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IPhO-37_Expt_Q - 37th International Physics Olympiad...

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Unformatted text preview: 37th International Physics Olympiad Singapore 8 ­ 17 July 2006 Experimental Competition Wed 12 July 2006 Experimental Competition Page 2 List of apparatus and materials Label A ○ B ○ C ○ D ○ E ○ F ○ G ○ H ○ Component Microwave transmitter Quant it y 1 Label Component Lattice structure in a black box Gonio meter Quant it y 1 Microwave receiver 1 Transmitter/receiver ho lder Digital mult imeter 2 J ○ K ○ Prism ho lder 1 L ○ M ○ Rotating table 1 DC power supply for transmitter Slab as a “Thin film” sample Reflector (silver metal sheet) Beam splitter (blue Perspex) Vernier caliper (provided separately) 1 Lens/reflector holder 1 1 N ○ Plano­cylindrical lens 1 1 O ○ Wax prism 2 Blu­Tack 1 pack 1 1 I ○ 30 cm ruler (provided separately) 1 Experimental Competition Page 3 Caution: · The output power of the microwave transmitter is well within standard safety levels. Nevertheless, one should never look directly into the microwave horn at close range when the transmitter is on. · Do not open the box containing the lattice ○. I · The wax prisms ○ are fragile (used in Part 3). O Note: · It is important to note that the microwave receiver output (CURRENT) is proportional to the AMPLITUDE of the microwave. · Always use LO gain setting of the microwave receiver. · Do not change the range of the multimeter during the data collection. · Place the unused components away from the experiment to minimize interference. · Always use the component labels (○, ○, ○,…) to indicate the components A B C in all your drawings. Experimental Competition Black lead Page 4 Red lead The digital mult imeter should be used with the two leads connected as shown in the diagram. You should use the “2m” current setting in this experiment. Experimental Competition Page 5 Part 1: Michelson interferometer 1.1. Introduction In a Michelson interfero meter, a beam splitter sends an inco ming electromagnet ic (EM) wave alo ng two separate paths, and then brings the constituent waves back together after reflect ion so that they superpose, forming an interference pattern. Figure 1.1 illustrates the setup for a Michelson interferometer. An incident wave travels from the transmitter to the receiver along two different paths. These two waves superpose and interfere at the receiver. The strength of signal at the receiver depends on the phase difference between the two waves, which can be varied by changing the optical path difference. Receiver Beam splitter Transmitter Reflectors Figure 1.1: Schemat ic diagram o f a Michelson interfero meter. 1.2. List of components 1) Microwave transmitter ○with ho lder ○ A C 2) Microwave receiver ○ ith ho lder○ B w C 3) Gonio meter ○ J 4) 2 reflectors: reflector ○ ith ho lder ○ and thin film ○ acting as a reflector. G w M F 5) Beam splitter ○ th rotating table ○ acting as a holder H wi L 6) Digital mult imeter ○ D Experimental Competition Page 6 [2 marks] 1.3. Task: Determination of wavelength of the microwave Using only the experimental components listed in Sect ion 1.2, set up a Michelson interfero meter experiment to determine the wavelength l of the microwave in air. Record your data and determine l in such a way that the uncertaint y is ≤0.02 cm. Note that the “thin film” is partially transmissive, so make sure you do not stand or move behind it as this might affect your results. Part 2: “Thin film” interference 2.1. Introduction A beam o f EM wave incident on a dielectric thin film splits into two beams, as shown in Figure 2.1. Beam A is reflected from the top surface of the film whereas beam B is reflected fro m the bottom surface o f the film. The superposition o f beams A and B results in the so called thin film interference. A B q1 q1 n t q2 Figure 2.1: Schemat ic of thin film interference. The difference in the optical path lengths of beam A and B leads to constructive or destructive interference. The resultant EM wave intensit y I depends on the path difference of the two interfering beams which in turn depends on the angle of incidence, q1, of the Experimental Competition Page 7 incident beam, wavelength l of the radiat ion, and the thickness t and refract ive index n o f the thin film. Thus, the refract ive index n of the thin film can be determined fro m I­θ1 plot, using values of t and l. 2.2. List of components 1) Microwave transmitter ○with ho lder ○ A C 2) Microwave receiver ○ ith ho lder○ B w C 3) Plano­cylindrical lens ○ with ho lder ○ N M 4) Gonio meter ○ J 5) Rotating table ○ L 6) Digital mult imeter ○ D 7) Polymer slab act ing as a “thin film” sample ○ F 8) Vernier caliper 2.3. Tasks: Determination of refractive index of polymer slab [6 marks] 1) Derive expressio ns for the condit ions of constructive and destructive interferences in terms o f q1, t, l and n. [1 mark] 2) Using only the experimental co mponents listed in Section 2.2, set up an experiment to measure the receiver output S as a function of the angle of incidence θ1 in the o o range from 40 to 75 . Sketch your experimental setup, clearly showing the angles of incidence and reflect ion and the position o f the film on the rotating table. Mark all co mponents using the labels given on page 2. Tabulate your data. Plot the receiver output S versus the angle o f incidence θ1. Determine accurately the angles corresponding to constructive and destructive interferences. [3 marks] 3) Assuming that the refract ive index of air is 1.00, determine the order of interference m and the refract ive index of the polymer slab n. Write the values of m and n on the answer sheet. Experimental Competition Page 8 [1.5 marks] 4) Carry out error analysis for your results and estimate the uncertaint y of n. Write the value of the uncertainty Δn on the answer sheet. [0.5 marks] Note: · The lens should be placed in front of the microwave transmitter with the planar surface facing the transmitter to obtain a quasi­parallel microwave beam. The distance between the planar surface of the lens and the aperture of transmitter horn should be 3 cm. · For best results, maximize the distance between the transmitter and receiver. · Deviations of the microwave emitted by transmitter from a plane wave may cause o extra peaks in the observed pattern. In the prescribed range from 40 to 75 o, only one maximum and one minimum exist due to interference. Part 3: Frustrated Total Internal Reflection 3.1. Introduction The pheno menon of total internal reflection (TIR) may occur when the plane wave travels fro m an optically dense medium to less dense medium. However, instead of TIR at the interface as predicted by geo metrical optics, the inco ming wave in realit y penetrates into the less dense medium and travels for some distance parallel to the interface before being scattered back to the denser medium (see Figure 3.1). This effect can be described by a shift D of the reflected beam, known as the Goos­Hänchen shift. P rism n 1 q D 1 n 2 Air Figure 3.1: A sketch illustrating an EM wave undergoing total internal reflect ion in a prism. The shift D parallel to the surface in air represents the Goos­Hänchen shift Experimental Competition Page 9 Prism n 1 Transmitter q D 1 z n 2 Air n 1 d Receiver Prism Figure 1.2: A sketch of the experimental setup showing the prisms and the air gap of distance d. The shift D parallel to the surface in air represents the Goos­Hänchen shift. z is the distance fro m the tip of the prism to the central axis o f the transmitter. If another medium o f refractive index n1 (i.e. made of the same material as the first medium) is placed at a small distance d to the first medium as shown in Figure 3.2, tunneling of the EM wave through the second medium occurs. This intriguing pheno menon is known as the frustrated total internal reflection (FTIR). The intensit y o f the transmitted wave, It, decreases exponent ially wit h the distance d: I t = I 0 exp ( - g d ) 2 (3.1) where I0 is the intensit y of the incident wave and g is: g= 2p l 2 n1 sin 2 q1 - 1 2 n2 (3.2) where l is the wavelength of EM wave in medium 2 and n2 is the refract ive index of medium 2 (assume that the refract ive index of medium 2, air, is 1.00). 3.2. List of components 1) Microwave transmitter ○ with ho lder ○ A C Experimental Competition Page 10 2) Microwave receiver ○ with holder ○ B C 3) Plano­cylindrical lens ○ with ho lder ○ N M 4) 2 equilateral wax prisms ○ with ho lder ○ and rotating table ○ acting as a O K L ho lder 5) Digital mult imeter ○ D 6) Gonio meter○ J 7) Ruler 3.3. Description of the Experiment Using only the list of components described in Section 3.2, set up an experiment to investigate the variat ion of the intensit y It as a function of the air gap separat ion d in FTIR. For consistent results, please take note of the fo llo wing: · Use one arm of the gonio meter for this experiment . · Choose the prism surfaces carefully so that they are parallel to each other. · The distance fro m the centre of the curved surface of the lens should be 2 cm fro m the surface of the pr ism. · Place the detector such that its horn is in contact wit h the face of the prism. · For each value of d, adjust the posit ion of the microwave receiver alo ng the prism surface to obtain the maximum signal. · Make sure that the digital mult i­meter is on the 2mA range. Collect data starting fro m d = 0.6 cm. Discontinue the measurements when the reading of the mult imeter falls below 0.20 mA. 3.4. Tasks: Determination of refractive index of prism material [6 marks] Task 1 Sketch your final experimental setup and mark all components using the labels given at page 2. In your sketch, record the value of the distance z (see Figure 3.2), the distance fro m the tip of the pr ism to the central axis of the transmitter. [1 Mark] Task 2 Perform your experiment and tabulate your data. Perform this task twice. [2.1 Marks] Experimental Competition Page 11 Task 3 (a) By plotting appropriate graphs, determine the refractive index, n1, of the prism with error analys is. (b) Write the refract ive index n1, and its uncertaint y ∆n1, of the prism in the answer sheet provided. [2.9 Marks] Part 4: Microwave diffraction of a metal­rod lattice: Bragg reflection 4.1. Introduction Bragg’s Law The lattice structure of a real crystal can be examined using Bragg’s Law, 2d sin q = m l (4.1) where d refers to the distance between a set of parallel crystal planes that “reflect” the X­ray; m is the order of diffract ion and q is the angle between the incident X­ray beam and the crystal planes. Bragg’s law is also commo nly known as Bragg’s reflect ion or X­ray diffract ion. Experimental Competition Page 12 Metal­rod lattice Because the wavelength of the X­ray is co mparable to the lattice constant of the crystal, traditional Bragg’s diffraction experiment is performed using X­ray. For microwave, however, diffract ion occurs in lattice structures wit h much larger lattice constant, whic h can be measured easily wit h a ruler. a b d z y x Figure 4.1: A metal­rod lattice o f lattice constants a and b, and interplanar spacing d. a b d y x Figure 4.2: Top­view o f the metal­rod lattice shown in Fig. 4.1 (not to scale). The lines denote diagonal planes of the lattice. Experimental Competition Page 13 In this experiment, the Bragg law is used to measure the lattice constant of a lattice made of metal rods. An example of such metal­rod lattice is shown in Fig. 4.1, where the metal rods are shown as thick vert ical lines. The lattice planes along the diagonal direct ion of the xy­plane are shown as shaded planes. Fig. 4.2 shows the top­view (looking down along the z­axis) o f the metal­rod lattice, where the points represent the rods and the lines denote the diagonal lattice planes. 4.2. List of components 1) Microwave transmitter ○ with ho lder ○ A C 2) Microwave receiver ○ with holder ○ B C 3) Plano­cylindrical lens ○ with ho lder ○ N M 4) Sealed box containing a metal­rod lattice ○ I 5) Rotating table ○ L 6) Digital mult imeter ○ D 7) Gonio meter○ J a a z y x Figure 4.3: A simple square lattice. In this experiment, you are given a simple square lattice made o f metal rods, as illustrated in Fig. 4.3. The lattice is sealed in a box. You are asked to derive the lattice constant a of Experimental Competition Page 14 the lattice fro m the experiment. DO NOT open the box. No marks will be given to the experimental results if the seal is found broken after the experiment. 4.3. Tasks: Determination of lattice constant of given simple square lattice [6 Marks] Task 1 Draw a top­view diagram o f the simple square lattice shown in Fig. 4.3. In the diagram, indicate the lattice constant a of the given lattice and the interplanar spacing d of the diagonal planes. With the help of this diagram, derive Bragg’s Law. [1 Mark] Task 2 Using Bragg’s law and the apparatus provided, design an experiment to perform Bragg diffract ion experiment to determine the lattice constant a of the lattice. (a) Sketch the experimental set up. Mark all co mponents using the labels in page 2 and indicate clearly the angle between the axis o f the transmitter and lattice planes, q, and the angle between the axis o f the transmitter and the axis of the receiver, z. In your experiment, measure the diffract ion on the diagonal planes the direction o f which is indicated by the red line on the box. [1.5 Marks] (b) Carry out the diffraction experiment for 20° ≤ q ≤ 50°. In this range, you will only observe the first order diffract ion. In the answer sheet, tabulate your results and record both the q and z. [1.4 Marks] (c) Plot the quant ity proportional to the intensit y o f diffracted wave as a function o f q. [1.3 Marks] (d) Determine the lattice constant a using the graph and estimate the experimental error. [0.8 Marks] Experimental Competition Page 15 Note: 1. For best results, the transmitter should remain fixed during the experiment. The separation between the transmitter and the lattice, as well as that between lattice and receiver should be about 50 cm. 2. Use only the diagonal planes in this experiment. Your result will not be correct if you try to use any other planes. 3. The face of the lattice box with the red diagonal line must be at the top. 4. To determine the position of the diffraction peak with better accuracy, use a number of data points around the peak position. ...
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