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Diffraction Grating Formula

A diffraction grating defines an optical component with a periodic structure  which is splits the light into various beams that travels in different directions. Its an alternative way to observe spectra other than prism. Generally, when light is incident on grating, the splitted light will have maxima at an angle θθ.  The formula for diffraction grating is used to calculate this angle.

Diffraction grating formula

λλ = d sin θnd sin θn

Where,

n = order of grating,

D = distance between two fringes or spectra

Question 1: A diffraction grating is of width 5 cm and produces a deviation of 22∘∘ in the second order with light of wavelength 580 nm. Find the slit spacing.

Solution:

Given: Angle θθ = 22 ∘∘, order n = 2, wavelength λλ = 580 nm.

The slit spacing is given by,

d = nλsin θnλsin θ

d = 2×580nmsin 22∘2×580nmsin 22∘

d = 3 ×× 10−6−6 m

Question 2: A diffraction grating has 24000 lines /cm separates a dark line at an angle of 32 ∘∘. Find the wavelength of light.

Solution:

Given: Separation between the slits d = 1/36000 = 2.77 ×× 10−5−5/cm = 2.77 ×× 10−7−7/m

Wavelength, λλ = d sin θθ

λλ = 2.77 ×× 10−7−7 sin 32 ∘∘

λλ = 146.7 nm

Therefore, the wavelength of light is146.7 nm.