1 University of California, Berkeley Physics H7C Spring 2011 (Yury Kolomensky) THE FINAL EXAM Monday, May 9, 7 10pm Maximum score: 200 points NAME: SID #: You are given 180 minutes for this exam. You are allowed three 8.5 11 sheets containing any information you wish on both sides. Derive all answers symbolically, then plug in the numbers, if appropriate. You may leave expressions such as 2, e, π unevaluated. Your description of the physics involved and symbolic answers are worth much more than the numerical answers. Show all work, and take particular care to explain what you are doing. Write directly on the exam, and if you need extra pages, make sure to put a note on the corresponding sheet. Cross out rather than erase parts of the problem you wish the grader to ignore. Box or circle final answers. There are six problems with points assigned as shown. Partial credit will be given for incomplete solutions, so attempt to do all problems. Some problems will take significantly longer than others, so judge time appropriately. At the beginning of the exam, please look through all problems and plan how you ll spend your time. If you need to ask a question, come to the proctor. Your question and the answer will be written on the board. Read all problems carefully. If you get stuck on a problem, move to another. Try to remain calm and work steadily. Good luck!
2 Useful formulae and values (to acceptable precision) Speed of light in vacuum c = 3 10 8 m/s Planck constant h = 6.63 10 34 J Planck constant, reduced h = h/2π = 1.055 10 34 J Electron charge e = 1.6 10 19 C Conversion constant hc = 197 ev nm Mass of the electron m e = 0.511 MeV/c 2 = 9.11 10 31 kg Mass of the proton m p = 938 MeV/c 2 = 1.67 10 27 kg Permittivity of free space ɛ 0 = 8.85 10 12 F/m Permeability of free space µ 0 = 4π 10 7 N/A 2 Fine structure constant α = e 2 /(4πɛ 0 hc) = 1/137 Classical electron radius r e = e 2 /(4πɛ 0 m e c 2 ) = 2.8 fm e Compton wavelength/2π λ e /2π = h/m e c = r e α 1 = 0.39 fm Bohr radius a 0 = 4πɛ 0 h 2 /m e e 2 = r e α 2 = 0.53 10 10 m Rydberg energy R = m e e 4 /2(4πɛ 0 ) 2 h 2 = m e c 2 α 2 /2 = 13.6 ev Bohr magneton µ B = e h/2m e = 5.8 10 11 MeV/T Wavefunction for the ground state of the hydrogen atom ψ 100 = (1/ πa 3 0 ) exp ( r/a 0)
3 1. (40 points) This is a little blitz problem to warm you up. This problem consists of ten questions, 4 points each. Circle correct answer for each part. (FYI: questions like these are frequently found on a Physics GRE exam) 1. If a charged pion that decays in 10 8 second in its own rest frame is to travel 30 meters in the laboratory before decaying, the pion s speed must be most nearly (a) 0.43 10 8 m/s (b) 2.84 10 8 m/s (c) 2.71 10 8 m/s (d) 2.98 10 8 m/s (e) 3.00 10 8 m/s 2. If the total energy of a particle of mass m is equal to twice its rest energy, then the magnitude of the particle s relativistic momentum is (a) mc/2 (b) mc/ 2 4. A beam of light has a small wavelength spread δλ about a central wavelength λ. The beam travels in vacuum until it enters a glass plate at an angle θ relative to the normal to the plate, as shown in the figure below. The index of refraction of the glass is given by n(λ). The angular spread δθ of the refracted beam is given by (a) δθ = 1 n δλ (b) δθ = dn λ δλ (c) δθ = 1 λ dn dλ δλ (d) δθ = sin θ (e) δθ = δλ sin θ λ tan θ n dn dλ δλ (c) mc (d) 3mc (e) 2mc 3. A spherical, concave mirror is shown in the figure below. The focal point F and the location of the object O are indicated. At what point will the image be located? (a) I (b) II (c) III (d) IV (e) V 5. Blue light of wavelength λ = 480 nm is most strongly reflected off a thin film of oil on a glass slide when viewed near normal incidence. Assuming that the index of refraction of the oil is n o = 1.2 and that of the glass is n g = 1.6, what is the minimum thickness of the oil film (other than zero)?
4 (a) 150 nm (b) 200 nm (c) 240 nm (d) 300 nm (e) 480 nm 6. Unpolarized light of intensity I 0 is incident on a series of three polarizing filters. The axis of the second filter is oriented at 45 to that of the first filter, while the axis of the third filter is oriented at 90 to that of the first filter. What is the intensity of the light transmitted through the third filter? (a) 0 (b) I 0 /8 (c) I 0 /4 (d) I 0 /2 (e) I 0 / 2 7. What causes dichroism? (a) Difference in absorption properties of the material depending on polarization of the incident light (b) Difference refraction indices of the material depending on polarization of the incident light (c) Difference in density of the material depending on polarization of the incident light (d) Deviation of the lens from the ideal, hyperbolic shape (e) Dependence of the index of refraction on wavelength of the incident light a surface of this metal, determine the kinetic energy of the photoelectrons. (a) 1 ev (b) 2 ev (c) 3 ev (d) 4 ev (e) 6 ev 9. A free particle with initial kinetic energy E and de Broglie wavelength λ enters a region in which it has potential energy V. What is the particle s new de Broglie wavelength? (a) λ(1 + E/V ) (b) λ(1 V/E) (c) λ/(1 E/V ) (d) λ 1 + V/E (e) λ/ 1 V/E 10. The energy required to remove both electrons from the helium atom in its ground state is 79.0 ev. How much energy is required to ionize helium (i.e., to remove one electron)? (a) 24.6 ev (b) 39.5 ev (c) 51.8 ev (d) 54.4 ev (e) 65.4 ev 8. In the photoelectric effect, the threshold wavelength for a particular metal is λ 0 = 300 nm. If light of wavelength λ = 200 nm is incident on
5 2. (35 points) A relativistic electron with energy E = 9.0 GeV collides head-on with a relativistic positron with energy E + = 3.1 GeV, traveling in the opposite direction to the electron. In the resulting annihilation, a particle Υ(4S) is created in a reaction e + e Υ(4S). Find the mass of Υ(4S) and its velocity in the LAB frame. 3. (45 points) An optical system consists of a thin converging lens with focal distance f = 50 cm, and a iris attached to the back of the lens. A screen is placed b = 75 cm behind the iris. The system is illuminated with plane light waves of wavelength λ = 0.5 µm. Find the possible values of the radius of the iris such that the intensity at the center of the screen is maximal. 4. (30 points) Neutron diffraction is one of the methods used to determine crystal structure of a material. Imagine that a beam on mono-energetic neutrons (of mass m and kinetic energy K) impinging on a crystal lattice with atomic spacing d at the inclination angle θ (it is customary in crystallography to measure the angle from the plane of the crystal, as opposed to the normal to the surface). If neutron energy is relatively small (thermal or cold neutrons), they would typically reflect off the nuclei of the crystal lattice elastically. (a) (20 pts) At what angles would the intensity of the reflected neutrons be maximal? (b) (10 pts) What is the smallest possible spacing d that this imaging using thermal neutrons (K = 0.026 ev, m = 939 MeV) could resolve? How do you think it compares to the typical crystal lattice spacing? K θ d 5. (25 points) A particle of mass m is confined in an infinite square well. The particle is in a stationary state with a probability density function P (1 cos αx), where α is a constant, and x is the distance from one edge of the well. Find the energy of the particle in this state. 6. (25 points) The wavefunction of an electron in 2P state of the hydrogen atom has a radial component R(r) r exp( r/2a 0 ), where a 0 is the Bohr radius. Find: (a) (10 points) The most probable distance between the electron and the proton r max ; (b) (10 points) Average distance between electron and proton r (c) (5 points) Explain qualitatively why the difference between energy levels of 2S 1/2 and 2P 1/2 states (known as Lamb shift).