A typical laboratory diffraction grating has 5.00 ×10^3 lines/cm, and these lines are contained

step 1 We are given a diffraction grating with line density 5 ,000 per centimeter and the width of the

A typical laboratory diffraction grating has 5.00 ×10^3...

A typical laboratory diffraction grating has 5.00 $\times$ 10$^3$ lines/cm, and these lines are contained in a 3.50-cm width of grating. (a) What is the chromatic resolving power of such a grating in the first order? (b) Could this grating resolve the lines of the sodium doublet (see Section 36.5) in the first order? (c) While doing spectral analysis of a star, you are using this grating in the $second$ order to resolve spectral lines that are very close to the 587.8002-nm spectral line of iron. (i) For wavelengths longer than the iron line, what is the shortest wavelength you could distinguish from the iron line? (ii) For wavelengths shorter than the iron line, what is the longest wavelength you could distinguish from the iron line? (iii) What is the range of wavelengths you could $not$ distinguish from the iron line?

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a regular pattern of closely spaced lines or slits that diffracts light into several beams traveling in different directions. The directions depend on the wavelength of the light and the grating’s spacing, and this property is exploited in spectroscopy to separate light into its component wavelengths.

Diffraction Order

The diffraction order is an integer (positive, negative, or zero) that indicates the series of maxima produced by a grating. Each order corresponds to a different angle at which constructive interference occurs for a given wavelength. Higher orders can provide greater angular separation between wavelengths but also come with trade?offs in intensity and potential overlap with other orders.

Resolving Power

Resolving power is a measure of an optical instrument’s ability to distinguish between two closely spaced wavelengths. For diffraction gratings, it is defined as the ratio of the wavelength to the smallest difference in wavelength that can be resolved, and it is proportional to both the diffraction order and the total number of lines illuminated. This concept is central in spectroscopy for determining how finely a grating can separate spectral lines.

Rayleigh Criterion

The Rayleigh criterion is a guideline used to determine the minimum angular separation at which two spectral lines can be considered distinct. In the context of diffraction gratings, it provides the threshold for resolving two nearby wavelengths based on their interference patterns, ensuring that the principal maximum of one line coincides with the first minimum of the other. This criterion underpins the design and application of resolving power in optical instruments.

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