ipho 2006 xxxvii singapore singaporeipho-37_expt_a_W

1、The 37th International Physics Olympiad SingaporeExperimental CompetitionWednesday, 12 July, 2006Sample SolutionPart 1a. A sketch of the experimental setup (not required)ReceiverRotating tableGoniometerMovable armGoniometer Fixed armBeam splitterTransmitterHolderReflectorb. Data sheet (not required)

2、2Position (cm)Meter reading (mA)Position (cm)Meter reading (mA)Position (cm)Meter reading (mA)Position (cm)Meter reading (mA)104.00.609100.91.01696.00.51491.00.925103.90.817100.851.06095.80.09890.91.094103.80.933100.81.09095.60.19290.81.245103.71.016100.70.99495.40.66990.71.291103.61.030100.60.94095

3、.30.87090.61.253103.50.977100.40.67395.21.00990.40.978103.40.890100.20.24995.11.11990.20.462103.30.738100.00.07495.01.13890.00.045103.20.54899.80.45794.91.08089.80.278103.10.31099.60.88394.70.78189.60.809103.00.14599.41.09594.50.40389.51.031102.90.07699.31.11194.30.04489.41.235102.80.17999.21.02294.

4、10.36489.31.277102.70.39299.00.78793.90.86089.21.298102.60.62398.80.35993.71.10389.11.252102.50.78698.60.07993.61.16089.01.133102.40.91898.40.41493.51.15988.80.684102.30.98898.20.86493.41.08388.60.123102.21.02698.01.12893.20.75388.50.020102.11.00697.91.18393.00.33188.40.123102.00.94597.81.13292.80.0

5、7388.20.679101.90.74797.71.01592.60.51588.01.116101.80.59797.50.71392.40.96887.91.265101.70.36397.20.09092.21.21787.81.339101.60.16197.00.34292.151.23487.71.313101.50.05596.80.71492.11.23087.61.190101.40.13996.61.00792.01.16587.40.867101.30.35796.51.08791.80.87187.20.3162.01.687.8 cm1.2103.6 cm0.80.

6、40.0 0.4889092949698100102104position (cm)3Meter reading (mA)101.20.58996.41.07091.60.35387.10.034101.10.78196.31.01891.40.01887.00.018101.00.95496.20.86591.20.39486.90.178104.00.609100.91.01696.00.51491.00.925103.90.817100.81.06095.80.09890.91.094103.80.933100.81.09095.60.19290.81.245103.71.016100.

7、70.99495.40.66990.71.291103.61.030100.60.94095.30.87090.61.253103.50.977100.40.67395.21.00990.40.978103.40.890100.20.24995.11.11990.20.462103.30.738100.00.07495.01.13890.00.045103.20.54899.80.45794.91.08089.80.278103.10.31099.60.88394.70.78189.60.809103.00.14599.41.09594.50.40389.51.031102.90.07699.

8、31.11194.30.04489.41.235102.80.17999.21.02294.10.36489.31.277102.70.39299.00.78793.90.86089.21.298102.60.62398.80.35993.71.10389.11.252102.50.78698.60.07993.61.16089.01.133102.40.91898.40.41493.51.15988.80.684102.30.98898.20.86493.41.08388.60.123102.21.02698.01.12893.20.75388.50.020102.11.00697.91.1

9、8393.00.33188.40.123102.00.94597.81.13292.80.07388.20.679101.90.74797.71.01592.60.51588.01.116101.80.59797.50.71392.40.96887.91.265101.70.36397.20.09092.21.21787.81.339101.60.16197.00.34292.151.23487.71.313101.50.05596.80.71492.11.23087.61.190101.40.13996.61.00792.01.16587.40.867101.30.35796.51.0879

10、1.80.87187.20.316101.20.58996.41.07091.60.35387.10.034101.10.78196.31.01891.40.01887.00.018101.00.95496.20.86591.20.39486.90.178From the graph (not required) or otherwise, the positions of the first maximum point and 12th maximum point are measured at 87.8 cm and 103.6 cm.The wavelength is calculate

11、d byl = 103.6 - 87.8 cm211Thus,Error analysis2l =d ,11Dd = 0.05 x2 cm = 0.1 cm.= 2 Dd11= 2 ? 0.10 = 0.01811Dlcm 0.02cm40.2 marksl = 2.87 cm1.8 marksnq2q1q1Part 2Deduction of interference conditions(a)ABntq2Assume that the thickness of the film is t and refractive index n. Let q1 be the incidentangle

12、 and q 2 the refracted angle. The difference of the optical paths DL is:DL = 2(nt / cosq2 - t tanq 2 sinq1 )Law of refraction:sin q1 = n sin q2ThusDL = 2t n 2 - sin 2 q 1Considering the 180 deg (p) phase shift at the airthin film interface for the reflectedbeam, we have interference conditions:2t n

13、2 - sin 2 q min = ml(m = 1,2,3,.)for the destructive peak2t n 2 - sin 2 q= (m ? 1 )landfor the constructive peakmax2If thickness t and wave length l are known, one can determine the refractive index of thethin film from I q 1spectrum (I is the intensity of the interfered beam).51 markq1q1(b)A sketch

14、 of the experimental setupReceiverGoniometerMovable armPlanoconvex cylinder lensRotating tableThin filmGoniometerFixed armTransmitterStudents should use the labeling on Page 2.(c)Data Set6X: q1 / degreeY: Meter reading S/mA40.00.30941.00.27042.00.22643.00.19644.00.16445.00.11446.00.06347.00.03648.00

15、.02249.00.03950.00.06651.00.13552.00.21553.00.26254.00.32155.00.39156.00.45457.00.5111 markU1 mA1o1.270.5o1.00.80.60.40.20.048o3540455055q ( o )6065707580From the data,min andmax can be found at 48oand 70.5 roespectively.To calculate the refractive index, the following equations are used:7Meter read

16、ing (mA)0.6 marks0.9 marks0.5 marks58.00.56659.00.62260.00.66461.00.69162.00.72263.00.75464.00.79665.00.83166.00.83667.00.86068.00.90469.00.97070.01.02271.01.01872.00.92673.00.80074.00.77075.0ncertainty: angle Dq0=.9?15 , current: ?0.000.52t n 2 - sin 2 48o = ml(m = 1, 2,3,.)(1)2t n 2 - sin 2 70.5o

17、= (m - 1 )land(2)2In this experiment, t = 5.28 cm, l = 2.85cm (measured using other method).Solving the simultaneous equations (1) and (2), we getsin 2 70.5o - sin 2 48om =+ 0.25l2t) 2(m = 4.83Substituting m = 5 in (1), we get n = 1.54m = 5Substituting m = 5 in (2), we also get n = 1.54Error analysi

18、s:n =sin 2 q + ( ml )22tm 2lm2 l21(sin 2q ? Dq +Dl -Dn =Dt)ml2t 22t 3sin 2 q + ()22tm 2lm2 l2= 1(sin 2q ? Dq +Dl -Dt)2t 22t 3nIf we take Dq = 0.5 o = 0.0087 rad, Dt = 0.05 cm, Dl = 0.02 cm, and q =52 ? 2.8552 ?2.8521Dn =(0.0087 sin 96 +o? 0.01 +?0.05) ? 0.022 ? 5.2822 ? 5.2831.54Thus,80.5 marksn + D

19、n = 1.54 0.020.5 marks1 markPart 3Sample SolutionTask 1Sketch your final experimental setup and mark all components using the labels given at page 2. In your sketch, write down the distance z (see Figure 3.2), where z is the distance from the tip of the prism to the central axis of the transmitter.T

20、ransmitterzLensPrismPrismdReceiver(Students should use labels on page 2.)Task 2Tabulate your data. Perform the experiment twice.Data Set# DS = 0.01 mA (for each set of current measurements)* S2 proportional to the intensity, It$ D(S2) = DIt = 2 S ? DS& DY = D(lnIt) = D(It)/It9X: d(cm)DX(cm)Set 1 S1

21、(mA)Set 2 S2(mA)Saverage(mA)DS(mA)#It (mA)2*D(It)$Y: ln(It (mA) 2)DY&0.600.050.780.780.7800.010.60800.0160.500.030.700.050.680.690.6850.010.46900.0140.760.030.800.050.580.590.5850.010.34200.0121.070.030.900.050.500.510.5050.010.25500.0101.370.041.000.050.420.420.4200.010.17600.0081.740.051.100.050.3

22、60.350.3550.010.12600.0072.070.061.200.050.310.310.3100.010.09610.0062.340.061.300.050.260.250.2550.010.06500.0052.730.081.400.050.210.220.2150.010.04620.0043.070.09Task 3By plotting appropriate graphs, determine the refractive index, n1, of the prism with error analysis. Write the refractive index

23、n1, and its uncertainty n1, of the prism in the answer sheet provided.Least Square Fitting10X = d(cm)DX(cm)Y = ln(It)DYDY2XYX2Y20.600.050.500.030.0010.2980.3600.2470.700.050.760.030.0010.5300.4900.5730.800.051.070.030.0010.8580.6401.1500.900.051.370.040.0021.2300.8101.8671.000.051.740.050.0021.7351.

24、0003.0101.100.052.070.060.0032.2781.2104.2901.200.052.340.060.0042.8111.4405.4871.300.052.730.080.0063.5531.6907.4691.400.053.070.090.0094.3041.9609.451SX =SY =SDY =S(DY)2 =SXY =SX2 =SY2 =9.0015.6480.4690.02917.5969.60033.544= I0 exp(-2g d ), taking natural log on both sides, we obtain:ln(It ) = -2g

25、 d + ln(I0 )which is of the form y = mx + c.FromItTo calculate the gradient, the following equation was used, where N is the number of data points:m = N ?( XY ) - (? X )(?Y ) = -3.247N ? X 2 - (? X )2To calculate the standard deviation sY of the individual Y data equation was used:values, thefollowi

26、ng?(DY )2o Y = 0.064N - 2Hence the standard deviation in the slope can be calculated:NN ? X 2 - (? X )2o= s= 0.082mYFrom the gradient:2g = 3.247? 0.082? 3.25 ? 0.08Using:k + g22n =21k sin q21where q1 = 60o, k2= 2p/l ? 2.20 (using the wavelength determinedfrom earlier part(using l = (2.85 ? 0.02)cm),

27、 we obtain:11n1 ? Dn1 = 1.434 ? 0.016? 1.43 ? 0.02Error Analysis for refractive index of n1?()?()? 1 1+ g+ g222 22k2k2dd?Dk?DgDn =+2?dkk sinqdgk sinq122?21?21?(22 )- 12(2 )12 ?(22 )- 12 ?k + gk + gg k+ g2= ?Dk+ ?Dg222Dn-?sinqk sinqk sinq12212121= 0.016? 0.02where:Dk = - 2p Dl = -0.0152l2Note: Other

28、methods of error analysis are also accepted.12Part 4Task 1Topview of a simple square lattice.aFigure 4.1: Schematic diagram of a simple square lattice with lattice constant a and interplaner d of the diagonal planes indicated.Deriving Braggs LawConditions necessary for the observation of diffraction

29、 peaks:1. The angle of incidence = angle of scattering.2. The pathlength difference is equal to an integer number of wavelengths.130.5 marksdFigure 4.2: Schematic diagram for deriving Braggs law.h = d sinq(1).The path length difference is given by,2h = 2d sinq(2).For diffraction to occur, the path d

30、ifference must satisfy,m = 1, 2, 3.(3).Figure 4.3 Illustration of the lattice used in the experiment (this Figure is not required)140.5 marksad2 d sinq = ml,Fig. 4.4 Actual lattice used for the experiment (this Figure is not required)Fig. 4.4 The actual lattice used in the experiment (not required)T

31、ask 2 (a)Lattice BoxonLIRotating TablePlanocylindrical Lens on HolderINJqDigital MultimeterMicrowave Transmitter on HolderDz = 180? 2qBMicroware Receiver on HolderFig. 4.5 Sketch of the experimental set up151.5 marksA0.140.12)0.12AWW (m(ymmst 0.08 iytiytinsstenene ni tteInIn0.06v iteveval titialalRe

32、ee 0.040.02016 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56Theta (degree)Task 2(b) & 2(c)Data Set0.140.120.10.080.060.040.02016 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56Theta (degree)Task 2(d)From eq 3 and let m = 1,2d sinqmax = l(4)From Fig. 4.3,16Relative intensity

33、 (mA)22.7 marksq/?z/?Output current S (mA)Intensity I=S2 (mA)220.0140.00.0230.00052921.0138.00.0380.00144422.0136.00.0700.004923.0134.00.1090.01188124.0132.00.1630.02656925.0130.00.2010.04040126.0128.00.2330.05428927.0126.00.2750.07562528.0124.00.3200.102429.0122.00.3500.122530.0120.00.3530.12460931

34、.0118.00.3580.12816432.0116.00.3540.12531633.0114.00.3420.11696434.0112.00.3210.10304135.0110.00.3030.09180936.0108.00.2800.078437.0106.00.2410.05808138.0104.00.2000.0439.0102.00.1830.03348940.0100.00.1620.02624441.098.00.1390.01932142.096.00.1200.014443.094.00.1090.01188144.092.00.0860.00739645.090.00.0660.00435646.088.00.0670.00448947.086.