Uncertainty propagation visible light spectrum

  1. 1. The problem statement, all variables and given/known data
    I have conducted an experiment which attempts to calculate the range of the visible light spectrum. Basically white light was shined through a diffraction grating (300 lines/mm) and diffraction theory is applied to calculate the wavelength.

    So, here are the variables:


    [tex]\Delta l=0.001[/tex]


    [tex]\Delta y=0.005[/tex]

    2. Relevant equations


    3. The attempt at a solution
    I combined these equations to end up with:
    The problem is that I don't know how to estimate an uncertainty for this equation. I know that for simple equations like [tex]y=q\times r[/tex] the uncertainty is [tex]\Delta y=\left(\frac{\Delta q}{q}+\frac{\Delta r}{r}\right)\times y[/tex]. Unfortunately I don't know how to apply this to a more complex equation. If anyone could lead me in the right direction as to an equation which would give the uncertainty for [tex]\lambda=d\times\sin\left(\arctan\left(\frac{y}{l}\right)\right)[/tex], it would be greatly appreciated.
  2. jcsd
  3. Redbelly98

    Redbelly98 12,053
    Staff Emeritus
    Science Advisor
    Homework Helper

    It can be done using calculus, if you've had calculus. But first I would get rid of the trig functions.

    What's an equivalent expression for sin(arctan(x)) ?
  4. [tex]\frac{x}{x^{2}+1}[/tex]
    Substituting [tex]\frac{y}{l}[/tex] for [tex]x[/tex] gives:
    [tex]\frac{\frac{y}{\left|y\right|}\times l}{\sqrt{y^{2}+l^{2}}}[/tex]

    I hadn't thought of doing this, so it seems to be a step in the right direction. I have done limited calculus, I'm just finishing the first year of IB Math HL so we're starting on integration right now. I looked briefly at the wikipedia page for error propagation and didn't really understand it.
  5. diazona

    diazona 2,156
    Homework Helper

    Hmm... yeah, Wikipedia is being ridiculously detailed about this.

    FYI, here's the usual case: if you have a function [tex]f(x, y, z)[/tex] and the uncertainties in the arguments are [tex]\delta x[/tex], [tex]\delta y[/tex], and [tex]\delta z[/tex], then the uncertainty in [tex]f[/tex] is
    [tex]\delta f = \sqrt{\left(\frac{\partial f}{\partial x}\delta x\right)^2 + \left(\frac{\partial f}{\partial y}\delta y\right)^2 + \left(\frac{\partial f}{\partial z}\delta z\right)^2}[/tex]
    Of course, there are some conditions on that formula, i.e. small, independent (uncorrelated) uncertainties and Gaussian distributions, but probably 99% of the time that formula is good enough.
  6. OK, thanks for the help so far. I applied the above formula to my equation and received the following result:
    [tex]\delta f = \sqrt{{\delta l}^{2}\,{\left( -\frac{d\,\left| l\right| \,y}{{l}^{2}\,\sqrt{{y}^{2}+{l}^{2}}}+\frac{d\,y}{\left| l\right| \,\sqrt{{y}^{2}+{l}^{2}}}-\frac{d\,\left| l\right| \,y}{{\left( {y}^{2}+{l}^{2}\right) }^{\frac{3}{2}}}\right) }^{2}+{\delta y}^{2}\,{\left( \frac{d\,\left| l\right| }{l\,\sqrt{{y}^{2}+{l}^{2}}}-\frac{d\,\left| l\right| \,{y}^{2}}{l\,{\left( {y}^{2}+{l}^{2}\right) }^{\frac{3}{2}}}\right) }^{2}}[/tex]

    Substituting with the variables in my first post returns the result:
    [tex]\delta f = 7.79\times 10^{-8}[/tex]

    Which is exactly what I received when I tried using an online uncertainty calculator! Thank you so much!
    Last edited: May 28, 2009
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