Final answer:
a) The angle of diffraction for the second-order peak is approximately 6.57 degrees. b) The wavelength of light for the first-order diffraction peak is approximately 783.3 nm. c) The angular separation between adjacent orders is approximately 37.67 degrees.
Step-by-step explanation:
a) To find the angle of diffraction for the second-order peak, we can use the formula for the angle of diffraction for a diffraction grating: θ = sin^(-1)(mλ/d), where m is the order of the diffraction peak, λ is the wavelength of light, and d is the distance between adjacent slits on the grating. In this case, m = 2, λ = 400 nm, and d = 1/1000 mm = 1/1000000 m. Plugging these values into the formula gives θ = sin^(-1)(2 * 400 * 10^(-9) / (1/1000000)). Solving for θ gives θ ≈ 0.1146 radians, or approximately 6.57 degrees.
b) To determine the wavelength of light for the first-order diffraction peak, we can rearrange the formula for the angle of diffraction and solve for λ: λ = d * sin(θ) / m. In this case, m = 1, θ = 6.57 degrees, and d = 1/1000 mm = 1/1000000 m. Plugging these values into the formula gives λ = (1/1000000) * sin(6.57 degrees) / 1. Solving for λ gives λ ≈ 783.3 nm.
c) The angular separation between adjacent orders can be found using the formula Δθ = sin^(-1)(mλ/d), where m is the order of one of the peaks and λ is the wavelength of light. In this case, m = 1, λ = 400 nm, and d = 1/1000 mm = 1/1000000 m. Plugging these values into the formula gives Δθ = sin^(-1)(1 * 400 * 10^(-9) / (1/1000000)). Solving for Δθ gives Δθ ≈ 0.6585 radians, or approximately 37.67 degrees.
d) The distance between the slits on the grating can be found using the formula d = 1/(lines per mm). In this case, we are given that there are 1000 slits per mm, so the distance between the slits is d = 1/1000 mm = 1/1000000 m.