Light emitted by element X passes through a diffraction grating that has 1200 slits/mm. The interference pattern is observed on a screen 77.0 cm behind the grating. First-order maxima are observed at distances of 58.0 cm , 65.4 cm , and 94.5 cm from the central maximum. What are the wavelengths of light emitted by element X?

500 nm, 530 nm, 650 nm

Explanation:

Let's say that there is diffraction grating observed with a slit spacing of s. Respectively we must determine the angle θ which will help us determine the 3 wavelengths ( λ ) of the light emitted by element X. This can be done applying the following formulas,

s( sin θ ) = m λ, such that y = L( tan θ ) - where y = positioning, or the distance of the first - order maxima, and L = constant, of 77 cm

Now the grating has a slit spacing of -

s = 1 / N = 1 / 1200 = 0.833 10⁻³ mm

The diffraction angles of the " positionings " should thus be -

θ = tan⁻¹ ( 0.58 / 0.77 ) = 37°,

θ = tan⁻¹ ( 0.654 / 0.77 ) = 40°,

θ = tan⁻¹ ( 0.945 / 0.77 ) = 51°

The wavelengths of these three bright fringes should thus be calculated through the formula : λ = s( sin θ ) -

λ = 0.833 10⁻³ sin( 37° ) = ( 500 10⁻⁹ m )

λ = 0.833 10⁻³ sin( 40° ) = ( 530 10⁻⁹ m )

λ = 0.833 10⁻³ sin( 51° ) = ( 650 10⁻⁹ m )

Wavelengths : 500 nm, 530 nm, 650 nm

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