Download Common Exam - 2003 Department of Physics University of Utah August 23, 2003

Common Exam - 2003
Department of Physics
University of Utah
August 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., 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
COMMON EXAM DATA SHEET
e = - 1.60 × 10-19 C = - 4.80 × 10 -10 esu
c = 3.00 × 108 m/s = 3.00 × 10 10 cm/s
h = 6.64 × 10 -34 JAs = 6.64 × 10 -27 ergAs = 4.14 × 10 -21 MeVAs
S = 1.06 × 10 -34 JAs = 1.06 × 10 -27 ergAs = 6.59 × 10
-22
MeVAs
k = 1.38 × 10 -23 J/K = 1.38 × 10 -16 erg/K
g = 9.80 m/s2 = 980 cm/s 2
G = 6.67 × 10-11 NAm2/kg2 = 6.67 × 10 -8 dyneAcm 2/g 2
NA = 6.02 × 1023 particles/gmAmole = 6.02 × 10 26 particles/kgAmole
g o(SI units) = 8.85 × 10 -12 F/m
: o(SI units) = 4B × 10 -7 H/m
m(electron) = 9.11 × 10 -31 kg = 9.11 × 10 -28 g= 5.4859 × 10 -4 AMU = 511 keV
M(proton) 1.673 × 10 -27 kg = 1.673 × 10 -24 g = 1.0072766 AMU = 938.2 MeV
M(neutron) 1.675 × 10 -27 kg = 1.675 × 10 -24 g = 1.0086652 AMU = 939.5 MeV
M(muon) = 1.88 × 10 -28 kg = 1.88 × 10 -25 g
1 mile = 1609 m
1 m = 3.28 ft
1 eV = 1.6 × 10 -19 J = 1.6 × 10 -12 ergs
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
current I(t) = Io cos(T t) 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 a
coil 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 of
parameters 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 length
inside the coil?
Problem 2: Quantum Mechanics
Consider a particle of mass m in a one-dimensional box with walls (of infinite potential) at x = 0
and x = L.
(a)
[5 pts.] Write down the Schrodinger equation and boundary conditions that determine
the 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? With
what 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 an
angle of 2 with respect to the direction opposite the incident. The situation is shown in the
diagram below. Assume the wavelength of the light is much smaller than the perfectly spherical
droplet.
(a)
[5 pts] Find expressions for the angle of incidence " , and the angle of refraction $ (with
respect to the normal), on entry into the droplet (step A), in terms of the impact
parameter b, the radius of the droplet R, and the index refraction, n, of the light in the
droplet.
(b)
[10 pts] Find expressions for the angles of deflection 2 A, on the initial entry (step A), 2 B,
upon the internal reflection (step B), and 2 C, on exit from the droplet (step C), in terms of
" and $ . Hence find an expression for the reflection angle, 2 = 180° - (2 A + 2 B + 2 C), in
terms of b, R, and n. For each 2 A, 2 B, 2 C, the angle of deflection is the amount by which
the light ray is rotated clockwise from its prior direction.
(c)
[5 pts.] Consider the "reflection" cross-section F . Clearly dF = 2B bdb. Differentiating
with respect to 2 we obtain
Calculate d2 /db and show that it vanishes for a specific angle 2 0 which depends
only on n. At this angle then, the differential reflection cross-section diverges and
most of the reflected light is peaked in this direction. Find the expression for 2 0
in terms of n
(d)
[5 pts.] The index of refraction for visible light in water is about n = 4/3. Find 2 0.
Problem 4: Quantum Mechanics/Modern Physics
An organic light emitting diode (OLED) consists of an organic layer, O, sandwiched in between
two metallic electrodes, C and A. One electrode (C) is the cathode for injecting electrons, e
(particles of charge q = -e, and spin se = ½), whereas the other electrode (A) is the anode for
injecting holes, h (particles of charge q = +e and spin sh = ½), as shown in Figure 1. Under
forward 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
electrodes (see Figure 1). The electrons and holes meet in the organic layer to form excitons, Ex,
which are composite particles, each having one electron and one hole. The total spin quantum
number of the exciton Ex is S, which is composed of se and sh. Some of the excitons then emit
light with photon energy, hf0.
(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 are
called singlet excitons).
(c)
[5 pts.] Suppose that only singlet excitons may emit light. What is the maximum
quantum efficiency, hmax [hmax = average percentage or fraction of incident electron-hole
pairs 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.
At what forward bias threshold, V0 would the device start to emit light?
Fig. 1. The anode (A) and cathode (C) inject holes (h) and
electron (e), respectively. e-h pair up to form excitons E x that
may give up light, hf0 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. As
this 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
coordinates x1 and x2 describe the vertical elongation, a shown, of each of the springs from
equilibrium. Please note that “equilibrium” refers to a configuration where the springs are
compressed equally by the weight of the bar such that the net force on the bar is zero.
(a)
[2 pts.] Find (DL)0, the amount by which the length of the spring is changed (note this
should be a negative value) by the weight of the bar in the equilibrium configuration.
(b)
[5 pts.] Write down the Lagrangian for the motion of the bar, in terms of x1 and x2 and
their time-derivatives. DO NOT assume that the motions are small.
(c)
[3 pts.] Write down the simplified Lagrangian applicable to small motion about
equilibrium (x1, x2 << L).
(d)
[5 pts.] From the small motion Lagrangian, find the equations of motion (coupled
differential equations) for x1 and x2. These equations should not explicitly involve the
gravitational acceleration constant g.
(e)
[7 pts.] Find the two normal mode (where both springs stretch and compress with the
same frequency) angular frequencies, w1, w2, for the small motion of the system. If you
are not sure how to start, try setting x1=A1cos(wt), and x2=A2cos(wt), with the same w,
and substitute into your equations of motion.
(f)
[3 pts.] Suppose at time t=0, we have x1 = a and x2 = 0 and both time derivatives are
zero. Find x1(t), the value of x1 as a function of time. Note that for only 3 points, the
solution to this part can be very long and tedious.
Common Exam - 2003
Department of Physics
University of Utah
August 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., 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
COMMON EXAM DATA SHEET
e = - 1.60 × 10-19 C = - 4.80 × 10 -10 esu
c = 3.00 × 108 m/s = 3.00 × 10 10 cm/s
h = 6.64 × 10 -34 JAs = 6.64 × 10 -27 ergAs = 4.14 × 10 -21 MeVAs
S = 1.06 × 10 -34 JAs = 1.06 × 10 -27 ergAs = 6.59 × 10
-22
MeVAs
k = 1.38 × 10 -23 J/K = 1.38 × 10 -16 erg/K
g = 9.80 m/s2 = 980 cm/s 2
G = 6.67 × 10-11 NAm2/kg2 = 6.67 × 10 -8 dyneAcm 2/g 2
NA = 6.02 × 1023 particles/gmAmole = 6.02 × 10 26 particles/kgAmole
g o(SI units) = 8.85 × 10 -12 F/m
: o(SI units) = 4B × 10 -7 H/m
m(electron) = 9.11 × 10 -31 kg = 9.11 × 10 -28 g= 5.4859 × 10 -4 AMU = 511 keV
M(proton) 1.673 × 10 -27 kg = 1.673 × 10 -24 g = 1.0072766 AMU = 938.2 MeV
M(neutron) 1.675 × 10 -27 kg = 1.675 × 10 -24 g = 1.0086652 AMU = 939.5 MeV
M(muon) = 1.88 × 10 -28 kg = 1.88 × 10 -25 g
1 mile = 1609 m
1 m = 3.28 ft
1 eV = 1.6 × 10 -19 J = 1.6 × 10 -12 ergs
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 another
particle of much larger mass, M. Derive from Newton’s Law of Universal Gravitation
the 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 the
form
Show that angular momentum about the gravitating center is conserved. Prove that the
path 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 as
a function of
and of effective potential
For
sketch the effective potential as a function of r. On your sketch indicate
the radius r = rcircular and energy E = Ecircular that corresponds to a circular orbit. If E is less
than zero but greater than E = Ecircular (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
angle 2 i from medium 1 (with phase velocity V1) onto a planar interface
with a second medium 2 (phase velocity V2). See the figure to the right
and note that V1 > V2, and both media are lossless (no absorption of
power). At the interface, part of the wave is reflected back into
medium 1 and part is transmitted into medium 2.
(a)
[5 pts.] Determine the wavelengths 8 1 and 8 2 of the acoustic
plane waves in medium 1 and medium 2, in terms of parameters
given.
(b)
[8 pts.] Derive an expression for the angle of transmission 2 t as defined in the figure, in
terms of V1, V2 and 2 i.
(c)
[6 pts.] If the incident beam is square (cross-sectional) with area Ai, and its dimensions
are very large compared to the wavelengths involved, determine the area At of the
transmitted beam in terms of Ai, 2 i and 2 t.
(d)
[6 pts.] The intensities of the incident, reflected and transmitted waves are Ii, Ir and It.
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 an
electron.
(a)
[5 pts.] How many linearly independent states are there for the spin configurations of the
hydrogen atom including both the nucleus and the electron?
(b)
[10 pts.] Now, denote the spin-up and spin-down states (along a certain quantization axis
called the z-axis), by *8 ,e>, *9 ,e> (or *8 ,p>, *9 ,p>) for the electron (or the proton). Write
down the spin states of the hydrogen atom in which the total spin Ö 2 and Ö z have definite
values. Here
and
, with
the spin angular
momentum operators for the electron and the proton, respectively. Indicate the value of
S2 amd Sz for each state.
(c)
[10 pts.] Calculate the expectation value of
obtained in (b).
in each of the four states you
Problem 9: Thermodynamics
Note: Treat parts [(a)], [(b), (c)], [(d)] as independent problems.
(a)
[5 pts.] Suppose that for some particular gas, cV, the specific heat (or “heat capacity”) at
constant volume, is given by
where " is some constant, and R is the usual gas constant. Find the entropy S of this gas
as a function of temperature.
(b)
[5 pts.] The ground state of some system is at energy E = 0. An excited state of the
system is at E = 0.1 eV. The system is at room temperature (kT = 1/40 eV). The
probability of finding a particle in the ground state is p0 at this temperature. What is the
probability of finding a particle in the excited state? Assume that the degeneracy of the
excited state is the same as that of the ground state, and assume that Maxwell-Boltzmann
statistics applies.
(c)
[5 pts.] Repeat part (b), but now assume that Fermi-Dirac statistics applies, and that the
Fermi energy is 1/40 eV.
(d)
[10 pts.] A certain gas consists of simple diatomic molecules, like N2. Transitions
among 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
specific heat cV of this gas as a function of temperature. Include a range of temperatures
broad enough to include all interesting changes. Indicate clearly at what temperature
changes occur in cV, and clearly indicate the values of cV at the various temperatures
between changes. (Hint: At very low temperatures the value of cV 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
(a)
case corresponds to a right-handed circular polarization.
[5 pts.] Assuming the electrons to be at rest on average, write down the equation of motion of an
electron under the influence of the wave and the B E field:
. Assume E 0 to be so small
that the electrons are non-relativistic, and B wave << B E so that only B E 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
a given electron). Substitute this into the equation of motion and solve for r 0. Use the
substitution T E = eB E/m(= eB E/mc in CGS or Gaussian units). T E is the Larmor frequency of the
electrons in the Earth’s magnetic field, and r0 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
plasma frequency T p 2 = Ne 2/(, 0 m) (= 4B Ne 2/m in CGS units).
(d)
[5 pts.] From your answer in part (c), find the electric susceptibility
(=
where
in CGS units). It turns out that for the ionosphere, TE . Tp . 10 7
s -1. Write down the frequency-dependent index of refraction
Remember that
(
in the limit T << T E , T p.
in CGS units).
For the remainder of the problem, use the limit T << T E, T p.
(e)
[5 pts.] Using the fact that the phase velocity is given by v p = 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.