for first line, n is the order of diffraction and       λ    1   is the wavelength of first line.Substitute the known values in the formula to calculate the angle.  sin  ⁡      θ    1    =            (      1      )      (      5890                        A          ˙                    )      (                                    10                          −              10                                 m                          1                                    A              ˙                                          )              2.02      ×              10                  −          6                     m              θ    1    =      sin          −      1        ⁡  (  0.29158  )    =      16.953    ∘  The formula to calculate the angle of diffraction for second line is,  sin  ⁡      θ    2    =            n              λ        2              d  Here,       θ    2   is the angle of diffraction for second line and       λ    2   is the wavelength of second line.Substitute the known values in the formula to calculate the angle.  sin  ⁡      θ    1    =            (      1      )      (      5896                        A          ˙                    )      (                                    10                          −              10                                 m                          1                                    A              ˙                                          )              2.02      ×              10                  −          6                     m              θ    1    =      sin          −      1        ⁡  (  0.29188  )    =      16.971    ∘  The formula to calculate the angular width between two lines is,   △  θ  =      θ    2    −      θ    1  Here,   △  θ is the angular width between two lines.Substitute the known values in the formula to calculate the angular between two lines.  △  θ  =  16.971  −  16.953    =      0.018    ∘  Thus, the angular separation between two lines is       0.018    ∘   which is very small so the two sodium lines cannot be seen distinctively.

Light incident normally on a grating of total ruled width of 5.1*10^-3m with 2520 lines in all

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