Съдържание

IPhO 2014, Експеримент — Задача 4. Part 2.3 Liquid crystal cell

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Всички материали от IPhO 2014

Experimental competition. Thursday, 17 July 2014 — Experiment. To see invisible! (20 points) · 17 юли 2014 г. · 4,5 т.

Условие

Light transmission through LCC

Place the LCC between the polarizers as shown in Fig. 6B.

The experimental dependence under investigation is strongly nonmonotonic with domains of quite abrupt changes. Take this into account when taking measurements.
To measure the AC voltage of the cell power supply and the DC voltage of the photodetector, connect the appropriate leads directly to the multimeter.

2.3.1 Measure the intensity of the transmitted light as a function of the voltage across the LCC. Plot the corresponding graph. (2.0 points)

2.3.2 Calculate the phase difference between the ordinary and extraordinary waves $\Delta\varphi_0$ when the power supply is disconnected with the LCC. (1.5 points)

2.3.3 In a sufficiently wide range of the voltage drop across the LCC, the phase difference between the ordinary and extraordinary waves depends on the voltage applied by the power law

\Delta\varphi = CU^{\beta}.

Using the data obtained, plot the graph which allows one to determine the applicability range of the above formula and calculate the exponent $\beta$ . Specify that range of applicability and evaluate the numerical value of the parameter $\beta$ . (1.0 point)

Отговори

Решение

Part 2.3 Liquid crystal cell

2.3.2 Light transmission through LCC

2.3.1 Results of the intensity measurements as a functions of voltage $U_{LC}$ are shown in table 5 $^{1}$ . Graph of the obtained function is drawn in the figure below.

Table 5.

$U_{LC}, V$	$U, mV$	$(\Delta\varphi)'$	$\Delta\varphi$	$\ln U_{LC}$	$\ln\Delta\varphi$
0	207	1,961	10,606	-	2,361
0,86	207	1,961	10,606	-0,151	2,361
0,91	211	1,990	10,577	-0,094	2,359
0,93	226	2,102	10,464	-0,073	2,348
0,94	237	2,190	10,377	-0,062	2,340
1,02	294	2,858	9,709	0,020	2,273
1,07	297	2,941	9,625	0,068	2,264
1,09	294	2,858	9,709	0,086	2,273
1,11	285	2,691	8,974	0,104	2,194
1,18	201	1,918	8,201	0,166	2,104
1,23	110	1,301	7,584	0,207	2,026
1,27	51	0,850	7,133	0,239	1,965
1,29	26	0,598	6,881	0,255	1,929
1,31	10	0,367	6,650	0,270	1,895
1,32	5	0,259	6,542	0,278	1,878
1,36	2	0,163	6,447	0,307	1,864
1,39	12	0,403	5,880	0,329	1,772
1,42	28	0,621	5,662	0,351	1,734
1,46	66	0,976	5,307	0,378	1,669
1,5	102	1,245	5,038	0,405	1,617
1,55	156	1,611	4,672	0,438	1,542
1,63	232	2,149	4,134	0,489	1,419
1,68	261	2,404	3,879	0,519	1,356
1,71	275	2,556	3,727	0,536	1,316
1,78	289	2,756	3,527	0,577	1,260
1,83	294	2,858	3,425	0,604	1,231
1,93	295	2,883	3,401	0,658	1,224
2,01	294	2,858	2,858	0,698	1,050
2,11	287	2,722	2,722	0,747	1,001
2,24	273	2,532	2,532	0,806	0,929
2,34	258	2,375	2,375	0,850	0,865
2,51	229	2,125	2,125	0,920	0,754
2,65	202	1,925	1,925	0,975	0,655
2,72	191	1,848	1,848	1,001	0,614
2,85	169	1,698	1,698	1,047	0,529
2,92	159	1,631	1,631	1,072	0,489
3,05	141	1,511	1,511	1,115	0,413
3,16	128	1,424	1,424	1,151	0,353
3,22	121	1,376	1,376	1,169	0,319
3,34	109	1,294	1,294	1,206	0,258
3,45	98	1,217	1,217	1,238	0,196
3,59	86	1,130	1,130	1,278	0,122
3,66	81	1,093	1,093	1,297	0,089
3,75	74	1,039	1,039	1,322	0,039
3,83	70	1,008	1,008	1,343	0,008
3,91	65	0,968	0,968	1,364	-0,032
4,03	58	0,911	0,911	1,394	-0,094
4,21	50	0,841	0,841	1,437	-0,173
4,43	41	0,757	0,757	1,488	-0,278
4,79	30	0,644	0,644	1,567	-0,441
4,98	25	0,586	0,586	1,605	-0,535
5,15	21	0,536	0,536	1,639	-0,625
5,51	15	0,451	0,451	1,707	-0,796
5,79	11	0,385	0,385	1,756	-0,954
6,07	8	0,328	0,328	1,803	-1,115
6,59	4	0,231	0,231	1,886	-1,463
6,94	2	0,163	0,163	1,937	-1,811

$^{1}$ We do not expect that participants can take the same number of measurements, 15-20 points are enough. It is principally important to find the dip in the graph.

Graph U, mV (0–350) versus U_LC, V (0–7): constant about 207 mV up to 0,9 V, a sharp peak near 1,07 V, a dip to almost zero near 1,36 V, a second maximum near 1,9 V and a slow decrease to zero at 7 V.

It is important to choose correct roots of equation (2) in order to adequately calculate phase shifts. In this case it is rather obvious because at large values of $U_{LC}$ the voltage difference tends to zero, $\Delta\varphi \to 0$ . Other solutions and corresponding equations are shown in the figure below.

The U, mV versus U_LC, V graph matched by arrows to a sin² curve of Δφ below it, with callouts for the branches Δφ = 4π − 2 arcsin√(U/U_max), Δφ = 2 arcsin√(U/U_max) + 2π, Δφ = 2π − 2 arcsin√(U/U_max), Δφ = 2 arcsin√(U/U_max) and Δφ → 0.

Results of calculation of $(\Delta\varphi)' = \arcsin\sqrt{\frac{U}{U_{max}}}$ and correct values of phase shift $\Delta\varphi$ are shown in the figure below.

Two graphs versus U_LC, V (0–7): (Δφ)' (0,0–1,6) with a scattered double peak, and below it the corrected Δφ (up to about 10,6) decreasing smoothly toward zero.

This figure is drawn for the sake of understanding. (It’s not required to draw it for participants).

2.3.2 The value of the phase shift at zero voltage is $\Delta\varphi_0 \approx 10.6$ .

2.3.3 In order to check applicability of the power function $\Delta\varphi = CU^{\beta}$ it is recommended to redraw the last graph logarithmically, as shown in the figure below.

Graph ln Δφ (−2,0 to 2,5) versus ln U_LC (−0,5 to 2,0): points lie on a straight falling line from about ln U_LC = 0 to 1,6, with deviations at both ends.

It can be seen from the graph that in the range of 1 V to 5 V the function is almost linear, which justifies the applicability of the power law. The power in that equation is equal to the slope of the graph, its numerical value is $\beta \approx 1.75$ .

Оригинал в Архива: 2014_IPhO_Experimenta_Competition.pdf · официални решения: 2014_IPhO_Solution_to_Experimental_Competition.pdf