550-nm light passes through a diffraction grating with 3000 lines per centimeter. the screen is 115 cm away from the grating. at what angle, measured in degrees, is the 2nd-order bright fringe located?

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Answer 1

The angle measured in degrees at which the 2nd-order bright fringe located is located is; 58.78°

What is the angle of the diffraction grating?

Formula for diffraction grating is; dsinθ = mλ

where;

d = grating spacing = 1/(3000 lines per cm) = 1/3 × 10⁻⁵ m

m = order of fringe

λ = wavelength of light = 550 nm = 550 × 10⁻⁹ m.

Now, tanθ = x/D

where;

x = distance of nth order fringe from central maximum.

D = distance of screen from grating = 115 cm = 1.15 m

Now sinθ = d/mλ

θ is small and so we can approximate as sinθ ≅ tanθ

Thus;

d/mλ = x/D

for a second order bright fringe, m = 2.

d/2λ = x/D

x = dD/2λ

For a dark fringe, we have

d/(m + 1/2)λ = x'/D

where;

x' is the distance of the fringe from the central maximum.

For a second-order dark fringe, m = 2. Thus;

d/(2 + 1/2)λ = x'/D

d/(5/2)λ = x'/D

2d/5λ = x'/D

x' = 2dD/5λ

So, the distance between the 2nd order bright fringe and the 2nd dark fringe is; x" = dD/2λ - 2dD/5λ

x" = dD/10λ

Plugging in the values of the variables, we have;

x" = 1/3 × 10⁻⁵ m × 1.15 m/(10 × 550 × 10⁻⁹ m)

x" = 1.15/165 × 10² m

x" = 0.697 m

x'' = 69.7 cm

Thus;

tan θ = x/D = 115/69.7

θ = tan⁻¹(115/69.7)

θ = 58.78°

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