Final answer:
A 30,000-line-per-centimeter grating will not produce a maximum for visible light. The longest wavelength for which it does produce a first-order maximum is 33.3 micrometers. The greatest number of lines per centimeter a diffraction grating can have and produce a complete second-order spectrum for visible light is 700 lines per centimeter.
Step-by-step explanation:
(a) For visible light, the longest wavelength is red light which has a wavelength of approximately 700 nm. To determine if a diffraction grating can produce a maximum for a given wavelength of light, we can use the equation:
sin(θ) = mλ/d
sin(θ) = (1)(700 x 10^-9 m) / (1/30000 m)
solve for sin(θ) gives:
sin(θ) = 2.33 x 10^-2
sin(θ) = mλ/d
Let's assume we want to find the longest wavelength for which a first-order maximum occurs, which means m = 1 and d = 1/30000 m. Plugging these values into the equation:
sin(θ) = (1)(λ) / (1/30000 m)
solving for λ gives:
λ = sin(θ) x (1/30000) m
The largest value of sin(θ) for a 30,000-line-per-centimeter grating is 1 (when θ = 90 degrees). Plugging this value into the equation gives:
λ = (1)(1/30000) m
λ = 3.33 x 10^-5 m = 33.3 μm
mλ = 2d
Let's assume we want a complete second-order spectrum, which means m = 2. Rearranging the equation to solve for d gives:
d = mλ / 2
Let's use the longest wavelength of visible light (700 nm) for λ. Plugging these values into the equation gives:
d = (2)(700 x 10^-9 m) / 2
d = 700 x 10^-9 m
Therefore, the greatest number of lines per centimeter a diffraction grating can have and produce a complete second-order spectrum for visible light is 700 lines per centimeter.