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Theory Question 1 Page 1 of 5 Theory Question 1: Gravity in a Neutron Interferometer Enter all your answers into the Answer Script. a a a a 2 θ 2 θ BS BS M M BS - Beam Splitters M - Mirror IN OUT2 OUT1 Figure 1a φ OUT1 IN OUT2 Figure 1b Physical situation We consider the situation of the famous neutron-interferometer experiment by Collela, Overhauser and Werner, but idealize the set-up inasmuch as we shall assume perfect beam splitters and mirrors within the interferometer. The experiment studies the effect of the gravitational pull on the de Broglie waves of neutrons. The symbolic representation of this interferometer in analogy to an optical interferometer is shown in Figure 1a. The neutrons enter the interferometer through the IN port and follow the two paths shown. The neutrons are detected at either one of the two output ports, OUT1 or OUT2. The two paths enclose a diamond-shaped area, which is typically a few cm 2 in size. The neutron de Broglie waves (of typical wavelength of 10 −10 m) interfere such that all neutrons emerge from the output port OUT1 if the interferometer plane is horizontal. But when the interferometer is tilted around the axis of the incoming neutron beam by angle φ (Figure 1b), one observes a φ dependent redistribution of the neutrons between the two output ports OUT1 and OUT2. #1 _IPO_2006, Sưu tầm và đóng gói bởi, Gv. Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 Theory Question 1 Page 2 of 5 Geometry For °= 0 φ the interferometer plane is horizontal; for °= 90 φ the plane is vertical with the output ports above the tilt axis. 1.1 (1.0) How large is the diamond-shaped area A enclosed by the two paths of the interferometer? 1.2 (1.0) What is the height H of output port OUT1 above the horizontal plane of the tilt axis? Express A and H in terms of , a θ , and φ . Optical path length The optical path length N opt (a number) is the ratio of the geometrical path length (a distance) and the wavelength λ . If λ changes along the path, N opt is obtained by integrating along the path. 1− λ 1.3 (3.0) What is the difference ΔN opt in the optical path lengths of the two paths when the interferometer has been tilted by angle φ ? Express your answer in terms of , a θ , and φ as well as the neutron mass M, the de Broglie wavelength 0 λ of the incoming neutrons, the gravitational acceleration g, and Planck’s constant . h 1.4 (1.0) Introduce the volume parameter 2 2 gM h V = and express ΔN opt solely in terms of A , V , 0 λ , and φ . State the value of V for M = 1.675×10 −27 kg, g = 9.800 m s −2 , and h = 6.626 × 10 −34 J s. 1.5 (2.0) How many cycles — from high intensity to low intensity and back to high intensity — are completed by output port OUT1 when φ is increased from °−= 90 φ to °= 90 φ ? Experimental data The interferometer of an actual experiment was characterized by a = 3.600 cm and °= 10.22 θ , and 19.00 full cycles were observed. 1.6 (1.0) How large was 0 λ in this experiment? 1.7 (1.0) If one observed 30.00 full cycles in another experiment of the same kind that uses neutrons with 0 λ = 0.2000 nm, how large would be the area A ? Hint: If 1<<x α , it is permissible to replace by () α x+1 x α + 1 . #2 _IPO_2006, Sưu tầm và đóng gói bởi, Gv. Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 Answer Script Theory Question 1 Page 3 of 5 Country Code Student Code Question Number 1 For Examiners Use Only 1.0 1.0 Answer Script Geometry 1.1 The area is =A 1.2 The height is = H #3 _IPO_2006, Sưu tầm và đóng gói bởi, Gv. Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 Answer Script Theory Question 1 Page 4 of 5 Country Code Student Code Question Number 1 Optical path length 1.3 In terms of , a θ , φ , M , 0 λ , g , and : h ΔN opt = 1.4 In terms of A , V , 0 λ , and φ : ΔN opt = The numerical value of V is =V 1.5 The number of cycles is # of cycles = For Examiners Use Only 3.0 0.8 0.2 2.0 #4 _IPO_2006, Sưu tầm và đóng gói bởi, Gv. Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 Answer Script Theory Question 1 Page 5 of 5 Country Code Student Code Question Number 1 Experimental data For Examiners Use Only 1.0 1.6 The de Broglie wavelength was = 0 λ 1.7 The area is =A 1.0 #5 _IPO_2006, Sưu tầm và đóng gói bởi, Gv. Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 SOLUTIONS to Theory Question 1 Geometry Each side of the diamond has length L = a cos θ and the dis- tance between parallel sides is D = a cos θ sin(2θ) = 2a sin θ. The area is the product thereof, A = LD, giving 1.1 A = 2a 2 tan θ . The height H by which a tilt of φ lifts OUT1 above IN is H = D sin φ or 1.2 H = 2a sin θ sin φ . Optical path length Only the two parallel lines for IN and OUT1 matter, each having length L. With the de Broglie wavelength λ 0 on the IN side and λ 1 on the OUT1 side, we have ∆N opt = L λ 0 − L λ 1 = a λ 0 cos θ 1 − λ 0 λ 1 . The momentum is h/λ 0 or h/λ 1 , respectively, and the statement of energy conservation reads 1 2M h λ 0 2 = 1 2M h λ 1 2 + MgH , which implies λ 0 λ 1 = 1 − 2 gM 2 h 2 λ 2 0 H . Upon recognizing that (gM 2 /h 2 )λ 2 0 H is of the order of 10 −7 , this simplifies to λ 0 λ 1 = 1 − gM 2 h 2 λ 2 0 H , and we get ∆N opt = a λ 0 cos θ gM 2 h 2 λ 2 0 H or 1 #6 _IPO_2006, Sưu tầm và đóng gói bởi, Gv. Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 1.3 ∆N opt = 2 gM 2 h 2 a 2 λ 0 tan θ sin φ . A more compact way of writing this is 1.4 ∆N opt = λ 0 A V sin φ , where 1.4 V = 0.1597 × 10 −13 m 3 = 0.1597 nm cm 2 is the numerical value for the volume parameter V . There is constructive interference (high intensity in OUT1) when the optical path lengths of the two paths differ by an integer, ∆N opt = 0, ±1, ±2, . . ., and we have destructive interference (low intensity in OUT1) when they diff er by an integer plus half, ∆N opt = ± 1 2 , ± 3 2 , ± 5 2 , . . . . Changing φ from φ = −90 ◦ to φ = 90 ◦ gives ∆N opt φ=90 ◦ φ=−90 ◦ = 2λ 0 A V , which tell us that 1.5 of cycles = 2λ 0 A V . Experimental data For a = 3.6 cm and θ = 22.1 ◦ we have A = 10.53 cm 2 , so that 1.6 λ 0 = 19 × 0.1597 2 × 10.53 nm = 0.1441 nm . And 30 full cycles for λ 0 = 0.2 nm correspond to an area 1.7 A = 30 × 0.1597 2 × 0.2 cm 2 = 11.98 cm 2 . 2 #7 _IPO_2006, Sưu tầm và đóng gói bởi, Gv. Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 Theory Question 2 Page 1 of 5 Theory Question 2: Watching a Rod in Motion Enter all your answers into the Answer Script. pinhole camera D ro d x v 0 Physical situation A pinhole camera, with the pinhole at 0 = x and at distance from the D x axis, takes pictures of a rod, by opening the pinhole for a very short time. There are equidistant marks along the x axis by which the apparent length of the rod, as it is seen on the picture, can be determined from the pictures taken by the pinhole camera. On a picture of the rod at rest, its length is . However, the rod is not at rest, but is moving with constant velocity L υ along the x axis. Basic relations A picture taken by the pinhole camera shows a tiny segment of the rod at position . x ~ 2.1 (0.6) What is the actual position x of this segment at the time when the picture is taken? State your answer in terms of x ~ , , , D L υ , and the speed of light =3.00×10c 8 m s -1 . Employ the quantities c υ β = and 2 1 1 β γ − = if they help to simplify your result. 2.2 (0.9) Find also the corresponding inverse relation, that is: express x ~ in terms of x , , D L , υ , and . c Note: The actual position is the position in the frame in which the camera is at rest Apparent length of the rod The pinhole camera takes a picture at the instant when the actual position of the center of the rod is at some point . 0 x 2.3 (1.5) In terms of the given variables, determine the apparent length of the rod on this picture. 2.4 (1.5) Check one of the boxes in the Answer Script to indicate how the apparent length changes with time. #8 _IPO_2006, Sưu tầm và đóng gói bởi, Gv. Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 Theory Question 2 Page 2 of 5 Symmetric picture One pinhole-camera picture shows both ends of the rod at the same distance from the pinhole. 2.5 (0.8) Determine the apparent length of the rod on this picture. 2.6 (1.0) What is the actual position of the middle of the rod at the time when this picture is taken? 2.7 (1.2) Where does the picture show the image of the middle of the rod? Very early and very late pictures The pinhole camera took one picture very early, when the rod was very far away and approaching, and takes another picture very late, when the rod is very far away and receding. On one of the pictures the apparent length is 1.00 m, on the other picture it is 3.00 m. 2.8 (0.5) Check the box in the Answer Script to indicate which length is seen on which picture. 2.9 (1.0) Determine the velocity υ . 2.10 (0.6) Determine the length L of the rod at rest. 2.11 (0.4) Infer the apparent length on the symmetric picture. #9 _IPO_2006, Sưu tầm và đóng gói bởi, Gv. Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 Answer Script Theory Question 2 Page 3 of 5 Country Code Student Code Question Number 2 For Examiners Use Only 0.6 0.9 1.5 1.5 Answer Script Basic Relations 2.1 x value for given value: x ~ x = 2.2 x ~ value for given x value: x ~ = Apparent length of the rod 2.3 The apparent length is )( ~ 0 xL = 2.4 Check one: The apparent length increases first, reaches a maximum value, then decreases. decreases first, reaches a minimum value, then increases. decreases all the time. increases all the time. #10 _IPO_2006, Sưu tầm và đóng gói bởi, Gv. Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 [...]... your data and determine l in such a way that the uncertainty 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 of EM wave incident on a dielectric thin film splits into two beams, as shown in Figure 2.1. Beam A is reflected from the ... Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 Experimental Competition Page 7 incident beam, wavelength l of the radiation, and the thickness t and refractive index n of the thin film. Thus, the refractive index n of the thin film can be determined from Iθ1 plot, using values of t and l. 2.2. List of components 1) Microwave transmitter ○ A with holder ○ C 2) Microwave receiver ... boiling water to the egg through the surface of the egg This gives 2 #26 _IPO_2006, Sưu tầm và đóng gói bởi, Gv Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 3.7 P = 4πR2 J = 4πκR(T1 − T0 ) ≈ 19.3 W for the amount of energy transferred to the egg per unit time From this we get an estimate for the time τ required for the... the slab has volume V (z) = Ah(z) where A is the cross-sectional area and h(z) is the thickness of the slab At any given height z, we combine the ideal gas law pV = N kT (N is the number of molecules in the slab) with the adiabatic law pV γ = const or (pV )γ ∝ pγ−1 to conclude that pγ−1 ∝ T γ Upon differentiation this gives (γ −1) dT dp =γ , p T so that 4 #28 _IPO_2006, Sưu tầm và đóng gói bởi, Gv... Llate L= ˜ ˜ Learly Llate = 1.73 m , this gives the length on the symmetric picture as 2.11 ˜ ˜ 2Learly Llate ˜ L= = 1.50 m ˜ ˜ Learly + Llate 4 #16 _IPO_2006, Sưu tầm và đóng gói bởi, Gv Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 Theory Question 3 Page 1 of 8 Theory Question 3 This question consists of five independent... from the top surface of the film whereas beam B is reflected from the bottom surface of the film. The superposition of beams A and B results in the so called thin film interference. A B q1 q1 n t q2 Figure 2.1: Schematic 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 intensity... skyscraper, the outside temperature is Tbot= 30°C The objective is to estimate the outside temperature Ttop at the top Consider a thin slab of air (ideal nitrogen gas with adiabatic coefficient γ = 7/5) rising slowly to height z where the pressure is lower, and assume that this slab expands adiabatically so that its temperature drops to the temperature of the surrounding air 3.14 (0.5) How is the fractional... THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 In conjunction with 2 L L = x+ − x− = γL − βγ D 2 + γx0 + ˜ ˜ γ 2 + βγ D 2 + γx0 − L 2 2 this tells us that D 2 + γx0 ± L 2 2 = 2γ 2 x0 ± (γL − L/γ) γx0 βL = ± 2βγ β 2 As they should, both the version with the upper signs and the version with the lower signs give the same answer for x0 ,... (or smallest value F = 2) and assume λ = 500 nm for the typical wavelength of daylight The digital resolution is given by the distance between the center of two neighboring pixels For our 5 Mpix camera this distance is roughly = L = 15.65 µm Np Ideally we should match the optical and the digital resolution so that neither aspect is overspecified Taking the given optical resolution in the expression for... = 14.34 N0 1.22λ N0 1 #25 _IPO_2006, Sưu tầm và đóng gói bởi, Gv Phan Hồ Nghĩa, THPT Chuyên Hùng Vương, PleiKu, Gia Lai Website: wWw.HoNghia.Net E-mail: Info@HoNghia.Net Mobile: +84914 171 299 Since this F value is not available, we choose the nearest value that has a higher optical resolution, 3.3 F0 = 11 When looking at a picture at distance z from the eye, the (small) subtended angle between two . analogy to an optical interferometer is shown in Figure 1a. The neutrons enter the interferometer through the IN port and follow the two paths shown. The neutrons are detected at either one of. mirrors within the interferometer. The experiment studies the effect of the gravitational pull on the de Broglie waves of neutrons. The symbolic representation of this interferometer in analogy. neutron de Broglie waves (of typical wavelength of 10 −10 m) interfere such that all neutrons emerge from the output port OUT1 if the interferometer plane is horizontal. But when the interferometer