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IZhO 2023, Експеримент — Задача 3. Monochromatic radiation with unknown wavelength

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Всички материали от IZhO 2023

XIX International Zhautykov Olimpiad/Experimental Competition · 3 февруари 2023 г. · 5 т.

Внимание: Бележка към темата

Figures 1–3 and Tables 1–3 with the registered signals were printed on the separate Writing sheets, which are not in the archive.

Please read this first:

February 3, 2023

The duration of the experimental competition is 4 hours. There is one problem.
You can use your own calculator for numerical calculations.
You are provided with white sheets of paper for drafts of your solutions, but these sheets will not be graded. Your final solutions must only be written on the Writing sheets. You can write very short explanations using equations, numbers, letters.
Use only the front side of Writing sheets. Write only inside the boxed areas.
Graphs must be drawn on Writing sheets within the graph paper area. All drawings must be done with a pen, not a pencil! If you need additional Writing sheets with the graph paper, you can find them at the end of your worksheet.
Fill in the boxes at the top of each Writing sheet with your country (Country), your student code (Student Code), problem number (Question Number), the progressive number of each Writing sheet (Page Number), and the total number of Writing sheets used (Total Number of Pages). If you use some blank Writing sheets that you do not wish to be graded, put a large X across the entire sheet and do not include it in your numbering.

Условие

The interferometer is illuminated by a monochromatic radiation with an unknown wavelength that you have to determine.

Figure 2 shows the dependence of the light intensity on the time for this case. Table 2 shows the values of the extrema coordinates for this particular case.

3.1 Plot the dependence of the mirror coordinates $x_m$ , at which the intensity is extreme, as a function of the number of the extremum .

3.2 Using the plotted graph, determine with maximum accuracy the value of the wavelength $\lambda$ of the light source. Estimate the error $\Delta\lambda$ of the obtained value.

Отговори

Решение

3.1 From the table of extrema, we select symmetrical points (according to the formulated criteria)

Table 2.

$m$	$t_m$	$x_m$
1	138	-4,088
5	240	-3,382
10	320	-2,476
15	384	-1,586
20	442	-0,698
24	486	0,000
28	529	0,682
33	586	1,556
38	653	2,489
43	730	3,363
47	830	4,071

3.2 For each extremum, using formula (13), we calculate the value of the mirror coordinate $x_m$ , after which we plot the dependence of the mirror coordinate on the extremum number $m$ . This dependence is described by the formula

x_m = m\frac{\lambda}{4}. \qquad (15)

The resulting graph confirms this dependence (shift along the number axis $m$ in this case does not play a role and is due to a different numbering of extrema). The coefficient of the slope of the graph calculated by the least squares is equal to

a = \left(0{,}1771 \pm 0{,}0008\right)\ \mu\mathrm{m}.

It follows from the form of function (15) that the radiation wavelength is equal to

\lambda = 4a = \left(0{,}709 \pm 0{,}003\right)\ \mu\mathrm{m}. \qquad (16)

№	Content	For the part	points
	3. Monochromatic radiation with unknown wavelength	5,0
3.1	Choice of points: - 10 or more points used 0,3 (5 or more -0,1); - utmost points excluded - 0,2; - maximum range used - 0,3; - points are roughly symmetrical - 0,2;		1,0
	extrema coordinates calculation: - using the correct formula for coordinates – 0,5; - the coordinates of extrema calculated with an error of no more than 10% - 1,0 ;		1,5
	graph plotting: (graded if calculations are graded); - axes signed and ticked – 0,1; - all points are plotted in accordance with the table 0,2; - linear dependence is obtained -0.2; - smoothing straight line drawn – 0,2;		0,7
3.2	wavelength calculation: - LSM used – 0,5 (averaging over all point – 0,3 ; 1-2 points used – 0,2); - numerical value in the range 0,70 – 0,72 μm – 0,8 (in the range of 0,68 -0,74 μm - 0,4, out of range – 0);		1,3
	wavelength error calculation: - LSM used – 0,2 (other reasonable method – 0,1); - value of the order $10^{-2}\ \mu m$ - 0,3;		0,5

Оригинал в Архива: IZhO 2023 PhysExp_eng.pdf · официални решения: IZhO-2023-Exp_multi_sol.pdf