Two spectral lines at wavelengths 589 nm and 590 nm are produced by a sodium lamp. They are to be resolved using a diffraction grating with line spacing 2 µm. Using the first order maxima (m = 1) and Rayleigh’s criterion, find the minimum number of lines N required on the grating to resolve these two lines. Explain the principle behind the calculation.
Answer
590 lines
The correct answer is 590 lines — Explain: Rayleigh’s criterion states two spectral lines are just resolved when the maximum of one coincides with the first minimum of the other; for gratings R = mN gives the number of lines needed to achieve that resolution.
Step-by-step solution
- Compute average wavelength: λ ≈ (589 + 590)/2 = 589.5 nm.
- Compute resolving power needed: R = λ/Δλ = 589.5 nm / 1 nm = 589.5.
- Recall grating resolving power: R = mN. With m = 1, N = R/m ≈ 589.5.
- Since N must be an integer, minimum N = 590.
- Explain: Rayleigh’s criterion states two spectral lines are just resolved when the maximum of one coincides with the first minimum of the other; for gratings R = mN gives the number of lines needed to achieve that resolution.
Common mistake: Ignoring the order number m in R = mN or using λ/Δλ without considering line spacing
Concept tested: Rayleigh criterion & grating resolving power
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