Two parallel slits are illuminated by light composed of two wavelengths, one of which is 657 nm. On a viewing screen, the light whose wavelength is known produces its third dark fringe at the same place where the light whose wavelength is unknown produces its fourth-order bright fringe. The fringes are counted relative to the central or zeroth-order bright fringe. What is the unknown wavelength

     λ = 5.75 10⁻⁷ mm

Explanation:

This is a slit interference exercise, we analyze each wavelength separately                    

            λ = 657 nm                                     indicate that the third dark pattern

          a sin θ = (m + ½) lam

          a sin θ = (3 + ½) 657 10⁻⁹

          a sin θ = 2299.5 10⁻⁹ nm

for the other wavelength in the same place we have m = 4 bright

          a sin θ = m lam

we substitute

           2299.5 10⁻⁹ = 4 λ

           λ =

           λ = 5.75 10⁻⁷ mm