Question

The two most prominent wavelengths in the light emitted by a hydrogen discharge lamp are 656 nm(red) and 486 nm (blue). Light from a hydrogen lamp illuminates a diffraction grating with 500 lines/mm , and the light is observed on a screen 1.3 m behind the grating.

What is the distance between the first-order red and blue fringes? Answer in terms of cm

Express your answer to two significant figures and include the appropriate units.

Answer 1

givens:
λ_red = 656 nm
λ_blue = 486 nm
line density = 500 / mm
L = 1.3 m
d = 1 mm / 500 lines = 2⋅10^-3 mm
m = 1

the positions can be determined by using the formula θ=sin^-1(mλ/d) & Ltan(θ)=y
θ_red = sin^-1(1⋅λ_red/d) = sin^-1(656 nm / 2⋅10^-3 mm)
θ_red = 19.147°
y_red=Ltan(θ_red) = (1.3 m)tan(19.147°) = 0.4513 m
θ_blue = sin^-1(1⋅λ_blue/d) = sin^-1(486 nm / 2⋅10^-3 mm)
θ_blue = 14.063°
y_blue=Ltan(θ_blue) = (1.3 m)tan(14.063°) = 0.325 m

so the blue fringe begins at 0.325 m and the red fringe begins at 0.