a. Find an expression for the positions y₁ of the first-order fringes of a diffraction grating if the line spacing is large enough for the small-angle approximation tan θ ­≈ sin θ ≈­ θ to be valid. Your expression should be in terms of d, L, and λ. 

b. Use your expression from part a to find an expression for the separation Δy on the screen of two fringes that differ in wavelength by Δλ.

c. Rather than a viewing screen, modern spectrometers use detectors—similar to the one in your digital camera—that are divided into pixels. Consider a spectrometer with a 333 line/mm grating and a detector with 100 pixels/mm located 12 cm behind the grating. The resolution of a spectrometer is the smallest wavelength separation Δλ_min that can be measured reliably. What is the resolution of this spectrometer for wavelengths near 550 nm, in the center of the visible spectrum? You can assume that the fringe due to one specific wavelength is narrow enough to illuminate only one column of pixels.