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final_exam_solutions

final_exam_solutions - Physics 214 FINAL Ream,‘,3“...

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Unformatted text preview: Physics 214 FINAL December 11, 2007 Ream,‘ ,3“ L“: "x. - f? ; A?” £33 a“. Q 4» L2 W a... .3, a. 3. Name: Signature: Section # and TA: Score: Part Possible Score I oints This examination is closed books and closed notes. No calculators or other electronic devices are permitted. A formula sheet and scratch paper are provided. Use the space provided after each question for your answers and write your final answers in the boxes provided. If you need more room for a problem, write "over" and continue on the back. Cross out anything that should notkbe considered part of your answer. The scratch . paper will not be graded. In all but Part I, you must show your work to get full credit. Make sure you have all 14 pages. Please write your name on each page. By signing this exam you certify that you adhere to the Cornell academic integrity code. Good luck! Page 1 of 14 Part I: Multiple Choice (4 points each for 10 questions) You do not need to Show work. Just give your answer in the boxes. 1). Which of the following is an example of tunneling (list all that apply)? (A) An electron is liberated from a metal when an electric field is applied, even though its potential energy should have it classically trapped according to the value of the material’s work function. (B) A beam of light strikes a piece of metal and sends an electron speeding away from its surface. (C) By means of electrons, electricity is conducted through a single piece of metal. (D) A flash of light appears on an old glow-in-the—dark clock dial indicating that an alpha particle has left a nucleus. 2) Which of the following is not true of a photon? (A)p=h//t (B)f=c//l (C) Its dispersion relation is not linear. (D)E=pC Page 2 of 14 3) Light in air is reflected from a lens that has an antireflective coating ; a of thickness twith index of refraction n = 1.38. The glass under the Ff coating has n’ = 1.50. What is the condition for destructive interference in the reflected light? 10 is the wavelength in air and m is any positive integer. (Assume normal incidence.) (A) t= mnflo (B) t= mAO/n (C) t = (m - mm (D) t= (m — ‘/2)/io/n (E) 2t = mn/io (F) 2t = m/io/n (G) 2t = (m — ‘/2)n/io (H) 2t = (m — ‘/2)/io/n 4) A transmission grating with slit spacing d = 2.1 pm and slit width 9:; a = 0.7 pm is illuminated with light of wavelength 0.6 pm. What is i ' * the total number of principal maxima that can be seen? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 (F) 7 (G) 8 5) In a photoelectric effect experiment, the stopping potential depends 33% 01] g W33 (A) what metal is being used as target (B) the frequency of the incident light (C) the intensity of the incident light (D) both (A) and (C) but not (B) (E) both (A) and (B) but not (C) (F) both (B) and (C) but not (A) Page 3 of 14 6) In a photoelectric effect experiment, which of the following quantities is directly proportional to the work function of the target? 55: List all that apply. (A) threshold frequency (B) threshold wavelength (C) stopping potential (D) intensity (E) current 7) In a Compton scattering experiment with incident x-rays of wavelength 1.00 pm, what is the maximum possible wavelength of a scattered photon? (A) 1.00 pm (B) 2.00 pm (C) 2.43 pm (D) 3.43 pm (E) 4.43 pm (F) 4.86 pm (G) 5.86 pm (H) 6.86 pm 8) A particle is confined to a 1D quantum well: U(x) = 0 inside the well (0 < x < L) and U(x) = U0 > 0 outside. Which statements are true for bound states both for finite U0 and for U0 = 00? List all that apply. (A) the probability density inside the well is non-uniform (B) the probability density outside the well is zero (C) the energy levels are discrete (i.e.not continuous) (D) the number of bound states is infinite (E) the minimum possible energy is nonzero (F) the wavefunction inside the well is sinusoidal (G) the uncertainty in p; is zero Page 4 of 14 9) The three lines (violet, blue—green, red) in the hydrogen spectrum that you saw in Lab 6 result from transitions from ni = 3, 4, or 5 to nf = 2. What is the wavelength of the Ed line? (E 1 represents the ground state energy for hydrogen.) (A) —(hc/E1)*(36/5) (B) —(hc/E1)*6 (C) —(hc/E1)*(100/21) (D) —(hc/E1)*(10/3) (E) ~(E1/hc)*(36/5) (F) -(E1/hC)*(6) (G) —(E1/hc)*(l 00/21) (H) —(E1/hc)*(10/3) 10) Imagine that a radar system with a dish type antenna of 1 m diameter aperture size uses 3 cm wavelength microwaves to scan for targets, Approximately how wide is the beam at a distance of 10 km from the station? (Small angle approximations are fine). (A) 0.4 gm" ha (B) {gm {W (C) 40 cm gm (D) 400 gm” gs}. Page 5 of 14 Part 11: Using an Interferometer to Weigh Neutrons (24 points) You do need to Show work. Consider the following design for an interferometer to measure the quantum phase shift of a neutron due to gravity. As shown below, slow moving (i.e. nonrelativistic) neutrons of mass m from a reactor having initial kinetic energy T O * enter the interferometer at beam splitter A and either take the high road with vertical height y = 0 along the path AB or the low road with vertical height y = -H along path CD. The two beams recombine at D and proceed to a neutron detector which gives a signal proportional to beam intensity. Treat the beams as plane waves of definite wavelength and frequency in the following discussion (you shouldn’t have to work with wavefunctions to give proper answers). [*We are using the symbol T for kinetic energy to avoid confusion with the wavenumber.] Page 6 of 14 (a) Explain why despite the variation in potential energy and hence kinetic energy with height, the quantum phase shifi along AC is the same as along BD. z twig“ k a: (b) Use conservation of energy to argue whether the neutron’s wavenumber, k is greater 6: less‘ 3 along the high road than along the low road. (Circle your choice (greater or less) in the prece fig“ sentence) “Maw \ A {k 1' F i f Asa; T‘k a Ragga“ E‘ «a; ,» a 5’? «:3 ,. i 1 ,7 i r h.“ «5 Wire .a t Y , ‘ x 3;; A e -x §a 5’ *3; w ‘5 w H 5171 it!“ “w” " i 5““; $.15“ ‘13“ ‘5"? ‘2‘ mx @§*ka Page 7 of 14 (c) Give a qualitative sketch of how the intensity in the detector should behave as a function of H assuming that the neutron beam from the reactor is very intense. (Here one somehow keeps the interferometer perfectly aligned while varying H . The actual experiment done involved a fixed reactor beam and rotation of the interferometer/detector assembly about a horizontal direction along the incoming beam.) rm (d) Regarding the difference in wavenumber of part (b), Ak as slight, differentiate the relation between kinetic energy and wavenumber to show that AT / T0 0C Ak / k and find the constant of proportionality. am‘mwwmy ,_ warsm f M» Page 8 of 14 (6) Assuming that all that matters is the change in wavenumber use the result of part ((1) to estimate the slight phase difference (A¢) in the two paths as a function of L, H, m, g (the acceleration of gravity at the earth’s surface), T0 and any constants needed. The best answer is the simplest possible. . was (53% i sags sinus sis § :2: was in! f E U s W" 3 l :5} x 6‘2... 5:1; i»:- I: is 5 5w 5;; “' gs gamma“ We {swam , Wmfiw' “a 5’ s its a a» M a i ,. 1" W 5am» ”jag , grit... if f x? was i :93 at) My,“ . s s g. sf , s! I is... s. «us EM: :2“ We“ is» , ,_. New 6 aa- 5&5 “ 4’ WW > c.» f R m» wwssasflwm was “mgflw “I 2 M “i“ v» ffiflm we) ,w. 2553* t, :3 "2 3&3“ g 5$ a”, f", a _ if? «:25 gyms if , (0 Suppose your souréce is so weak that neutrons arrive one at a time in individual clicks in a loudspeaker hooked up to the detector). Suppose that you get an experimental answer to (c) by scanning H back and forth in little steps covering the same range in H over and over again while spending equal time at each step. Qualitatively explain briefly how your answer to (c) will develop as time passes and you accumulate data. Page 9 of 14 Part III: A Quantum Particle Encounters a Barrier (24 points) You do need to Show Work. Consider the behavior of a beam of particles each of mass m traveling along a single spatial dimension (x) given potential energy equal to 0 for x < 0 and U 0 for x > 0. The particles are streaming in from a source at x = --00. E is the energy of the incident particles and we will begin by assuming that E < U 0 . “Na N‘ Page 10 of 14 (b) Assuming that the incident beam is a plane wave of unit amplitude, find the complete wavefunction ‘1’“ x, t ) for all space and time, (Note that it will be necessary to give different expressions for x < 0 and x > 0), There is much work to be done in this part of the problem. Make judicious choices for the form of the wavefunction, introducing and defining as needed new parameters. Your final answer needs to be expressed in terms of the starting quantities m , E , U 0 and universal constants as needed. U Page ll ofl4 m e: a 3 3m «3.31». 3 . Wwa W W’MW; «a WWmemwg m ‘ 3 3,, [Even If you did not get part (b), you may be able to answer parts (c) and (d). ] 3-2:: a“? a, :3 33133» (0) Explain the qualitative change that occurs in the wavefunction for x > 0 if E starts to exceed fie e: ”333% 2333 ((1) Without going into along mathematical discussion, completely describe the wavefunction over all space in the limitE / U 0 —> oo. 1‘ “ 5 f i P? - 3 A1 l , 3 3%; swam :3” gawk / 53:: 33/1 353$?“ 55'" if" 331:3 3 ~ i" 1“ s n 3 f ,- dmtfi 13:3“:qu W“? ,.. 3 3 A X. ,— x" " 3 f \ L 3‘: W” \.F 9% ‘ M. r’ {5.)} a j} ,3 fw- ‘ Page 12 of 14 Part IV: The Case of the Lucky Radio Engineer (12 points) You do need to show work. You’ve been tasked with the design and assembly of a linear antenna array to be used to beam radio broadcasts from a transmitting station to two distant cities. You go traditional and drive in phase a line of N vertical wire antennas spaced (1 = l / 2 apart laid out over a total end to end distance L . Your array is positioned exactly between the two cities along a perpendicular as shown below (hence the name “broadside” array). ’7; o la 7—; /—-v/A/4/47¢.'//¢ :-’F 4 149,47”)!wa é—-—-——~— 9° 3e ¢ ”*3 b C) (a) Estimate the angular width of the beam A 6 directed at either city in the limit that N is large. Use a well drawn and labeled phasor diagram to derive and explain your result. 55a :3: 5’ 533:; 7, “My \ "K. 6;" WM L fica} «K g “a 9&3 mi as a? it f A ' E: r Page 13 of 14 (b) Expecting a bonus for your work you are outraged when it doesn’t come through. In anger that night, you go to the array, dig up all the antennas and replant them. While leaving them along the original line and spanning the original distance L you fiendishly use a random spacing between the antennas that had been formerly exactly half a wavelength apart. The next morning you receive field engineering reports that the signal strength in both your target cites is still strong. Explain what’s going on. Are people at work goofing with you? Your answer should be short and concise. g l f g fl ,5 i s 1}; .«fr it] \ ‘4 a éiw it} 'Q 5 ‘ S 5‘ wwgm V leg... W“ 5* 5’5 é. rm 51* W» a. Effie. a. «’1 g ’9‘} a it“ 53.4 as» ”a.“ _ 5 6 ii 775?: ggwm ‘ m g» a , a A .r : f 1 ,. W“ 5‘“; “A We ‘5‘ ‘e ”5: «a 15 a M55” :53 area a ,3 5 5’ 2 . , g 33:3 35‘ {ya ”a; '5 k‘ i; ii if ~ R ‘ if??? ii} a vatfa: \Efiig‘matf g r : «fix? a“: x m ’a' s ‘ 39"» . ‘3 xi»? its. : §fi$fifi i s’ f g“ Page 14 of 14 ...
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