The human eye can readily detect wavelengths from about 400 nm to 700 nm. If white light illuminates a diffraction grating having 750 lines/mm, over what range of angles does the visible m = 1 spectrum extend?

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TettetoxDetnhte5 · Accepted Answer

Expression for the bright fringes in a diffraction grating is as follows:  d  sin  ⁡  θ  =  m  λRearrange the above equation for   θ                                  sin        ⁡        θ        =                              m            λ                    d                                                  θ        =                              sin                          −              1                                                (                                                    m                λ                            d                                )                    Substitute 1 for m, 400 nm for   λ and       (                  1                  750                      )              m      m       for d in the above equation.                                  θ        =                              sin                          −              1                                                (                                                    m                λ                            d                                )                                                  =                              sin                          −              1                                                (                                                                      (                  1                  )                                                  (                                      400                    ×                                                                                            10                                                                          −                          9                                                                                                                                                        m                                                                              )                                                                              (                                                            1                                              750                                                                              )                                                                      m                    m                                                                                )                                                  =                              sin                          −              1                                                (                                                                      (                  1                  )                                                  (                                      400                    ×                                                                                            10                                                                          −                          9                                                                                                                                                        m                                                                              )                                                                              (                                      1.33                    ×                                                                                            10                                                                          −                          6                                                                                                                                                        m                                                                              )                                                              )                                                  =                  17.5          ∘                    Step 2Expression for the bright fringes in a diffraction grating is as follows:  d  sin  ⁡  θ  =  m  λRearrange the above equation for   θ  .                                  sin        ⁡        θ        =                              m            λ                    d                                                  θ        =                              sin                          −              1                                                (                                                    m                λ                            d                                )                    Substitute 1 for m, 700 nm for   λ and       (                  1                  750                      )              m      m       for d in the above equation.                                  θ        =                              sin                          −              1                                                (                                                    m                λ                            d                                )                                                  =                              sin                          −              1                                                (                                                                      (                  1                  )                                                  (                                      700                    ×                                                                                            10                                                                          −                          9                                                                                                                                                        m                                                                              )                                                                              (                                                            1                                              750                                                                              )                                                                      m                    m                                                                                )                                                  =                              sin                          −              1                                                (                                                                      (                  1                  )                                                  (                                      700                    ×                                                                                            10                                                                          −                          9                                                                                                                                                        m                                                                              )                                                                              (                                      1.33                    ×                                                                                            10                                                                          −                          6                                                                                                                                                        m                                                                              )                                                              )                                                  =                  31.75          ∘                    The range of angles the visible m = 1 spectrum extend is from       17.5    ∘                t      o      31                          .75              ∘

The human eye can readily detect wavelengths from about 400

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