Download 2002 - University of Utah Physics

Common Exam - 2002
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
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
Trig Identities
cos(" + $) = cos " cos $ - sin " sin $
sin(" + $) = sin " cos $ + cos " sin $
sin θ / 2 =
1 − cos θ
2
cos θ / 2 =
1 + cos θ
2
Table of Integrals and Other Formulas
Spherical Harmonics
Conic Section
Normal Distribution
Cylindrical Coordinates (orthonormal bases)
Spherical Coordinates (orthonormal bases)
Maxwell Equations (Rationalized MKS)
G G 1
∇⋅E = ρ
∈o
G
G G
∂B
∇×E = −
∂t
G G
∇⋅B = 0
G
G G
G
∂E
∇ × B = µ o J + µ o ∈o
∂t
Maxwell Equations (Gaussian Units)
G G
∇ ⋅ E = 4 πρ
G
G G 1 ∂B
∇×E =
c ∂t
G G
∇⋅B = 0
G
G G 4π G 1 ∂E
∇×B =
J+
c
c ∂t
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)
(c)
(d)
G
[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).
[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.)
G
[8 pts.] Find the electric field E , as a function of x, for points on the x axis.
Problem 2: General Physics
The circuit below initially has no charges on the capacitors, no currents flowing, and switches S1
and S2 are both open.
(a)
(b)
(c)
[7 pts.] At time t = 0, switch S1 is closed. What is the current through the 30 S resistor
immediately after S1 is closed?
[8 pts.] A long time after S1 is closed, what is the voltage across capacitor C1?
[10 pts.] Switch S2 is eventually also closed. Find the charge on C1 an infinitely long
time after S2 is closed.
C1 = 10 :f
C2 = 40 :f
C3 = 40 :f
C4 = 80 :f
C5 = 30 :f
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?
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/sec2 at a time 11 seconds after car A passes it. How far will car B go
before catching car A?
Problem 5: Thermodynamics
A system that consist of n moles of a monoatomic ideal gas (CV = 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. To, So, n and R are known values.
(a)
(b)
(c)
(d)
[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.
[5 pts.] Draw schematically the closed cycle with processes I, II and III on a suitable PV co-ordinate system. Be sure to indicate the location of points a, b, and c in your
diagram.
Common Exam - 2002
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
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
Trig Identities
cos(" + $) = cos " cos $ - sin " sin $
sin(" + $) = sin " cos $ + cos " sin $
sin θ / 2 =
1 − cos θ
2
cos θ / 2 =
1 + cos θ
2
Table of Integrals and Other Formulas
Spherical Harmonics
Conic Section
Normal Distribution
Cylindrical Coordinates (orthonormal bases)
Spherical Coordinates (orthonormal bases)
Maxwell Equations (Rationalized MKS)
G G 1
∇⋅E =
ρ
∈o
G
G G
∂B
∇×E = −
∂t
G G
∇⋅B = 0
G
G G
G
∂E
∇ × B = µ o J + µ o ∈o
∂t
Maxwell Equations (Gaussian Units)
G G
∇ ⋅ E = 4 πρ
G
G G 1 ∂B
∇×E =
c ∂t
G G
∇⋅B = 0
G
G G 4π G 1 ∂E
∇×B =
J+
c
c ∂t
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, rn.
[4 pts.] What is the approximate value of the smallest orbital radius, known as the Bohr
radius a0.
[8 pts.] Express the formula for the wavelength of radiation emitted for transition
between energy levels nf and ni. What is the approximate wavelength of photons
resulting from the n = 3 to n = 2 transition? (Hint: Note that the Rydberg constant,
R = 1.0973 × 107 m-1 is within the 1% of the combination of fundamental constants given
by ke2/2a0hc .)
Problem 7: General Physics
Show all work.
(a)
(b)
(c)
[8 pts.] A certain cosmic ray proton (mass = 939 MeV) has an energy of 1019 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 1025 kg and a radius of around 107 meters. Use
dimensional analysis to give a rough estimate of the pressure near the center of the earth.
(Be sure to specify units.)
Problem 8: Quantum Mechanics
G
Consider a spin-½ particle, say an electron, in a static magnetic field B o in the z-direction. The
Hamiltonian is given by
ˆ z Bo
H o = −µσ
where : is Bohr's magneton, σ̂ z is one of the Pauli matrices:
0 1
σˆ x = 
,
1
0


 0 −i 
σˆ y = 
,
i
0


1 0 
σˆ z = 

 0 −1
(a)
[5 pts.] Write down the energy levels of Ho and the corresponding wave function in spin
space.
(b)
[8 pts.] Now a small static magnetic field B1 is applied in the x-direction. The
G
Hamiltonian of the system is changed to
H = H o + H1 ,
(c)
H1 = −µσˆ x B1
with B1 << Bo. Using first order perturbation theory, calculate the energy levels of H and
the corresponding wave functions in spin space.
[12 pts.] Without assuming B1 << Bo, find the exact energy levels and spin-space wave
functions for H.
Problem 9: Lagrange Mechanics
Swinging in a plane are two pendula which are attached to each other. The first pendulum
consists of a mass m1 suspended from the ceiling by a massless rod of length R1. The second
pendulum has a mass m2 and is attached to mass m1 by a massless rod of length R2. Do not
assume small amplitudes except in part (d).
(a)
(b)
(c)
(d)
[5 pts.] For the case m2 = 0 write down the Lagrangian for the system, and find the
equation of motion for N1(t).
[5 pts.] For arbitrary values of m1 and m2, 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 m1 is forced so that it becomes a small amplitude
oscillation:
N1(t) = N0 sin (T0t) .
Write the equation of motion for N2.(t), assuming that N2(t) is small.
Problem 10: Electromagnetism: Waves and Maxwell Equations
Let x, y, z be Cartesian coordinates and let ˆi, ˆj, kˆ 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
G
E = E 0 sin ( β z + ωt ) ˆi
where t is time. If the numerical value of T is 10 Hz, what is the value of $? (Specify
units.)
(b)
(c)
(d)
[7 pts.] In terms of E0, $, T and c give all components of B̂ , the oscillating magnetic
field vector, as functions of x, y, z and t.
[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 E0, $, T and c.)
[6 pts.] Suppose now that the electric vector lies in the xz plane, and its x-component is
(
E x = E 0 sin β  x + 3z  + ωt
Find the z-component of the oscillating electric vector.
).