(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A diffraction grating has 4,200 rulings/cm. On a screen 2.00 m from the grating, it is found that for a particular order m, the maximum corresponding to two closely spaced wavelengths of sodium (589.0 nm and 589.6 nm) are separated by 1.54 mm. Determine the value of m.

2. Relevant equations

dsinθ = mλ (for bright fringes)

3. The attempt at a solution

I spent about 15 minutes working this problem out under the assumption that sinθ ≈ tanθ = y/L. First, I drew a diagram of the corresponding bright fringes, and I made λ_{1}correspond to the wavelength that is higher on the screen, while λ_{2}is lower on the screen. Since λ_{1}has a larger angle in my diagram, I knew that λ_{1}should be 589.6 nm for my equations and λ_{2}should be 589.0 nm. Also, I knew that y_{2}= y_{1}- 1.54*10^{-3}. I came up with two equations:

dsinθ_{1}= mλ_{1}→ dy_{1}/L = mλ_{1}

dsinθ_{2}= mλ_{2}→ dy_{2}/L = mλ_{2}→ d(y_{1}- 1.54*10^{-3}) = mλ_{2}

I divided these two equations, went through some algebra, solved for y_{1}, and plugged it back into the first equation I found to solve for m; I got m = 3. However, I checked the back of the book and it says m = 2. So, I checked over my work again and realized that the small angle approximation I made does not hold because the angles are not small for diffraction gratings (correct me if I'm wrong). Now, I'm back to square one and honestly not sure where to go. I know sinθ = y/hypotenuse, but it seems like this route could get very algebraically sloppy, and I don't want to go down that road before getting some help.

Thank you!

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# Homework Help: Diffraction grating problem involving two wavelengths.

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