Diffraction gratings with 10,000 lines per centimeter are readily available.

Isaac Barry 2022-09-04 Answered
Diffraction gratings with 10,000 lines per centimeter are readily available. Suppose you have one, and you send a beam of white light through it to a screen 2.00 m away.Find the angles for the first-order diffraction of the shortest and longest wavelengths of visible light (380 and 760 nm).
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Answers (1)

Sanaa Hudson
Answered 2022-09-05 Author has 7 answers
Given information:
The number of lines on the grating, N = 10000  lines/cm = 10000 × lines 10 2  m = 10 6  lines/m
The distance between the screen and slits, D = 2.00  m
The shortest wavelength, λ min = 380  nm = 380 × 10 9  m
The longest wavelength, λ max = 760  nm = 760 × 10 9  m
To find
the angles corresponding to first-order diffraction from these wavelengths.
The grating equation is given by
d sin ( θ ) = n λ sin ( θ ) = n λ d θ = sin 1 n λ d
where,
d=width of the slit
n=order of diffraction
λ = wavelength of incident light
θ = angle of diffraction
The width of the slit is calculated as follows
d = 1 N d = 1 10 6 d = 10 6  m
Apply the above formula for the shortest wavelength
θ min = sin 1 ( 1 × 380 × 10 9 10 6 ) = sin 1 ( 0.38 ) = 22.33
Now, apply the above formula for the longest wavelength
θ max = sin 1 ( 1 × 760 × 10 9 10 6 ) = sin 1 ( 0.76 ) = 49.46
Answer:
The angle corresponding to the shortest wavelength is, θ min = 22.33
The angle corresponding to the longest wavelength is, θ max = 49.46
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