White light (400–700 nm) incident on a 600 lines/mm diffraction grating produces rainbows of diffracted light. What is the width of the first-order rainbow on a screen 2.0 m behind the grating?

Verified step by step guidance

1. Step 1: Understand the problem. The diffraction grating splits white light into its constituent wavelengths, creating a spectrum. The goal is to calculate the width of the first-order rainbow on a screen 2.0 m behind the grating. This involves using the diffraction grating equation and geometry.
2. Step 2: Recall the diffraction grating equation: \( d \sin \theta = m \lambda \), where \( d \) is the distance between adjacent lines on the grating (grating spacing), \( \theta \) is the diffraction angle, \( m \) is the diffraction order, and \( \lambda \) is the wavelength of light. Convert the grating density (600 lines/mm) into grating spacing: \( d = \frac{1}{600 \times 10^3} \, \text{m} \).
3. Step 3: Calculate the diffraction angles for the shortest wavelength (400 nm) and longest wavelength (700 nm) in the first-order spectrum (\( m = 1 \)). Use \( \sin \theta = \frac{m \lambda}{d} \) for each wavelength. Ensure \( \lambda \) is converted to meters (e.g., \( 400 \, \text{nm} = 400 \times 10^{-9} \, \text{m} \)).
4. Step 4: Determine the positions of the diffracted light on the screen using the geometry of the setup. The position \( y \) on the screen is given by \( y = L \tan \theta \), where \( L \) is the distance from the grating to the screen (2.0 m). Calculate \( y \) for both \( \theta_{400} \) and \( \theta_{700} \).
5. Step 5: Find the width of the first-order rainbow by subtracting the position of the shortest wavelength from the position of the longest wavelength: \( \text{Width} = y_{700} - y_{400} \). This gives the physical separation of the spectrum on the screen.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that disperses light into its component wavelengths. It consists of multiple closely spaced lines or slits, which cause constructive and destructive interference of light waves. The angle at which light is diffracted depends on the wavelength and the spacing of the grating lines, allowing for the separation of different colors in the spectrum.

Order of Diffraction

The order of diffraction refers to the integer value (m) that indicates the number of wavelengths by which light is out of phase after passing through a grating. The first-order rainbow corresponds to m=1, where the light is diffracted at a specific angle that produces the first visible spectrum. Higher orders (m=2, 3, etc.) correspond to additional spectra that appear at greater angles.

Angle of Diffraction

The angle of diffraction is the angle at which light is bent as it passes through a diffraction grating. It can be calculated using the grating equation, d sin(θ) = mλ, where d is the distance between grating lines, θ is the angle of diffraction, m is the order of diffraction, and λ is the wavelength of light. This angle is crucial for determining the position of the diffracted light on a screen.