One day, after pulling down your window shade, you notice that sunlight is passing through a pinhole in the shade and making a small patch of light on the far wall. Having recently studied optics in your physics class, you're not too surprised to see that the patch of light seems to be a circular diffraction pattern. It appears that the central maximum is about across, and you estimate that the distance from the window shade to the wall is about . Estimate (a) the average wavelength of the sunlight (in ) and (b) the diameter of the pinhole (in ).
a) The average wavelength of the sunlight
b) The diameter of the pinhole
vaguely remember that when using λ and aperture diameter D, the expression for the wavelength of the beams emanating from a completely circular probe at position L from the pupil is
This means that the beam diameter is linearly proportional to the wavelength used. Therefore, if we want to consider an average wavelength used, assuming the intensity per wavelength was constant, this average will be just the simplest arithmetic mean; in our case, half the sum of the lowermost and uppermost wavelengths of the visible spectrum . That is,
They can already obtain the aperture's length as
Because sound is a wave, it's possible to make a diffraction grating for sound from a large board of sound-absorbing material with several parallel slits cut for sound to go through. When sound waves pass through such a grating, listeners from the grating report "loud spots" on both sides of center. What is the spacing between the slits? Use for the speed of sound.
FIGURE shows light of wavelength incident at angle on a reflection grating of spacing . We want to find the angles um at which constructive interference occurs.
a. The figure shows paths and along which two waves travel and interfere. Find an expression for the path-length difference . 3
b. Using your result from part a, find an equation (analogous to Equation for the angles at which diffraction occurs when the light is incident at angle . Notice that m can be a negative integer in your expression, indicating that path is shorter than path .
c. Show that the zeroth-order diffraction is simply a “reflection.” That is,
d. Light of wavelength 500 nm is incident at on a reflection grating having reflection lines/mm. Find all angles at which light is diffracted. Negative values of are interpreted as an angle left of the vertical.
e. Draw a picture showing a single light ray incident at and showing all the diffracted waves at the correct angles.
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.
FIGURE Q33.1 shows light waves passing through two closely spaced, narrow slits. The graph shows the intensity of light on a screen behind the slits. Reproduce these graph axes, including the zero and the tick marks locating the double-slit fringes, then draw a graph to show how the light-intensity pattern will appear if the right slit is blocked, allowing light to go through only the left slit. Explain your reasoning.
A Michelson interferometer uses light from a sodium lamp. Sodium atoms emit light having wavelengths and . The interferometer is initially set up with both arms of equal length , producing a bright spot at the center of the interference pattern. How far must mirror be moved so that one wavelength has produced one more new maximum than the other wavelength?
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