5. A grating with 400 lines per mm is illuminated with light of wavelength 600.0 nm.
(a) Determine the angle at which the maxima are observed
(b) Determine the largest order (n) that can be seen with this grating and this wavelength.
To determine the angle at which the maxima are observed and the largest order that can be seen with this grating and wavelength, we can use the formula for the maximum order of diffraction:
nλ = d*sin(θ)
where:
- n is the order of the maxima
- λ is the wavelength of light
- d is the grating spacing (the distance between the adjacent slits or lines on the grating)
- θ is the angle at which the maxima are observed.
In this case, we are given:
- λ = 600.0 nm (nanometers)
- d = 1/400 mm = 0.0025 mm = 2.5 μm (micrometers) (since there are 400 lines in 1 mm, the spacing between the lines is the reciprocal of that, which is 1/400 mm)
(Note: We converted mm to μm since wavelength is usually measured in micrometers)
Let's solve the problem step by step:
(a) Determine the angle at which the maxima are observed:
- Rearrange the formula to solve for θ: θ = sin^(-1)(nλ/d)
- Substitute the given values: θ = sin^(-1)((1)(600.0 nm)/(2.5 μm))
- Evaluate: θ ≈ 14.5° (rounded to one decimal place)
Therefore, the angle at which the maxima are observed is approximately 14.5°.
(b) Determine the largest order (n) that can be seen with this grating and wavelength:
- Rearrange the formula to solve for n: n = (d*sin(θ))/λ
- Substitute the given values: n = ((2.5 μm)*sin(14.5°))/(600.0 nm)
- Evaluate: n ≈ 0.0092
- Since the order has to be a whole number, we round down to the nearest whole number.
- Therefore, the largest order that can be seen with this grating and wavelength is 0.
Please note that the largest order is 0 in this case because the wavelength (600.0 nm) is larger than the grating spacing (2.5 μm), resulting in no observable diffraction maxima.
sinθ = lambda/d
so θ =sin-1(lambda/d)
You'll have to figure out d given there are 400 lines in one mm (so how far apart are the lines?)