Diffraction: grating and sodium lamp spectrum

In summary, the question asks for the distance between two yellow components of light from a sodium lamp, given a grating with 10,000 lines per centimeter and a screen 1.00 m away. Using the grating equation and assuming m=0, the angles are found to be 36.13 degrees and 36.09 degrees for the 589 nm and 588 nm components respectively. The difference in tangents is used to calculate the distance between the two components, resulting in a value of 0.0011m. This method seems valid, but the question does hint at using a specific equation for spectroscopy, which was not used in this solution.
  • #1
Breedlove
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Homework Statement


Light from a laboratory sodium lamp has two strong yellow components at 589.5923 nm and 588.9953 nm. How far apart in the first-order spectrum will these two lines be on a screen 1.00 m from a grating having 10,000 lines per centimeter?


Homework Equations


Grating equation
a sin (theta) = m (lambda)

The Attempt at a Solution


Okay, so... I'm using Hecht's fourth edition of Optics and am struggling with question 33 as reproduced above. I'm not really sure if I need anything else besides the grating equation but through my inspection I'm pretty sure it's the only one that seems valid for the information given and what I want to achieve. I might be wrong though. Anyway, my supposed method to go about this is to find the angle difference. So I'm going to solve for theta using the two different lambdas, and then using the angles I think I should be able to come up with how far apart they are given that it is 1.00 m from the screen and use trigonometry. Does that sound good? Is that viable? I did it and ended up with...

36.13 degrees for the 589 nm and 36.09 for the 588nm. I took the difference of the tangents, and got .0011m.

Did I do this correctly or should I have used an equation specific to spectroscopy, given that the question hinted about "first-order spectrum"? I just took that to mean that m = 1.

Thanks for any help you can provide!
 
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  • #2
I believe first order in this context means m=0. So it looks like you did it right.
 

1. What is a diffraction grating?

A diffraction grating is an optical component that has a series of parallel lines or grooves etched onto its surface. When light passes through the grating, it is diffracted into different directions, creating a spectrum of colors.

2. How does a diffraction grating work?

The parallel lines on the grating act as a series of narrow slits, causing the light waves to interfere with each other. This interference results in the separation of the different wavelengths of light, creating a spectrum.

3. What is the purpose of using a sodium lamp in diffraction experiments?

Sodium lamps emit a very specific wavelength of light (589.3 nanometers) which makes them useful in diffraction experiments. This wavelength is easily distinguishable and can help calibrate the diffraction grating.

4. What factors affect the diffraction pattern produced by a grating?

The spacing of the lines on the grating, the wavelength of the light, and the angle at which the light hits the grating can all affect the resulting diffraction pattern.

5. How is a diffraction grating different from a prism?

A prism separates white light into its different wavelengths by refraction, while a diffraction grating separates light by diffraction. A grating can produce a more precise and detailed spectrum compared to a prism.

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