Common Exam - 2003Department of Physics

University of UtahAugust 23, 2003

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., usebooklet #1 for problem #1, etc.).

To receive full credit, not only should the correct solutions be given, but a sufficientnumber 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 helpfulto give a clear outline of the steps you intended to complete to reach a solution. In someof the questions with multiple parts you will need the answer to an earlier part in order towork a later part. If you fail to solve the earlier part you may represent its answer with analgebraic symbol and proceed to give an algebraic answer to the later part. This is aclosed book exam: No notes, books, or other records should be consulted. YOU MAYONLY USE THE CALCULATORS PROVIDED. The total of 250 points is dividedequally among the ten questions of the examination.

All work done on scratch paper should be NEATLY transferred to answer booklet.

SESSION 1

COMMON EXAM DATA SHEET

e = - 1.60 × 10 C = - 4.80 × 10 esu-19 -10

c = 3.00 × 10 m/s = 3.00 × 10 cm/s8 10

h = 6.64 × 10 JAs = 6.64 × 10 ergAs = 4.14 × 10 MeVAs -34 -27 -21

S = 1.06 × 10 JAs = 1.06 × 10 ergAs = 6.59 × 10 MeVAs -34 -27 -22

k = 1.38 × 10 J/K = 1.38 × 10 erg/K -23 -16

g = 9.80 m/s = 980 cm/s2 2

G = 6.67 × 10 NAm /kg = 6.67 × 10 dyneAcm /g-11 2 2 -8 2 2

AN = 6.02 × 10 particles/gmAmole = 6.02 × 10 particles/kgAmole23 26

og (SI units) = 8.85 × 10 F/m -12

o: (SI units) = 4B × 10 H/m -7

m(electron) = 9.11 × 10 kg = 9.11 × 10 g= 5.4859 × 10 AMU = 511 keV -31 -28 -4

M(proton) 1.673 × 10 kg = 1.673 × 10 g = 1.0072766 AMU = 938.2 MeV -27 -24

M(neutron) 1.675 × 10 kg = 1.675 × 10 g = 1.0086652 AMU = 939.5 MeV -27 -24

M(muon) = 1.88 × 10 kg = 1.88 × 10 g -28 -25

1 mile = 1609 m

1 m = 3.28 ft

1 eV = 1.6 × 10 J = 1.6 × 10 ergs -19 -12

hc = 12,400 eVAD

Trig Identities

cos(" + $) = cos " cos $ - sin " sin $

sin(" + $) = sin " cos $ + cos " sin $

Table of Integrals and Other Formulas

Spherical Harmonics

Conic Section

Normal Distribution

Cylindrical Coordinates (orthonormal bases)

Spherical Coordinates (orthonormal bases)

Maxwell Equations (Rationalized MKS)

Maxwell Equations (Gaussian Units)

Problem 1: Electricity and Magnetism

A long solenoid coil with its axis along the z direction has a radius r, N turns(N >> 1), length R (R >> r) and is made of superconducting wire. An AC

ocurrent I(t) = I cos(Tt) is flowing through the leads of the coil as shown.

(a) [5 pts.] Calculate the on-axis magnetic field in the coil as a

function of time (magnitude and direction).

(b) [5 pts.] Write the general expression for the self inductance L of acoil in terms of its magnetic flux and current through the coil.

(c) [5 pts.] Calculate the value of the inductance of the coil described above in terms ofparameters provided.

(d) [5 pts.] Calculate the induced EMF(t) across the coil (as a function of time).

(e) [5 pts.] At the peak of the current in the coil, what is the magnetic energy per unit lengthinside the coil?

Problem 2: Quantum Mechanics

Consider a particle of mass m in a one-dimensional box with walls (of infinite potential) at x = 0and x = L.

(a) [5 pts.] Write down the Schrodinger equation and boundary conditions that determinethe energy levels of the particle

(b) [5 pts.] Determine the energy levels and corresponding wave functions for the particle. You need not normalize the wave function.

(c) [5 pts.] Suppose at the initial time t = 0, the wave function of the particle is known to be

If one measures the energy of this state, what are the possible measured values? Withwhat probabilities will these values be found in the measurement?

(d) [5 pts.] What is the expectation value for the energy in this state?

(e) [5 pts.] Write down the wave function at t > 0 for the initial state specified in (c) above.

Problem 3: General Physics

Rainbows are formed when (A) light rays from the Sun refract into a water droplet, (B)internally reflect (not totally) on the far side, and (C) refract out again. On exit, the ray makes anangle of 2 with respect to the direction opposite the incident. The situation is shown in thediagram below. Assume the wavelength of the light is much smaller than the perfectly sphericaldroplet.

(a) [5 pts] Find expressions for the angle of incidence ", and the angle of refraction $ (withrespect to the normal), on entry into the droplet (step A), in terms of the impactparameter b, the radius of the droplet R, and the index refraction, n, of the light in thedroplet.

A B(b) [10 pts] Find expressions for the angles of deflection 2 , on the initial entry (step A), 2 ,

Cupon the internal reflection (step B), and 2 , on exit from the droplet (step C), in terms of

A B C" and $. Hence find an expression for the reflection angle, 2 = 180° - (2 + 2 + 2 ), in

A B Cterms of b, R, and n. For each 2 , 2 , 2 , the angle of deflection is the amount by whichthe light ray is rotated clockwise from its prior direction.

(c) [5 pts.] Consider the "reflection" cross-section F. Clearly dF = 2Bbdb. Differentiatingwith respect to 2 we obtain

0Calculate d2/db and show that it vanishes for a specific angle 2 which dependsonly on n. At this angle then, the differential reflection cross-section diverges and

0most of the reflected light is peaked in this direction. Find the expression for 2in terms of n

0(d) [5 pts.] The index of refraction for visible light in water is about n = 4/3. Find 2 .

Problem 4: Quantum Mechanics/Modern Physics

An organic light emitting diode (OLED) consists of an organic layer, O, sandwiched in betweentwo metallic electrodes, C and A. One electrode (C) is the cathode for injecting electrons, e

e(particles of charge q = -e, and spin s = ½), whereas the other electrode (A) is the anode for

hinjecting holes, h (particles of charge q = +e and spin s = ½), as shown in Figure 1. Underforward bias, a voltage V is applied to the anode A and zero voltage is applied to the cathode C. Under this condition, electrons e and holes h are injected into the organic layer O from opposite

xelectrodes (see Figure 1). The electrons and holes meet in the organic layer to form excitons, E ,which are composite particles, each having one electron and one hole. The total spin quantum

x e hnumber of the exciton E is S, which is composed of s and s . Some of the excitons then emit

0light with photon energy, hf .

(a) [5 pts.] What are the possible quantum numbers for the exciton spin S?

(b) [8 pts.] What is the probability, p that the excitons are formed with S = 0 (these arecalled singlet excitons).

(c) [5 pts.] Suppose that only singlet excitons may emit light. What is the maximum

max maxquantum efficiency, h [h = average percentage or fraction of incident electron-holepairs which generate an exciton leading to the emission of a photon]?

(d) [7 pts.] Suppose that the two metallic electrodes are in fact made from the same metal.

0At what forward bias threshold, V would the device start to emit light?

Fig. 1. The anode (A) and cathode (C) inject holes (h) and

xelectron (e), respectively. e-h pair up to form excitons E that

0may give up light, hf under forward bias, V.

Problem 5: Lagrangian Mechanics

As shown in the figure below, a symmetric bar of length L is supported by two vertical springs;each has a spring constant k. The left end and right ends of the bar can move up and down. Asthis happens the center of the bar does not move horizontally, and both springs remain vertical.The mass of the bar is M and the moment of inertia, of the bar bout its center of mass, is I. The

1 2coordinates x and x describe the vertical elongation, a shown, of each of the springs fromequilibrium. Please note that “equilibrium” refers to a configuration where the springs arecompressed equally by the weight of the bar such that the net force on the bar is zero.

0(a) [2 pts.] Find (DL) , the amount by which the length of the spring is changed (note thisshould be a negative value) by the weight of the bar in the equilibrium configuration.

1 2 (b) [5 pts.] Write down the Lagrangian for the motion of the bar, in terms of x and x andtheir time-derivatives. DO NOT assume that the motions are small.

(c) [3 pts.] Write down the simplified Lagrangian applicable to small motion about

1 2equilibrium (x , x << L).

(d) [5 pts.] From the small motion Lagrangian, find the equations of motion (coupled

1 2differential equations) for x and x . These equations should not explicitly involve thegravitational acceleration constant g.

(e) [7 pts.] Find the two normal mode (where both springs stretch and compress with the

1 2same frequency) angular frequencies, w , w , for the small motion of the system. If you

1 1 2 2are not sure how to start, try setting x =A cos(wt), and x =A cos(wt), with the same w,and substitute into your equations of motion.

1 2(f) [3 pts.] Suppose at time t=0, we have x = a and x = 0 and both time derivatives are

1 1zero. Find x (t), the value of x as a function of time. Note that for only 3 points, thesolution to this part can be very long and tedious.

Common Exam - 2003Department of Physics

University of UtahAugust 23, 2003

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., usebooklet #1 for problem #1, etc.).

To receive full credit, not only should the correct solutions be given, but a sufficientnumber 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 helpfulto give a clear outline of the steps you intended to complete to reach a solution. In someof the questions with multiple parts you will need the answer to an earlier part in order towork a later part. If you fail to solve the earlier part you may represent its answer with analgebraic symbol and proceed to give an algebraic answer to the later part. This is aclosed book exam: No notes, books, or other records should be consulted. YOU MAYONLY USE THE CALCULATORS PROVIDED. The total of 250 points is dividedequally among the ten questions of the examination.

All work done on scratch paper should be NEATLY transferred to answer booklet.

SESSION 2

COMMON EXAM DATA SHEET

e = - 1.60 × 10 C = - 4.80 × 10 esu-19 -10

c = 3.00 × 10 m/s = 3.00 × 10 cm/s8 10

h = 6.64 × 10 JAs = 6.64 × 10 ergAs = 4.14 × 10 MeVAs -34 -27 -21

S = 1.06 × 10 JAs = 1.06 × 10 ergAs = 6.59 × 10 MeVAs -34 -27 -22

k = 1.38 × 10 J/K = 1.38 × 10 erg/K -23 -16

g = 9.80 m/s = 980 cm/s2 2

G = 6.67 × 10 NAm /kg = 6.67 × 10 dyneAcm /g-11 2 2 -8 2 2

AN = 6.02 × 10 particles/gmAmole = 6.02 × 10 particles/kgAmole23 26

og (SI units) = 8.85 × 10 F/m -12

o: (SI units) = 4B × 10 H/m -7

m(electron) = 9.11 × 10 kg = 9.11 × 10 g= 5.4859 × 10 AMU = 511 keV -31 -28 -4

M(proton) 1.673 × 10 kg = 1.673 × 10 g = 1.0072766 AMU = 938.2 MeV -27 -24

M(neutron) 1.675 × 10 kg = 1.675 × 10 g = 1.0086652 AMU = 939.5 MeV -27 -24

M(muon) = 1.88 × 10 kg = 1.88 × 10 g -28 -25

1 mile = 1609 m

1 m = 3.28 ft

1 eV = 1.6 × 10 J = 1.6 × 10 ergs -19 -12

hc = 12,400 eVAD

Trig Identities

cos(" + $) = cos " cos $ - sin " sin $

sin(" + $) = sin " cos $ + cos " sin $

Table of Integrals and Other Formulas

Spherical Harmonics

Conic Section

Normal Distribution

Cylindrical Coordinates (orthonormal bases)

Spherical Coordinates (orthonormal bases)

Maxwell Equations (Rationalized MKS)

Maxwell Equations (Gaussian Units)

Problem 6: Mechanics

(a) [5 pts.] Consider a particle of negligible mass, :, in a circular orbit about anotherparticle of much larger mass, M. Derive from Newton’s Law of Universal Gravitationthe relationship between the orbital period, T and orbital radius, R.

(b) [10 pts.] The gravitational force is a central force. That is, it can be expressed in theform

Show that angular momentum about the gravitating center is conserved. Prove that thepath of the particle must line in a plane.

(c) [10 pts.] Express the total energy, E, for a particle in a (not necessarily circular) orbit asa function of and of effective potential

For sketch the effective potential as a function of r. On your sketch indicate

circular circularthe radius r = r and energy E = E that corresponds to a circular orbit. If E is less

circularthan zero but greater than E = E (with L fixed), what will be the shape of the orbit? Indicate on your sketch the minimum and maximum allowed orbital radii.

Problem 7: General Physics

A longitudinal acoustic plane wave of frequency f is incident at a small

i 1angle 2 from medium 1 (with phase velocity V ) onto a planar interface

2with a second medium 2 (phase velocity V ). See the figure to the right

1 2and note that V > V , and both media are lossless (no absorption ofpower). At the interface, part of the wave is reflected back intomedium 1 and part is transmitted into medium 2.

1 2(a) [5 pts.] Determine the wavelengths 8 and 8 of the acousticplane waves in medium 1 and medium 2, in terms of parametersgiven.

t(b) [8 pts.] Derive an expression for the angle of transmission 2 as defined in the figure, in

1 2 iterms of V , V and 2 .

i(c) [6 pts.] If the incident beam is square (cross-sectional) with area A , and its dimensions

tare very large compared to the wavelengths involved, determine the area A of the

i i ttransmitted beam in terms of A , 2 and 2 .

i r t(d) [6 pts.] The intensities of the incident, reflected and transmitted waves are I , I and I . Write an equation which established the exact relation between the three intensities.

Problem 8: Quantum Mechanics

Consider the spin states of a hydrogen atom consisting of a proton (atomic nucleus) and anelectron.

(a) [5 pts.] How many linearly independent states are there for the spin configurations of thehydrogen atom including both the nucleus and the electron?

(b) [10 pts.] Now, denote the spin-up and spin-down states (along a certain quantization axiscalled the z-axis), by *8,e>, *9,e> (or *8,p>, *9,p>) for the electron (or the proton). Write

zdown the spin states of the hydrogen atom in which the total spin Ö and Ö have definite2

values. Here and , with the spin angular

momentum operators for the electron and the proton, respectively. Indicate the value of

zS amd S for each state.2

(c) [10 pts.] Calculate the expectation value of in each of the four states youobtained in (b).

Problem 9: Thermodynamics

Note: Treat parts [(a)], [(b), (c)], [(d)] as independent problems.

V(a) [5 pts.] Suppose that for some particular gas, c , the specific heat (or “heat capacity”) atconstant volume, is given by

where " is some constant, and R is the usual gas constant. Find the entropy S of this gasas a function of temperature.

(b) [5 pts.] The ground state of some system is at energy E = 0. An excited state of thesystem is at E = 0.1 eV. The system is at room temperature (kT = 1/40 eV). The

0probability of finding a particle in the ground state is p at this temperature. What is theprobability of finding a particle in the excited state? Assume that the degeneracy of theexcited state is the same as that of the ground state, and assume that Maxwell-Boltzmannstatistics applies.

(c) [5 pts.] Repeat part (b), but now assume that Fermi-Dirac statistics applies, and that theFermi energy is 1/40 eV.

2(d) [10 pts.] A certain gas consists of simple diatomic molecules, like N . Transitionsamong the vibrational states of this molecule give off light with a frequency around 1014

Hz. Rotational transitions give off light 100 times lower in frequency. Sketch the

Vspecific heat c of this gas as a function of temperature. Include a range of temperaturesbroad enough to include all interesting changes. Indicate clearly at what temperature

V Vchanges occur in c , and clearly indicate the values of c at the various temperatures

Vbetween changes. (Hint: At very low temperatures the value of c is 1.5 R.)

Problem 10: Electrodynamics

The Earth’s ionosphere can be treated as a dilute plasma (i.e., the interactions between electrons, and

between electrons and ions can be neglected) of uniform density N and in a uniform magnetic field

in the +z direction. In this problem we consider the propagation of circularly polarized plane

radio waves in a direction parallel to the magnetic field:

The case corresponds to a right-handed circular polarization.

(a) [5 pts.] Assuming the electrons to be at rest on average, write down the equation of motion of an

E 0electron under the influence of the wave and the B field: . Assume E to be so small

wave E Ethat the electrons are non-relativistic, and B << B so that only B need to be included in the

Lorentz force.

(b) [5 pts.] Assume a steady state solution of the form

where you can assume the motion in the z direction to be negligible (i.e., treat z as a constant for

0a given electron). Substitute this into the equation of motion and solve for r . Use the

E E E Esubstitution T = eB /m(= eB /mc in CGS or Gaussian units). T is the Larmor frequency of the

0electrons in the Earth’s magnetic field, and r is the Larmor radius.

(c) [5 pts.] The quantity measures the separation of the electron from its average

position. Write an expression for the polarization density (which is the average dipole

moment per unit volume) induced by the passage of the wave. Substitute into your answer the

p 0plasma frequency T = Ne /(, m) (= 4B Ne /m in CGS units).2 2 2

(d) [5 pts.] From your answer in part (c), find the electric susceptibility where

TE (= in CGS units). It turns out that for the ionosphere, . Tp . 107

E ps . Write down the frequency-dependent index of refraction in the limit T << T , T . -1

Remember that ( in CGS units).

E pFor the remainder of the problem, use the limit T << T , T .

p(e) [5 pts.] Using the fact that the phase velocity is given by v = T/k = c/n, find the group velocities

for the two polarizations. Is there something strange about one of the two polarizations?

Explain what you think this means.