The yellow light of a sodium lamp is not a single wavelength…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
The yellow light of a sodium lamp is not a single wavelength. It has two very close lines, of wavelengths nm and nm. A teacher wants to show the two lines as separate bright lines with a diffraction grating that has lines in each millimetre. She asks how good the grating must be. The lamp gives a wide parallel beam that falls on the whole grating, and she is free to use the first order or the second order of the pattern. The resolving power of a grating is , and it equals , where is the order and is the number of lines that the light falls on. Find the resolving power that is needed to separate the lines. Find the number of lines that must be lit in the second order. Find the width of the grating that must be lit in the first order.
Need a reminder? The key ideaShow
The Rayleigh limit of a circular aperture is . A microscope resolves . A grating has the resolving power .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The resolving power. The mean is nm and nm. So .
Step 2. The second order. We want , so lines.
Step 3. The first order. lines. At lines in each millimetre, the width is .
Answer: ; lines; mm.
Check. A modest grating, less than mm of lines, would do. The pattern is easy to resolve in practice, as long as the beam is wide enough to light that many lines.
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