Two gratings A and B have slit separations dA and dB, respectively. They are used with the same light and the same observation screen. When grating A is replaced with grating B, it is observed that the first-order maximum of A is exactly replaced by the second-order maximum of B. (a) Determine the ratio dB/dA (No units) of the spacings between the slits of the gratings. Then find the next two principla maxima of grating A and the principal maxima of B that exactly replace them when the gratings are switched [(b) and (c) for the first pair, (d) and (e) for the second pair]. Identify these maxima by their order numbers (No units), listing the maxima for grating B first.

a) dB / dA = 2 ,

b) Network B     Network A

        2                         1

        4                         2

        6                         3

Explanation:

a) The expression for grating diffraction is

         d sin θ = m λ

where d the distance between two slits, λ the wavelength and m an integer that represents the diffraction range

In this exercise we are told that the two spectra are in the same position, let's write the expression for each network

Network A

m = 1

         sin θ = 1 λ / dA

Network B

m = 2

        sin θ = 2 λ / dB

they ask us for the relationship between the distances, we match the equations

            λ / dA) = 2 λ / dB

            dB / dA = 2

b) let's write the equation of the networks

         sin θ = m_A  λ / dA

         sin θ = m_B  λ / dB

we equalize

           m_A  λ/ dA = m_B  λ / dB

we use that

          dB / dA = 2

           m_A 2 = m_B

therefore the overlapping orders are

Network B     Network A

   2                         1

   4                         2

    6                       3

b.

hope this : )