Download Common Exam - 2004 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 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
go(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
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 - Special Relativity
A particle of mass m hits a particle of the same mass that is at rest. They stick together and the
total energy of the composite particle is 4 mc2.
(a)
[10 pts.] Find the initial speed vi of the projectile particle.
(b)
[8 pts.] Find the mass of the composite particle.
(c)
[7 pts.] Find the velocity of the composite particle.
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 V0, with E > V0.
(a)
[9 pts.] Determine the stationary wave function N(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 =
(c)
[5 pts.] What is the probability of reflection if E =
(d)
[3 pts.] Write down the time-dependent wave function R(x,t) that corresponds to the
problem for all values of x and t.
V0.
V0?
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)
[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?
(b)
[5 pts.] How does the wavelength in water compare with that in air?
(c)
[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.
(d)
[5 pts.] Write an equation for the air particle speed in the tube at position x which shows
its time and space dependence when the maximum air displacement is 10-4 m.
(e)
[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
water is 1.48 × 106 Rayleighs.
Problem 4 - Modern Physics
A
[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).
B
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 8 << 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.
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 D0 and
force constant k. The configuration of the system is completely
described by a single angle 2 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)
[4 pts.] Write down the Cartesian coordinates (x1,y1) of the center of mass of the rod on
the right in terms of 2.
(b)
[4 pts.] Write down the total kinetic energy Tcm associated with the motion of the centers
of mass of the two rods in terms of 2 and/or
(c)
.
[4 pts.] Write down the total kinetic energy Trot associated with the rotation of the two
rods about their respective centers of mass in terms of 2 and/or
(d)
[4 pts.] Write down the gravitational potential energy, Vg, of the rods in terms of 2
and/or
(e)
. Take the height of the origin to be the reference zero for Vg.
[4 pts.] Write down the elastic potential energy Vs stored in the spring in terms of 2
and/or
(f)
.
. Take Vs to be zero when the spring is relaxed.
[5 pts.] Write down the Lagrangian of the system in terms of 2 and and write down
the (differential) equation of motion for 2. You do not need to solve the differential
equation.
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 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
go(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
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 - 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 2 to that of the
large loop.
For parts (a) and (b), a current of I(t) = I0 exp - t/J is circulating through the large loop. You can
assume that the time constant J is sufficiently large that the system is quasi-static (i.e., no
retardation effects.)
(a)
[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.
(b)
[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) = i0 cos T 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.)
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)
[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.
(b)
[9 pts.] Solve for the velocity v(t) of the particle as a function of time. Find the limit of
v at t 6 4.
(c)
[8 pts.] Determine the total amount of heat produced by the particle motion after it has
fallen for a time t = m/k.
Problem 8 - Quantum Mechanics
Consider the angular momentum operators in Cartesian coordinates
(a)
[10 pts.] Determine the commutation relations
terms of operators
(b)
and
. Obtain the commutation relations for
through their similarity to
. Recall that
.
[7 pts.] Show that no two values of the components of angular momentum Lx, Ly and Lz
can be simultaneously specified in a quantum mechanical state.
(c)
in
. You need only show the detailed algebraic steps
for the determination of
and
where
[8 pts.] Show that it is possible to simultaneously specify Lz and L2 where
.
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)
[5 pts.] Show that at high temperatures the total magnetic moment is: M = N:2H/kT.
(d)
[5 pts.] Show that the work done at constant T in the high T limit when the magnetic
field is changed from Hi to Hf is:
(e)
[5 pts.] Show that at low temperatures the magnetic moment can be written as:
Indicate what low temperature means.
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, 8max, 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 8max = 400 nm. Express the formula relating 8max and T. Determine the
constant of proportionality in this formula. What is the surface temperature of the star
when 8max = 800 nm?
(b)
[5 pts.] The energy flux, F, from the surface of the star is 1.56 × 108 W/m2 when
8max = 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 8max = 800 nm?
(c)
[5 pts.] The total luminosity (total power output), L0, of the star remains constant in
time. Express the formula for the radius of the star in terms of the star’s constant
luminosity L0 and surface temperature T. What is the value for the ratio of the star’s
maximum radius, Rmax, to minimum, Rmin, for the two surface temperatures determined in
part (a)?
(d)
[5 pts.] The sodium absorption line that is normally at 589.30 nm is found to shift in the
spectrum of this star from a minimum of 589.20 nm to a maximum of 589.40 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 Rmin to its maximum radius Rmax 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 vavg in your formula]. Determine the values of Rmin and Rmax.