FIGURE CP33.73 shows two nearly overlapped intensity peaks of the

FIGURE CP33.73 shows two nearly overlapped intensity peaks of the sort you might produce with a diffraction grating (see Figure 33.9b). As a practical matter, two peaks can just barely be resolved if their spacing Δy equals the width w of each peak, where w is measured at half of the peak’s height. Two peaks closer together than w will merge into a single peak. We can use this idea to understand the resolution of a diffraction grating.

FIGURE CP33.73 shows two nearly overlapped intensity peaks of the

a. In the small-angle approximation, the position of the m = 1 peak of a diffraction grating falls at the same location as the m = 1 fringe of a double slit: y1 = λL/d. Suppose two wavelengths differing by Δλ pass through a grating at the same time. Find an expression for Δy, the separation of their first-order peaks.
b. We noted that the widths of the bright fringes are proportional to 1/N, where N is the number of slits in the grating. Let’s hypothesize that the fringe width is w = y1/N. Show that this is true for the double-slit pattern. We’ll then assume it to be true as N increases.
c. Use your results from parts a and b together with the idea that Δymin = w to find an expression for Δλmin, 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 656.45 nm. In deuterium, which is a “heavy” isotope of hydrogen, the wavelength is 656.27 nm. What is the minimum number of slits in a diffraction grating that can barely resolve these two wavelengths in the first-order diffraction pattern?

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