Estimating measurement uncertainties

"Measurement uncertainties": Exercise

We want to determine the width "a" of a diffracting slit from the diffraction pattern observed on a screen located at a distance "D" of the split, which is lit by a laser source of wavelength λ (cf. figure 3).

figure 3 : Experimental situation for the measurement of the width "a" of the split [zoom...]

Question

What is the expression of the illumination I(θ) on the screen for a split that is considered of nearly infinite length in comparison to its width "a" ?

Question

Express the angle θ according to the distance between the split and the screen D and to the width L of the central lobe of diffraction measured on the screen.

Find another expression of θ from the apparent diameter of the central spot of diffraction deduced from I(θ).

Deduce from these two expressions of θ the relation between the split's width a and quantities L, D and λ .

Question

Using the propagation of uncertainties law, express the standard uncertainty u(a).

Question

The distance between the split and the screen D is measured five times in a row with a measuring tape of resolution 1mm. The measured values are the following:

Table 9 [zoom...]

Calculate u(D).

Question

The width L of the diffraction spot is measured 5 times in a row with a ruler of resolution 0.5mm. The measured values are the following:

Table 10 [zoom...]

Calculate u(L).

Question

Calculate a and u(a) assuming the hypothesis that the wavelength λ of the luminous source (λ=633nm) includes no uncertainty .

Question

Calculate the expanded uncertainty U(a) and express the measurement result of width a.