Finding Uncertainty in $\theta$ with Fixed $\lambda$

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The discussion focuses on calculating the uncertainty in the angle θ, defined as θ=sin^{-1}(nλ/d), with a fixed λ and no uncertainty in λ. The original poster attempted to derive the uncertainty using a total derivative but mistakenly used the wrong derivative notation and neglected to include the factor of n in their calculations. After clarifying that n=1, they realized their previous calculations were incorrect, leading to an exaggerated uncertainty of nearly 450 degrees. The conversation highlights the importance of correctly applying mathematical principles and checking calculations multiple times. Ultimately, the poster resolved their issue and achieved more accurate results.
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I'm trying to find the uncertainty in \theta where \theta is given by:

\theta=sin^{-1}\frac{n\lambda}{d}

in this case, I am assuming there is no uncertainty in \lambda.

This is what I tried:

\delta \theta=\sqrt{(\frac{d\theta}{dd})^2(\delta d)^2}

(the total derivative in there should be a partial derivative, but I don't know how to get that symbol)

\delta \theta=\sqrt{(\frac{\frac{\lambda}{d}}{\sqrt{d^2-\lambda^2}})^2\delta d^2}

I think that is right, but if I use the values \lambda=632.8 nm, d=1.08 \mu m and \delta d =.001 \mu m I get an uncertainty of almost 450 degrees. Where am I making my mistake?
 
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1.Click on this:\partial.
2.U assumed "n=1",else that derivative should have included the product n\lambda.
3.If n\neq 1,then you should include "n" in the derivative (under the square root) and redo your calculations.
4.If "n=1",then it's either the numbers are badly chosed,or u ****ed those calculations.

Daniel.
 
1. Thanks for the latex lesson
2. n=1
3. Turns out it was just my math. Although I did that calculation at least 4 or 5 times and kept getting the same answer before. I don't know what I was doing wrong, but it works much better now.
 
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