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

Transcription

1 Common Exam 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 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. YOU MAY ONLY USE THE CALCULATORS PROVIDED. 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 8 10 c = m/s = cm/s h = J s = erg s = MeV s = J s = erg s = MeV s k = J/K = erg/k 2 2 g = 9.80 m/s = 980 cm/s G = N m /kg = dyne cm /g N A = particles/gm mole = particles/kg mole -12 o(si units) = F/m -7 o(si units) = 4 10 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 ev = J = ergs hc = 12,400 ev

3 Table of Integrals and Other Formulas

4 Spherical Harmonics Conic Section Normal Distribution Cylindrical Coordinates (orthonormal bases)

5 Spherical Coordinates (orthonormal bases) Maxwell Equations (Rationalized MKS) Maxwell Equations (Gaussian Units)

6 Problem 1 - Special Relativity A particle of mass m hits a particle of the same mass that is at rest. They stick together and the 2 total energy of the composite particle is 4 mc. (a) (b) (c) [10 pts.] Find the initial speed v of the projectile particle. [8 pts.] Find the mass of the composite particle. [7 pts.] Find the velocity of the composite particle. i

7

8 Problem 2 - Quantum Mechanics A one dimensional quantum particle of mass m is incident from the left on the one dimensional potential barrier as shown below. The energy of the incident particle is E and the barrier height is V, with E > V. 0 0 (a) [9 pts.] Determine the stationary wave function (x) that corresponds to this problem for all values of x. Define all variables and coefficients in the wave function in terms of the parameters given above and fundamental constants. Set the amplitude of the incident wave to one. (b) [8 pts.] Calculate the probability that the particle will be reflected if E = V 0. (c) [5 pts.] What is the probability of reflection if E = V 0? (d) [3 pts.] Write down the time-dependent wave function (x,t) that corresponds to the problem for all values of x and t.

9

10

11 Problem 3 - Acoustics A resonance tube is employed to find the frequency of a tuning fork. The tube is open at one end and closed at the other end. The water level indicates the positions of resonance. Resonance is obtained when the length of air column is 0.52 m and 2.25 m; assume they are the shortest and next shortest lengths for resonances. The speed of sound in air is 344 m/s. (a) (b) (c) (d) (e) [5 pts.] What is the frequency of the tuning fork? Will the end effect at the open end of the tube have an impact on the calculation? Why? [5 pts.] How does the wavelength in water compare with that in air? [5 pts.] In the air portion of the tube, what is the phase of the gas particle velocity relative to the sound pressure when on resonance? Explain. [5 pts.] Write an equation for the air particle speed in the tube at position x which shows -4 its time and space dependence when the maximum air displacement is 10 m. [5 pts.] Compare the intensity of sound in air and in water for the same acoustic pressure. The characteristic impedance of air is 415 Rayleighs and in the impedance in 6 water is Rayleighs.

12

13

14 Problem 4 - Modern Physics A B [10 pts.] An infinite circular cylinder with a radius a is surrounded by a metal and is infiltrated by a dielectric with refractive index n. It forms a waveguide. Using the uncertainty principle concept, estimate the order of magnitude of the lowest frequency that can propagate through this waveguide (the cut-off frequency). A parallel beam of light goes through a circular opening in an opaque screen as shown in the figure below. The diameter of the opening is V. The wavelength of the light << V. (a) [10 pts.] Estimate the value of the diameter V such that at a given distance l from a opening the width of the beam W has a minimum as a function of V. Use the concept of uncertainty principle. (b) [5 pts.] Estimate the minimum width W.

15

16 Problem 5 - Classical Mechanics The figure below shows two uniform rods of mas M and length L joined in a hinge at point a. The hinge is attached to a vertical wire and is free to slide up and down without friction. The far ends of the rods (points b and c ) are attached to a common horizontal wire and are free to slide left and right without friction. The midpoints (centers of mass) of the rods are connected by an ideal, massless, spring of relaxed length D 0 and force constant k. The configuration of the system is completely described by a single angle made by the rod on the left to the horizontal as shown. Please assume a coordinate system as shown, with x along the horizontal and y along the vertical such that the origin is at the intersection between the two wires. (a) (b) (c) (d) (e) [4 pts.] Write down the Cartesian coordinates (x 1,y) 1 of the center of mass of the rod on the right in terms of. [4 pts.] Write down the total kinetic energy T associated with the motion of the centers of mass of the two rods in terms of and/or. [4 pts.] Write down the total kinetic energy T associated with the rotation of the two rods about their respective centers of mass in terms of and/or. [4 pts.] Write down the gravitational potential energy, V, of the rods in terms of and/or. Take the height of the origin to be the reference zero for V. [4 pts.] Write down the elastic potential energy V stored in the spring in terms of and/or. Take V to be zero when the spring is relaxed. s cm rot s g g (f) [5 pts.] Write down the Lagrangian of the system in terms of and and write down the (differential) equation of motion for. You do not need to solve the differential equation.

17

18

19

20 Common Exam 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 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. YOU MAY ONLY USE THE CALCULATORS PROVIDED. 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

21 COMMON EXAM DATA SHEET e = C = esu 8 10 c = m/s = cm/s h = J s = erg s = MeV s = J s = erg s = MeV s k = J/K = erg/k 2 2 g = 9.80 m/s = 980 cm/s G = N m /kg = dyne cm /g N A = particles/gm mole = particles/kg mole -12 o(si units) = F/m -7 o(si units) = 4 10 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 ev = J = ergs hc = 12,400 ev

22 Table of Integrals and Other Formulas

23 Spherical Harmonics Conic Section Normal Distribution Cylindrical Coordinates (orthonormal bases)

24 Spherical Coordinates (orthonormal bases) Maxwell Equations (Rationalized MKS) Maxwell Equations (Gaussian Units)

25 Problem 6 - Electricity and Magnetism The figure below shows a small circular wire loop of radius a placed at the center of a large circular loop of radius r. The symmetry axis of the small loop is tilted at angle to that of the large loop. For parts (a) and (b), a current of I(t) = I 0 exp - t/ is circulating through the large loop. You can assume that the time constant is sufficiently large that the system is quasi-static (i.e., no retardation effects.) (a) (b) [9 pts.] In the absence of the small loop, find the magnitude of the magnetic field, B at the center of the large loop as a function of the current, I. [8 pts.] Find the EMF v(t) induced in the small loop as a function of time by the current in part (a). Since a << r you may assume the field generated by the large loop to be uniform over the small loop. For part (c), the current I in parts (a) and (b) has been switched off. (c) [8 pts.] Now suppose a current of i(t) = i 0 cos t is circulating in the small loop. Find the induced EMF V(t) in the the large loop as a function of time. Again, you can assume a << r and quasi-static conditions. (Hint: You do not need to perform any integration for this part.)

26

27

28

29 Problem 7 - Mechanics A particle (mass m) under the influence of gravity (g) is dropped from rest in a long tube filled with a viscous medium. The magnitude of the viscous force on the particle is proportional to the magnitude of the particle s velocity. The proportionality constant is related to the viscosity of the medium and the size of the particle. In this problem, use the symbols v for the particle velocity and the symbol k for the proportionality constant. (a) (b) (c) [8 pts.] Write the differential equation describing the motion of the particle in the viscous medium. Be careful with signs and use the simple diagram below to define the coordinate system in the equation. [9 pts.] Solve for the velocity v(t) of the particle as a function of time. Find the limit of v at t. [8 pts.] Determine the total amount of heat produced by the particle motion after it has fallen for a time t = m/k.

30

31

32 Problem 8 - Quantum Mechanics Consider the angular momentum operators in Cartesian coordinates where (a) [10 pts.] Determine the commutation relations and in terms of operators for the determination of. You need only show the detailed algebraic steps. Obtain the commutation relations for and through their similarity to. Recall that. (b) (c) [7 pts.] Show that no two values of the components of angular momentum L, L and L can be simultaneously specified in a quantum mechanical state. 2 [8 pts.] Show that it is possible to simultaneously specify L and L where. z x y z

33

34

35 Problem 9 - Statistical Mechanics Consider a paramagnetic system of N magnetic ions in thermal equilibrium with an external temperature T. The magnetic ions have a magnetic moment. They are placed in a magnetic field H and can point only along the field or opposite to it. The energy of an ion is given by. (a) [5 pts.] What fraction of ions are pointing along the magnetic field? (b) [5 pts.] What is the total magnetic moment M as a function of H and T? (c) (d) 2 [5 pts.] Show that at high temperatures the total magnetic moment is: M = N H/kT. [5 pts.] Show that the work done at constant T in the high T limit when the magnetic field is changed from H to H is: i f (e) [5 pts.] Show that at low temperatures the magnetic moment can be written as: Indicate what low temperature means.

36

37

38 Problem 10 - General Physics Consider a pulsating star whose spectrum changes in a periodic manner. The spectrum at any given time can be assumed to be consistent with a black body spectrum. (a) [5 pts.] The wavelength of the peak intensity, max, of the spectrum varies from a minimum of 400 nm to a maximum of 800 nm. The surface temperature, T, of the star is 7245 K when max = 400 nm. Express the formula relating max and T. Determine the constant of proportionality in this formula. What is the surface temperature of the star when = 800 nm? max (b) (c) (d) 8 2 [5 pts.] The energy flux, F, from the surface of the star is W/m when max = 400 nm. Express the formula relating flux, F, and surface temperature, T. Determine the constant of proportionality in this formula. What is the value of the flux when = 800 nm? max [5 pts.] The total luminosity (total power output), L 0, of the star remains constant in time. Express the formula for the radius of the star in terms of the star s constant luminosity L 0 and surface temperature T. What is the value for the ratio of the star s maximum radius, R max, to minimum, R min, for the two surface temperatures determined in part (a)? [5 pts.] The sodium absorption line that is normally at nm is found to shift in the spectrum of this star from a minimum of nm to a maximum of nm. Assume that the star is at rest with respect to the earth. What is the maximum surface speed of the star as it expands? The surface speed is non-relativistic. (e) [5 pts.] The star expands from its minimum radius R min to its maximum radius R max in 12.7 hours. Assume that the average speed of the surface during this expansion is ½ of the maximum speed of expansion that you found in part (d) [if you did not find a value in part (d), use v in your formula]. Determine the values of R and R. avg min max

39

40