Final answer:
The width of the first-order spectrum on the screen is found by calculating the angles for the first-order diffraction for both the shortest and longest wavelengths of visible light, and then determining their positions on the screen by projecting at a distance of 2.80m away from the diffraction grating.
Step-by-step explanation:
The question involves calculating the width of the first-order spectrum produced by a diffraction grating when it is hit by white light containing wavelengths between 400 nm to 750 nm. To solve this, we will use the diffraction grating formula d sin(θ) = mλ, where d is the grating spacing, θ is the diffraction angle, m is the order of the spectrum, and λ is the wavelength of the light.
Firstly, the grating spacing d is the inverse of the number of lines per unit length. With 7800 lines/cm, we get d = 1 / (7800 × 102) m = 1.28205128205×10-6 m.
For the first-order spectrum, m = 1, and we calculate the angle for the shortest wavelength (400 nm) and the longest wavelength (750 nm).
The angles θ for each wavelength are found using the formula:
θ = arcsin(mλ/d), which yields two different angles for the two wavelengths. We can then use these angles in the formula y = L tan(θ), where L is the distance to the screen, to find the positions of the first-order maximum on the screen for each wavelength. The difference between these two positions gives us the width of the spectrum.
After calculating the angles and positions, we subtract the position of the red light (longest wavelength) from that of the violet light (shortest wavelength) to find the width of the first-order spectrum on the screen.