Yellow light of wavelength 5.5 x 10^-7 m is incident normally at a diffraction grating with 300 lines per mm. Calculate the angle between the first-order and second-order maxima.
To calculate the angle between the first-order and second-order maxima, we need to use the equation for the angle of diffraction from a diffraction grating.
The equation is given as:
dsinθ = mλ
where:
- d is the spacing between adjacent lines on the diffraction grating,
- θ is the angle of diffraction,
- m is the order of the maximum (first-order, second-order, etc.),
- λ is the wavelength of light.
First, we need to calculate the spacing between adjacent lines on the diffraction grating (d) using the information given in the question. The problem states that the diffraction grating has 300 lines per mm, so we need to convert this to meters:
d = 1 / (300 lines/mm) = 1 / (300 x 10^3 lines/m) = 3.33 x 10^-6 m
Substituting this value and the given wavelength (λ = 5.5 x 10^-7 m) into the equation, we can solve for the angles of diffraction for the first-order (m = 1) and second-order (m = 2) maxima:
For the first-order maximum:
dsinθ₁ = λ₁
(3.33 x 10^-6 m) sinθ₁ = 5.5 x 10^-7 m
Solving for sinθ₁:
sinθ₁ = (5.5 x 10^-7 m) / (3.33 x 10^-6 m) = 0.165
Taking the inverse sine of both sides to find θ₁:
θ₁ = arcsin(0.165) = 9.52°
For the second-order maximum:
dsinθ₂ = λ₂
(3.33 x 10^-6 m) sinθ₂ = 2(5.5 x 10^-7 m)
Solving for sinθ₂:
sinθ₂ = 2(5.5 x 10^-7 m) / (3.33 x 10^-6 m) = 0.330
Taking the inverse sine of both sides to find θ₂:
θ₂ = arcsin(0.330) = 12.73°
Finally, to find the angle between the first-order and second-order maxima, we subtract θ₁ from θ₂:
Angle between first-order and second-order maxima = θ₂ - θ₁
= 12.73° - 9.52°
= 3.21°
So the angle between the first-order and second-order maxima is 3.21°.