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Relative Error Propogation in Equations

  1. Nov 2, 2012 #1
    Greetings,

    In atomic spectra experiment I came across with error propogation in the nonlinear equation:
    [itex]\lambda=d\times\sin(\theta)[/itex] which gives the wavelength when first order constructive interference is observed at a given angle with respect to the normal of the plane of the grating. The relative error I am interested in is [itex]\frac{\Delta \lambda}{\lambda}[/itex]. In the laboratory manual it is stated without proof to be:

    [itex]\frac{\Delta \lambda}{\lambda}=\sqrt{(\frac{\Delta \theta}{\theta})^2+(\frac{\Delta d}\{d})^2}[/itex] I am pretty confused about it since I could not manage to verify it. I need a demonstration on why the relative error in wavelength is given by the preceding expression. I would be glad if anyone can guide me with references or suggestions.

    Thanks in advance
     
    Last edited: Nov 2, 2012
  2. jcsd
  3. Nov 2, 2012 #2

    haruspex

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    It's a bit unusual to write the error as [itex]\frac{\lambda}{\Delta\lambda}[/itex]. I'd expect [itex]\frac{\Delta\lambda}{\lambda}[/itex]. Is it possible you got the latex \frac parameters backwards?
    Anyway, the root-sum-squares formula results from the assumption that the underlying errors follow roughly a normal distribution, and that the magnitudes of those errors (delta/value) express a multiple of the standard deviation, and the same multiple for each. The root-sum-squares formula then gives you an estimate for that same number of standard deviations for the error in lambda.
    OTOH, if you want the absolute range of error in lambda then the correct way is to consider all possible errors in the measured quantities and see what range results. For the present case that would give [itex]\frac{\Delta\lambda}{\lambda} = \frac{\Delta\theta}{\theta}+\frac{\Delta d}{d}[/itex] (all errors assumed to be expressed as > 0).
     
  4. Nov 2, 2012 #3
    Thanks for the answer.You are definitely right, owing to the fact that I am a Latex Newbie, I got the parameters backwards, I would correct them ASAP. By the way I am not that familiar with standard deviatation etc. so could you provide some rigorous formulation which allows the author to arrive at that conclusion?

    Note: I cannot edit my first post so that the Latex code is displayed properly may use some help here too.
     
    Last edited: Nov 2, 2012
  5. Nov 2, 2012 #4

    haruspex

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  6. Nov 3, 2012 #5
    By the way the term "maximum relative error" is used in the laboratory manual which I forgot to mention earlier, I think that poses a contradiction.
     
  7. Nov 3, 2012 #6

    haruspex

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    How so?
     
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