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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 Î”y equals the width w of each peak, where w is measured at half of the peakâ€™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.

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: y_{1} = Î»L/d. Suppose two wavelengths differing by Î”Î» pass through a grating at the same time. Find an expression for Î”y, 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. Letâ€™s hypothesize that the fringe width is w = y_{1}/N. Show that this is true for the double-slit pattern. Weâ€™ll then assume it to be true as N increases.

c. Use your results from parts a and b together with the idea that Î”y_{min} = w to find an expression for Î”Î»_{min}, 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 â€œheavyâ€ 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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