From the diffraction grating to the screen is 1 m. When the grating is illuminated by monochromatic light with a wavelength of 500 nm, the distance between the central and first maxima on the screen is 1 cm.How many strokes per millimeter are there in this grating?
Question
From the diffraction grating to the screen is 1 m. When the grating is illuminated by monochromatic light with a wavelength of 500 nm, the distance between the central and first maxima on the screen is 1 cm.How many strokes per millimeter are there in this grating?
Solution
To solve this problem, we can use the formula for diffraction grating:
dsin(θ) = mλ
where:
- d is the distance between the slits in the grating,
- θ is the angle of the maxima from the central maximum,
- m is the order of the maxima (in this case, m=1 for the first maxima),
- λ is the wavelength of the light.
First, we need to find the angle θ. We can use the small angle approximation sin(θ) ≈ tan(θ) ≈ θ for small angles. The tangent of the angle is the opposite side (the distance between the central and first maxima, which is 1 cm) divided by the adjacent side (the distance from the grating to the screen, which is 1 m). So, tan(θ) = 1 cm / 1 m = 0.01, and therefore θ ≈ 0.01 radians.
Next, we can substitute the known values into the formula:
dsin(θ) = mλ d0.01 = 1500 nm d = (1*500 nm) / 0.01 d = 50000 nm = 50 µm
The number of strokes per millimeter is the reciprocal of the distance between the slits in micrometers, so there are 1 / 50 µm = 20 strokes per millimeter in the grating.
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