calculate the wavelength of laser light source for the first order diffraction. spacing between grating and screen is 50 cm note that grating has 2500 lines per inch
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Answer:
The analysis of multi-slit interference in Interference allows us to consider what happens when the number of slits N approaches infinity. Recall that N-2 secondary maxima appear between the principal maxima. We can see there will be an infinite number of secondary maxima that appear, and an infinite number of dark fringes between them. This makes the spacing between the fringes, and therefore the width of the maxima, infinitesimally small. Furthermore, because the intensity of the secondary maxima is proportional to 1\text{/}{N}^{2}, it approaches zero so that the secondary maxima are no longer seen. What remains are only the principal maxima, now very bright and very narrow ((Figure)).
(a) Intensity of light transmitted through a large number of slits. When N approaches infinity, only the principal maxima remain as very bright and very narrow lines. (b) A laser beam passed through a diffraction grating. (credit b: modification of work by Sebastian Stapelberg)
Figure a shows a graph of I versus sine theta. It has two vertical lines at sine theta equal to lambda by D and minus lambda by D. Figure b shows a bright red spot on a black background in the center. This is surrounded on either side by progressively dimmer spots, going outwards.
In reality, the number of slits is not infinite, but it can be very large—large enough to produce the equivalent effect. A prime example is an optical element called a diffraction grating. A diffraction grating can be manufactured by carving glass with a sharp tool in a large number of precisely positioned parallel lines, with untouched regions acting like slits ((Figure)). This type of grating can be photographically mass produced rather cheaply. Because there can be over 1000 lines per millimeter across the grating, when a section as small as a few millimeters is illuminated by an incoming ray, the number of illuminated slits is effectively infinite, providing for very sharp principal maxima.
A diffraction grating can be manufactured by carving glass with a sharp tool in a large number of precisely positioned parallel lines.
Figure shows a rectangular flat block with thin, parallel grooves. The grooves are cut at regular spacings d.
Diffraction gratings work both for transmission of light, as in (Figure), and for reflection of light, as on butterfly wings and the Australian opal in (Figure). Natural diffraction gratings also occur in the feathers of certain birds such as the hummingbird. Tiny, finger-like structures in regular patterns act as reflection gratings, producing constructive interference that gives the feathers colors not solely due to their pigmentation. This is called iridescence.
(a) Light passing through a diffraction grating is diffracted in a pattern similar to a double slit, with bright regions at various angles. (b) The pattern obtained for white light incident on a grating. The central maximum is white, and the higher-order maxima disperse white light into a rainbow of colors.
Figure shows a vertical line on the left. This has five grooves. A ray enters from the left and five rays emerge from the right, one from each groove. These point to squares which are labeled, from top to bottom: second order rainbow, first order rainbow, central white, first order rainbow, second order rainbow. The first order rainbows shown in the squares are brighter than the second order rainbows.
(a) This Australian opal and (b) butterfly wings have rows of reflectors that act like reflection gratings, reflecting different colors at different angles. (credit a: modification of work by “Opals-On-Black”/Flickr; credit b: modification of work by “whologwhy”/Flickr)