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Chapter 33: Q. 36 (page 956)
shows the light intensity on a screen behind a double slit. The slit spacing is and the wavelength of the light is . What is the distance from the slits to the screen?
The screen's length from the slits is
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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?
A Michelson interferometer is set up to display constructive interference (a bright central spot in the fringe pattern of Figure) using light of wavelength l. If the wavelength is changed to , does the central spot remain bright, does the central spot become dark, or do the fringes disappear? Explain. Assume the fringes are viewed by a detector sensitive to both wavelengths.
For your science fair project you need to design a diffraction grating that will disperse the visible spectrum over in first order.
How many lines per millimeter does your grating need?
What is the first-order diffraction angle of light from a sodium lamp ?
The intensity at the central maximum of a double-slit interference pattern is . The intensity at the first minimum is zero. At what fraction of the distance from the central maximum to the first minimum is the intensity ? Assume an ideal double slit.
shows the light intensity on a screen behind a double slit. Suppose one slit is covered. What will be the light intensity at the center of the screen due to the remaining slit?
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