# Error question

1. Apr 1, 2009

### shonen

I'm during an experiment involving the Michelson- Interferometer, now i need to find the uncertainty for lambda using the equation.

2 (path length difference) = N (lambda)

Now I no the above equation can be changed to

2(path length difference)/ N = Lambda; and from their the corresponding equation to find the uncertainty in lambda is

(uncertainty in lambda)/ lambda = ( (uncertainty in path length difference)/path length difference)^2 + ( (uncertainty in fringe count)/fringe count )^2 )^ (1/2)

Path length difference is a free variable meaning It's not a specific value and the same can be said of lambda. So what I decided to do is find the corresponding lambda for each path length difference and plug A conjugate pair into the equation. The problem is that when I do that I get different values for the uncertainty. What am I doing wrong.

2. Apr 1, 2009

### turin

If you want the expected error, then you should use the measured path length and measured number of fringes. You must estimate the errors in these two measurements based on the error sum of
- how precisely the measuring devices are able to make the measurements
- the standard deviation in the measurements that you made

The precision should be determined by the tick marks on the micrometer (or whatever graduation confines your measuring device) and how well you think that you can grade the fringes. The estimate for these errors should always be a fraction of the corresponding graduation. So, assuming that the micrometer has 10um tick marks, the estimated precision error should be less than 10um. The estimated fringe count error should be less than one fringe. (If you think that you missed a fringe count, then you should redo the experiment - you should NEVER use carelessness as a factor in the error estimate.)

As the experiment is repeated over and over again, you should see the statistical error approach the precision error. If the satistical error becomes significantly less than the precision error, then you likely have introduced some systematic error, and you should reconsider the integrity of the experiment.

3. Apr 1, 2009

### shonen

Path length was measured 8 different times and number of fringes counted for each trial was 20. The problem is determining what value to give lambda and path length difference for the equation listed above in order to determine the uncertainty for lambda.

Like say I have for my first trial

Path Length difference 0.0115 mm and calculated Lambda = 0.00115 mm

For my second trial

Path length difference 0.0065 mm and calculated Lambda = 0.00065 mm

If i use the first set I get a different value for the uncertainty as appose to using the second set. This is what I'm getting at. I apologize in advance if i missed anything in the explanation.

4. Apr 3, 2009

### turin

I will try to explain it in a slightly more specific way. However, your lab manual/instructor should explain all of this to you.

The wavelength is, theoretically, a function of path length and fringe count, and so, in your experiement, there is an experimental function of path length and fringe count that is equal to this theoretical function. However, experimentally, the value of the function is not really the wavelength, it is just some estimator of the wavelength. The size of the fluctutions in the value of this function due to fluctuations in the path length and fringe count are quantified by the expected error. So, if

Lth ≡ theoretical path length difference
Nth ≡ theoretical number of fringe changes for the given L (which can be fractional)
λth ≡ theoretical wavelength

then there is some theoretical function such that

λth = f(Lth,Nth)

So, if

Lav ≡ average measured path length difference
Nav ≡ average measured number of fringe changes for the given L (which you typically decide as an experimental parameter, so that it is assumed to be a whole number)

then f(Lav,Nav) is an estimator for the wavelength, i.e.

<λ> = <f(Lav,Nav)>

Experimentally, this is not the whole story. You also want to know how good this estimate is. Assuming that f depends only on the two variables, L and N, the uncertainty in the estimation of λex is determined by the uncertainties of L and N. To first order, and assuming no correlation, this is just given by the square root of the sum of the squares of the partial derivatives each multiplied by the error sum of the appropriate variable, evaluated at the average measured values. Loosely:

<δλ2> = (δL⋅∂f/∂L)2 + (δN⋅∂f/∂N)2

I would have to remind myself how to take correlations into account. Your experimental procedure probably avoids this issue by fixing the value of N. Theoretically, L and N are strongly correlated.