FIGURE shows two nearly overlapped intensity peaks of the sort you might produce with a diffraction grating . As a practical matter, two peaks can just barely be resolved if their spacing equals the width w of each peak, where is measured at half of the peak’s height. Two peaks closer together than will merge into a single peak. We can use this idea to understand the resolution of a diffraction grating.
a. In the small-angle approximation, the position of the peak of a diffraction grating falls at the same location as the fringe of a double slit: . Suppose two wavelengths differing by pass through a grating at the same time. Find an expression for localid="1649086237242" , the separation of their first-order peaks.
b. We noted that the widths of the bright fringes are proportional to localid="1649086301255" , where localid="1649086311478" is the number of slits in the grating. Let’s hypothesize that the fringe width is localid="1649086321711" Show that this is true for the double-slit pattern. We’ll then assume it to be true as localid="1649086339026" increases.
c. Use your results from parts a and b together with the idea that localid="1649086329574" to find an expression for localid="1649086347645" , the minimum wavelength separation (in first order) for which the diffraction fringes can barely be resolved.
d. Ordinary hydrogen atoms emit red light with a wavelength of localid="1649086355936" In deuterium, which is a “heavy” isotope of hydrogen, the wavelength is localid="1649086363764" What is the minimum number of slits in a diffraction grating that can barely resolve these two wavelengths in the first-order diffraction pattern?
