Common Exam Department of Physics University of Utah August 24, 2002

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1 Common Exam Department of Physics University of Utah August 24, 2002 Examination booklets have been provided for recording your work and your solutions. Please note that there is a separate booklet for each numbered question (i.e., use booklet #1 for problem #1, etc.). To receive full credit, not only should the correct solutions be given, but a sufficient number of steps should be given so that a faculty grader can follow your reasoning. Define all algebraic symbols that you introduce. If you are short of time it may be helpful to give a clear outline of the steps you intended to complete to reach a solution. In some of the questions with multiple parts you will need the answer to an earlier part in order to work a later part. If you fail to solve the earlier part you may represent its answer with an algebraic symbol and proceed to give an algebraic answer to the later part. This is a closed book exam: No notes, books, or other records should be consulted. NO CALCULATORS MAY BE USED. The total of 250 points is divided equally among the ten questions of the examination. All work done on scratch paper should be NEATLY transferred to answer booklet. SESSION 1

2 COMMON EXAM DATA SHEET e = C = esu c = m/s = cm/s h = JAs = ergas = MeVAs S = JAs = ergas = MeVAs k = J/K = erg/k g = 9.80 m/s 2 = 980 cm/s 2 G = NAm 2 /kg 2 = dyneacm 2 /g 2 N A = particles/gmamole = particles/kgamole g o (SI units) = F/m : o (SI units) = 4B 10-7 H/m m(electron) = kg = g= AMU = 511 kev M(proton) kg = g = AMU = MeV M(neutron) kg = g = AMU = MeV M(muon) = kg = g 1 mile = 1609 m 1 m = 3.28 ft 1 ev = J = ergs hc = 12,400 evad Trig Identities cos(" + $) = cos " cos $ - sin " sin $ sin(" + $) = sin " cos $ + cos " sin $ 1 cos θ 1+ cos θ sin θ / 2 = cos θ / 2 = 2 2

3 Spherical Harmonics Table of Integrals and Other Formulas

4 Conic Section Normal Distribution Cylindrical Coordinates (orthonormal bases)

5 Spherical Coordinates (orthonormal bases) Maxwell Equations (Rationalized MKS) 1 E = ρ o B E = t B= 0 E B=µ oj+µ o o t Maxwell Equations (Gaussian Units) E = 4πρ 1 B E = c t B= 0 4π 1 E B= J+ c c t

6 Problem 1: Electrostatics Two thin rings, each of radius a, are arranged as shown in the figure. The centers of the rings lie on the x axis. Ring #1 lies in the plane x = L, and Ring #2 lies in the plane x = - L. Each ring has positive charge Q distributed uniformly over it. (a) [6 pts.] For L >> a, find the force of repulsion between the rings. For the questions below do not assume that L >> a. (b) [4 pts.] What is the electrical field strength E at the point midway between the rings? (i.e., on the x axis, at x = 0). (c) [7 pts.] Find the electrical potential V(x) due to the rings, as a function of x, along the x axis. (Take the potential to be zero at 4.) (d) [8 pts.] Find the electric field E, as a function of x, for points on the x axis.

8 Problem 2: General Physics The circuit below initially has no charges on the capacitors, no currents flowing, and switches S 1 and S 2 are both open. (a) [7 pts.] At time t = 0, switch S 1 is closed. What is the current through the 30 S resistor immediately after S 1 is closed? (b) [8 pts.] A long time after S 1 is closed, what is the voltage across capacitor C 1? (c) [10 pts.] Switch S 2 is eventually also closed. Find the charge on C 1 an infinitely long time after S 2 is closed. C 1 = 10 :f C 2 = 40 :f C 3 = 40 :f C 4 = 80 :f C 5 = 30 :f

10 Problem 3: Quantum Mechanics Suppose we have two non-interacting particles, both of mass m, in a one-dimensional infinite well: 0 for a x a V(x) = for x > a (a) (b) (c) (d) [6 pts.] Write down the Schrödinger equation and the boundary conditions for the wave function of a single particle. [6 pts.] Obtain the energy levels, together with their degeneracy and corresponding wave functions for a single particle. [6 pts.] If the two particles are identical bosons with spin-0, what are the energy and degeneracy for the ground and the first excited states? [7 pts.] If the two particles are identical particles with spin one-half, what are the energy, degeneracy and corresponding wave functions for the ground and the first excited states?

13 Problem 4: General Physics (a) (b) (c) [9 pts.] On a certain trip you travel ¼ the distance at 60 m/sec, a the time at 40 m/sec, and the remainder at 50 m/sec. What was your average speed for the trip? [8 pts.] A bus travels 2000 m in 20 sec with constant acceleration. At the end of the 2000 m it is going 49 m/sec. What was its acceleration? [8 pts.] Car A goes at a constant speed of 25 m/sec. Car B starts moving with constant acceleration ½ m/sec 2 at a time 11 seconds after car A passes it. How far will car B go before catching car A?

15 Problem 5: Thermodynamics A system that consist of n moles of a monoatomic ideal gas (C V = 3/2 nr) undergoes the closed cycle abca along the processes I, II and III as displayed in the figure in a T-S coordinate system where T is temperature and S is entropy. T o, S o, n and R are known values. (a) (b) (c) [10 pts.] For each of the processes I, II and III find the change in the internal energy )U, the work )W done on the system and the heat )Q that enters the system. For a complete cycle, calculate the total work W done on the system, and the values for U, the total change in internal energy, and Q the total heat added. [5 pts.] Does the closed cycle represent an engine or a refrigerator? [5 pts.] If the cycle represents an engine, find the efficiency, :. If the cycle represents a refrigerator, find the efficiency of performance, 0. (d) [5 pts.] Draw schematically the closed cycle with processes I, II and III on a suitable P- V co-ordinate system. Be sure to indicate the location of points a, b, and c in your diagram.

17 Common Exam Department of Physics University of Utah August 24, 2002 Examination booklets have been provided for recording your work and your solutions. Please note that there is a separate booklet for each numbered question (i.e., use booklet #1 for problem #1, etc.). To receive full credit, not only should the correct solutions be given, but a sufficient number of steps should be given so that a faculty grader can follow your reasoning. Define all algebraic symbols that you introduce. If you are short of time it may be helpful to give a clear outline of the steps you intended to complete to reach a solution. In some of the questions with multiple parts you will need the answer to an earlier part in order to work a later part. If you fail to solve the earlier part you may represent its answer with an algebraic symbol and proceed to give an algebraic answer to the later part. This is a closed book exam: No notes, books, or other records should be consulted. NO CALCULATORS MAY BE USED. The total of 250 points is divided equally among the ten questions of the examination. All work done on scratch paper should be NEATLY transferred to answer booklet. SESSION 2

18 COMMON EXAM DATA SHEET e = C = esu c = m/s = cm/s h = JAs = ergas = MeVAs S = JAs = ergas = MeVAs k = J/K = erg/k g = 9.80 m/s 2 = 980 cm/s 2 G = NAm 2 /kg 2 = dyneacm 2 /g 2 N A = particles/gmamole = particles/kgamole g o (SI units) = F/m : o (SI units) = 4B 10-7 H/m m(electron) = kg = g= AMU = 511 kev M(proton) kg = g = AMU = MeV M(neutron) kg = g = AMU = MeV M(muon) = kg = g 1 mile = 1609 m 1 m = 3.28 ft 1 ev = J = ergs hc = 12,400 evad Trig Identities cos(" + $) = cos " cos $ - sin " sin $ sin(" + $) = sin " cos $ + cos " sin $ 1 cos θ 1+ cos θ sin θ / 2 = cos θ / 2 = 2 2

19 Spherical Harmonics Table of Integrals and Other Formulas

20 Conic Section Normal Distribution Cylindrical Coordinates (orthonormal bases)

21 Spherical Coordinates (orthonormal bases) Maxwell Equations (Rationalized MKS) 1 E = ρ o B E = t B= 0 E B=µ oj+µ o o t Maxwell Equations (Gaussian Units) E = 4πρ 1 B E = c t B= 0 4π 1 E B= J+ c c t

22 Problem 6: Modern Physics Consider Bohr's model of the hydrogen atom. The electron is assumed to orbit the proton in a circular orbit under the influence of a Coulomb attractive force. (a) (b) (c) (d) [5 pts.] What condition is imposed on the electron's orbital angular momentum in Bohr's model? Express this condition in terms of a mathematical equation involving an integer index n.. [8 pts.] This condition on orbital angular momentum results in a discrete set of stable orbits. Determine the allowed values of orbital radii, r n. [4 pts.] What is the approximate value of the smallest orbital radius, known as the Bohr radius a 0. [8 pts.] Express the formula for the wavelength of radiation emitted for transition between energy levels n f and n i. What is the approximate wavelength of photons resulting from the n = 3 to n = 2 transition? (Hint: Note that the Rydberg constant, R = m -1 is within the 1% of the combination of fundamental constants given by ke 2 /2a 0 hc.)

26 Problem 7: General Physics Show all work. (a) (b) (c) [8 pts.] A certain cosmic ray proton (mass = 939 MeV) has an energy of ev. Its speed is c(1 -,) where c is the speed of light and, is small. Find the approximate value of,. [8 pts.] What is the approximate energy (in ev) of X-rays used for X-ray crystallography? [9 pts.] The earth has a mass of around kg and a radius of around 10 7 meters. Use dimensional analysis to give a rough estimate of the pressure near the center of the earth. (Be sure to specify units.)

28 Problem 8: Quantum Mechanics Consider a spin-½ particle, say an electron, in a static magnetic field Hamiltonian is given by H = µσˆ B o z o B o in the z-direction. The where : is Bohr's magneton, ˆσ z is one of the Pauli matrices: i ˆ 1 0 σ x =, ˆ y, ˆ z 1 0 σ = σ = i (a) [5 pts.] Write down the energy levels of H o and the corresponding wave function in spin space. (b) [8 pts.] Now a small static magnetic field B is applied in the x-direction. The 1 Hamiltonian of the system is changed to H = H + H, H = µσˆ B o 1 1 x 1 (c) with B 1 << B o. Using first order perturbation theory, calculate the energy levels of H and the corresponding wave functions in spin space. [12 pts.] Without assuming B 1 << B o, find the exact energy levels and spin-space wave functions for H.

30 Problem 9: Lagrange Mechanics Swinging in a plane are two pendula which are attached to each other. The first pendulum consists of a mass m 1 suspended from the ceiling by a massless rod of length R 1. The second pendulum has a mass m 2 and is attached to mass m 1 by a massless rod of length R 2. Do not assume small amplitudes except in part (d). (a) (b) (c) (d) [5 pts.] For the case m 2 = 0 write down the Lagrangian for the system, and find the equation of motion for N 1 (t). [5 pts.] For arbitrary values of m 1 and m 2, write down the Lagrangian for the system. [ pts.] From the Lagrangian in part (b) derive the equations of motion of the system. [ pts.] Assume the motion of m 1 is forced so that it becomes a small amplitude oscillation: N 1 (t) = N 0 sin (T 0 t). Write the equation of motion for N 2.(t), assuming that N 2 (t) is small.

32 Problem 10: Electromagnetism: Waves and Maxwell Equations Let x, y, z be Cartesian coordinates and let ˆˆˆ i, j,k be unit vectors in the corresponding directions. (a) [5 pts.] In a vacuum region of spacetime, suppose that the electric field vector is given by E = E sin β z+ωt ˆi 0 ( ) where t is time. If the numerical value of T is 10 Hz, what is the value of $? (Specify units.) (b) [7 pts.] In terms of E 0, $, T and c give all components of ˆB, the oscillating magnetic field vector, as functions of x, y, z and t. (c) [7 pts.] What is the direction and magnitude of the time-averaged electromagnetic power per unit area carried by the fields? (Give the answer in terms of E 0, $, T and c.) (d) [6 pts.] Suppose now that the electric vector lies in the xz plane, and its x-component is ( ) Ex = E0 sin β x+ 3z +ωt Find the z-component of the oscillating electric vector..

Common Exam Department of Physics University of Utah August 28, 2004

Common Exam Department of Physics University of Utah August 28, 2004 Common Exam - 2004 Department of Physics University of Utah August 28, 2004 Examination booklets have been provided for recording your work and your solutions. Please note that there is a separate booklet