Final answer:
To calculate the width of the first-order spectrum on a screen 3.0 m from a diffraction grating, we use the formula nλ = d sin(θ) to find the positions of first-order maxima for 400 nm and 650 nm wavelengths, and then subtract the two positions to find the width.
Step-by-step explanation:
The student is asking about the width of the first-order spectrum produced on a screen that is 3.0 meters away when white light passes through a grating with 4000 lines per centimeter. To calculate this, we use the diffraction grating formula nλ = d sin(θ), where n is the order of the spectrum, λ is the wavelength of light, d is the grating spacing, and θ is the diffraction angle.
First, we calculate the value of d, which is the inverse of the number of lines per centimeter. With 4000 lines per centimeter, d = 1 / 400000 cm or d = 2.5 x 10⁻⁵ m. We can then calculate the angles for the first-order spectrums for 400 nm and 650 nm respectively using the formula for n=1. This allows us to find the positions on the screen using the tan(θ) ≈ θ approximation for small angles (in radians), where the position y is given by y = L tan(θ) for a distance L from the grating to the screen.
After calculating the angles, we can find the positions of the first-order spectrum for both 400 nm and 650 nm wavelengths and subtract one from the other to find the width of the first-order spectrum. As the formula involves trigonometric calculations and the precise value depends on the actual angle values obtained, the specific numerical answer is not provided here. The student would carry out these calculations to determine the width on the screen.