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

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

A typical laboratory diffraction grating has $5.00 \times$ $10^{3}$ lines $/ \mathrm{cm},$ and these lines are contained in a $3.50-\mathrm{cm}$ 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 $10^{3}$ lines $/ \mathrm{cm},$ and these lines are contained in a $3.50-\mathrm{cm}$ 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?

A typical laboratory diffraction grating has $5.00 \times$ $10^{3}$ lines $/ \mathrm{cm},$ and these lines are contained in a $3.50-\mathrm{cm}$ 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 $10^{3}$ lines $/ \mathrm{cm},$ and these lines are contained in a $3.50-\mathrm{cm}$ 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 instrument that consists of a large number of equally spaced slits or grooves. When light falls on the grating, it diffracts, and the interference between waves from successive slits leads to a pattern of bright and dark regions. This apparatus is used to separate incoming light into its constituent wavelengths, making it an essential tool in spectroscopy.

Resolving Power

Resolving power is the capability of an optical device to distinguish between two closely spaced wavelengths. In the case of a diffraction grating, it is quantitatively defined as R = ?/??, and is determined by the product of the diffraction order (m) and the number of illuminated lines (N) on the grating. This concept is crucial for determining the smallest wavelength difference that can be differentiated by the grating.

Diffraction Orders

Diffraction orders refer to the numbered maxima produced by a diffraction grating as a result of constructive interference of light, with the 0th order being the central maximum. Higher orders (indicated by higher values of m) correspond to light that has been diffracted to larger angles. The order chosen for use in an experiment significantly influences both the intensity of the diffraction pattern and the resolving power of the grating.

Spectral Analysis

Spectral analysis involves examining the spectrum of light emitted or absorbed by materials to determine their properties or composition. Diffraction gratings are often used in spectral analysis to resolve closely spaced spectral lines, such as those in doublet structures, by spreading out the light into its component wavelengths. Using higher diffraction orders can improve the resolution, making it possible to discern subtle differences in wavelength.

Transcript

00:02 Now, i know that the resolving power is n times m, whereas to use m equals 1, and n is going to be 5 ,000 lines per centimeter times 3 .5 centimeters.

00:26 So our answer is just going to be 5 ,000 times 3 .5.

00:34 Put that in a calculator.

00:37 Of course, i could probably do that in my head.

00:39 But anyway, the resolving power is 17 ,500.

00:47 Let's move one to b.

00:58 All right.

00:59 As i look in section 36 .5, i see that in order to resolve, the sodium doublet, you need to have a resolving power of 1 ,000.

01:15 17 ,500 is greater than 1 ,000.

01:21 So, yes, you can resolve that.

01:24 See, we've got here.

01:29 Okay, so it says that we're going to use the second order, and we're resolving spectral lines that are very close to 587.

01:46 8 .002 times 10 to the negative 9th power meter spectral line of iron.