(II) White light passes through a 640-slit/ mm diffraction grating. First-order and second-order visible spectra (“rainbows”) appear on the wall 32 cm away as shown in Fig. 35–40. Determine the widths ℓ₁ and ℓ₂ of the two “rainbows” (400 nm to 700 nm). In which order is the “rainbow” dispersed over a larger distance?

Verified step by step guidance

1. Step 1: Understand the problem. A diffraction grating with 640 slits per millimeter is used to produce first-order and second-order spectra of white light. The task is to calculate the widths (ℓ₁ and ℓ₂) of the rainbows for the first and second orders, corresponding to the wavelength range of 400 nm to 700 nm. The wall is 32 cm away from the grating.
2. Step 2: Convert the given grating information into a usable form. The number of slits per millimeter is 640, so the slit spacing (d) can be calculated as: d = 1 / (640 × 10³) meters. This gives the distance between adjacent slits in the grating.
3. Step 3: Use the diffraction grating equation to find the angular positions of the wavelengths at the edges of the spectrum for each order. The equation is: mλ = d sin(θ), where m is the order of diffraction, λ is the wavelength, d is the slit spacing, and θ is the diffraction angle. Solve for θ for both 400 nm and 700 nm wavelengths in the first order (m = 1) and second order (m = 2).
4. Step 4: Calculate the linear positions of the wavelengths on the wall. The linear position (y) on the wall is related to the angle θ by the formula: y = L tan(θ), where L is the distance from the grating to the wall (32 cm or 0.32 m). Compute y for both 400 nm and 700 nm wavelengths in the first and second orders.
5. Step 5: Determine the widths of the rainbows. The width of the rainbow (ℓ) is the difference between the linear positions of the 700 nm and 400 nm wavelengths: ℓ = y(700 nm) - y(400 nm). Perform this calculation for both the first and second orders to find ℓ₁ and ℓ₂. Compare the two widths to determine which order has the larger dispersion.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams traveling in different directions. The angle at which light is diffracted depends on the wavelength and the spacing of the slits. This principle is crucial for understanding how different wavelengths of light are separated, leading to the formation of visible spectra.

Order of Diffraction

The order of diffraction refers to the integer multiples of the wavelength that determine the angles at which light is constructively interfered. The first-order spectrum corresponds to the first angle where constructive interference occurs, while the second-order spectrum corresponds to the second angle. Higher orders result in wider dispersion of light, which is essential for analyzing the distances between the 'rainbows' produced.

Wavelength and Color Dispersion

Wavelength is a fundamental property of light that determines its color; shorter wavelengths correspond to blue light, while longer wavelengths correspond to red light. In a diffraction grating setup, different wavelengths are dispersed at different angles, leading to a spectrum of colors. Understanding how wavelength affects the dispersion helps in calculating the widths of the 'rainbows' and determining which order disperses over a larger distance.