Physics 113, Spring 2000, Test #3
Name:_____________________
Equations: Constants:
Wave equation: y(x,t) = ymsin(kx +/- wt)
Wavenumber: k = 2p/l
Frequency: w/2p = f = 1/T p =3.1415926
Wave speed: n = w/k = l/T = lf
String wave speeed: n =Öt/m
String av. power: = 0.5mnw2ym2
Interfering waves: y'(x,t) = 2ymcos(f/2)sin(kx - wt + f/2)
Standing waves: y'(x,t) = 2ymsin(kx)cos(wt)
Resonance: f = n/l = nn/2L
Sound wave speed: n =ÖB/r 343m/s in air, 1482m/s in water
Sound intensity: I = Power/Area = P/4pr2
Sound decibels: b=(10 dB) log(I/I0) I0=10-12W/m2
Beats: fbeat= f1 - f2
Doppler effect: f' = f (n +/- nD) / (n -/+ nS)
Shock wave: sin(q) = n/nS
Electromagnetic waves: E = Emsin(kx - wt)
B = Bmsin(kx - wt);
Speed of light in a vacuum: c = E/B = 1/ Öm0e0 e0 = 8.85e-12 F/m
Energy flow: S = 1/m0E x B m0 = 1.26e-6 H/m
Intensity of wave: I = <<>S<>> = 1/cm0 c = 3.00e+8 m/s
Radiation pressure: F = IA/c (total absorb) Power of sun = 3.9e26W
pressure: p = Force/Area
Snell's law: n1 sin(q1) = n2 sin(q2)
Total internal reflection: qc = arcsin(n2/ n1)
Brewster angle: qb = arctan(n2/ n1)
Spherical Mirror: 1/p + 1/i = 1/f = 2/r
Spherical Refractor: n1/p + n2/i = (n2 - n1)/r (single surface)
Thin lens: 1/p + 1/i = 1/f = (n - 1) (1/r1 - 1/r2)
Lateral magnification: m = -i/p = h'/h
Magnifying glass: mq = 25cm/f
Compound microscope: M = m mq = (-s/fOB)(25cm/fEYE)
Refracting telescope: m = - fOB/fEYE
Gravitational force: F = G m1 m2 / r2 G = 6.67e-11 N m2/kg2
Wavelength & Index of Refraction: ln=l/n
Young's Double Slit: d sinq =ml (maxima), d sinq =(m+1/2)l (minima)
Double Slit Intensity: I = 4 I0cos²(f/2), where f=(2pd/l ) sinq
Thin film interference: 2L = (m + ½) l/n2 (maxima)
Single Slit Diffraction: a sinq =ml (minima)
Single Slit Intensity: I = Im(sina/a)², where a=(pa/l ) sinq
Circular Aperture Diffraction: sinq=1.22 l/d (first minima)
Double+Single Slit Diffraction Intensity: I = Imcos²(f/2)(sina/a)²
Multple slit half-widths: Dq=l/(Nd cosq)
Diffraction grating Dispersion: D = Dq/Dl=m/(d cosq)
Diffraction grating Resolving Power: R = l/Dl = Nm
X-Ray Diffraction: 2d sin q = ml
Relativistic factors: b=v/c, g=1/Ö(1-b²)
Rel. Time Dilation: Dt = gDt0
Rel. Length Contraction: L = L0/g
Lorentz Transformation: z'=z; y'=y; x' = g(x-vt); t'=g(t-vx/c²)
Addition of Velocities: u = (u' + v)/ (1 + u'v/c²)
Rel. Longitudinal Doppler: f'= f [ (1 - b) / (1+b) ]1/2
Rel. Transverse Doppler: f' = f [ 1 - b²]1/2
Rel. Momentum: p = gmv
Rel. Kinetic Energy: K = mc²(g -1)
Rel. Total Energy: E = gmc²
Rel. Triangle rule: E² = (pc)² +(mc²)² and (pc)² = K²+ 2Kmc²
Photon Energy & Momentum: E = hf; p = hf/c = h/l
Photoelectric effect: hf = Kmax + F
Compton Shift: Dl = (h/mc)(1 - cos f)
deBoglie matter waves: l = h/p
Schroedinger Equation: d²y/dx² + 8p²m/h² [E - V(x)]y = 0
Heisenberg Uncertainty: Dx Dp S h/2p
Barrier Tunneling: T Y exp(-2kL); k=( 8p²m/h² [E - V(x)] )1/2
Electron in Infinite Well: En = (h² /8mL² ) n²
Wavefunctions for Infinite Well: yn = A sin(np/L)x
Probability Density: P = |yn |²
Hydrogen Atom: V = -1/4pe0 (e²/r)
Hydrogen Energy levels: En = (me4/ 8e0²h²) 1/n² = -13.6 eV/n²
H Radial Probability Density: P(r) = (4/a3) r² exp(-2r/a)
Masses of the isotopes:
hydrogen 1H = 1.007 825 u
deuterium 2H = 2.014 102 u
tritium 3H = 3.016 049 u
helium 3 3He = 3.016 029 u
helium / alpha particle 4He = 4.002 603 u
neutron n = 1.008 665 u
atomic mass unit u = 931.5 MeV
TNT energy equivalent 1T of TNT = 2.6e22 MeV
Bohr frequency condition hf = E_high - E_low
Orbital angular momentum L = [l(l+1)]1/2h/2p
Bohr magneton m_B = eh/4pm
spin angular momentum S = [s(s+1)]1/2 h/2p
K-shell X-ray l_min = h c / K_0
Decay rate N = N_0 e -lt
Hubble's Law v = Hr
Multiple Choice, answer all.
[1] Assume that lasers are available whose wavelengths can be precisely
tuned to anywhere in the visible range, that is, in the range 450 nm < l
< 650 nm. If every television channel occupies a bandwidth of 10 MHz,
about how many channels could be accomodated by free-space laser
communication within this wavelength range?
(A) 200 (B) 2,000 (C) 200,000 (D) 2,000,000 (E)20,000,000
[2] Consider the elements krypton and rubidium. (a) Which is more
suitable for use in a Stern-Gerlach experiment of the kind described in
the book? (Beam of atoms sent through a non-homogeneous magnet.) (b)
Which, if either, would not work at all?
(A)Kr,Rb won't work (B)Rb,Kr won't work (C)Kr,both work (D)Rb, both
work
[3] From which atom of each of the following pairs is it easier to remove
an electron: (a) Kr or Br? (b) Rb or Cs? (c) He or H?
(A) Kr,Rb,He (B)Br,Rb,He (C)Kr,Cs,He (D) Br,Cs,He (E) Kr,Rb,H
(F)Br,Rb,H (G)Kr,Cs,H (H) Br,Cs,H
[4] Which of the following statements, if any, are true (a) At low enough
temperatures, Si behaves as an insulator (b) At high enough temperatures,
Si becomes a good conductor. (c) At high enough temperatures, Si behaves
like a metal
(A)a (B)b (C)c (D) a,b,c (E)a,b (F) a,c (G)b,c (H) none
[5] Germanium (Ge, Z=32) has the same crystal structure and the same
bonding pattern as silicon. Is the net charge on a Ge ion within its
lattice,
(A) +e (B) +2e (C)+4e (D)+28e (E)+32e
[6] Of the elements arsenic (As), indium (In), tin (Sn), gallium (Ga),
antimony (Sb) and boron (B), which would produce n-type material if used
as a dopant in silicon (Si)?
(A) B (B) B,Ga,In (C)As (D)As,Sb (E) In,Sn,Sb
[7] In your body, is the ratio of protons to neutrons:
(A) about equal (B)greater than one (C) less than one (D) depends
on the number of electrons (explain)
[8] The radionuclides 209At and 209Po emit alpha particles with energies
of 5.65 MeV and 4.88 MeV respectively. Which nuclide has the longer
half-life?
(A) At (B) Po (C) not enough information given (explain)
[9] Radionuclides decay exponentially. Batteries, stars and students also
decay and burn out. Which statements describing these decays are true:
(a) All decays are exponential. (b) Half-lives only describe exponential
decay. (c) The half-life of students taking Physics is about 2 semesters
(30 weeks).
(A) a (B) b (C) c (D) a,b (E) a,c (F) b,c (G) none of the above
(explain)
[10] Suppose that a 238U nucleus "swallows" a neutron and theen decays
not by fission but by beta decay, emitting an electron and a neutrino.
Which nuclide remains after this decay:
(A) 239Pu (B) 238Np (C)239Np (D) 238Pa (E) None of
the above (explain)
[11] Which of these elements is NOT "cooked up" by equilibrium
thermonuclear fusion processes in stellar interiors:
(A) Carbon, C (B) Neon, Ne (C) Silicon, Si (D) Chromium, Cr
(E) Bromine, Br (F) None of the above (explain)
[12] Not only particles such as electrons and protons, but also entire
atoms can be classified as fermions or bosons, depending on whether their
overall spin quantum numbers are, respectively, half integral or
integral. Consider the helium isotopes 3He and 4He. Which of the
following statements is correct?
(A) both are fermions (B) both are bosons (C) 4He is a fermion and 3He a
boson (D) 4He is a boson and 3He a fermion.
[13] An electron cannot decay into two neutrinos. Which of the following
conservation laws would necessarily be violated if it did: (Circle all
that apply.)
(A) Energy (B) Angular Momentum (C) Charge (D) Lepton number (E) Linear
momentum (F) Baryon number (G) Strangeness
[14] If the entire universe we observe were compressed into a point not
much bigger than a proton, as the Big Bang theory would suppose occurred
at 10e-43 s, which of the following conservation laws would necessarily
be violated (and why):
(A) Energy (B) Angular Momentum (C) Charge (D) Lepton number (E) Linear
momentum (F) Baryon number (G) Strangeness
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[15] In the game "snap the whip", the child on the end of the chain of
children loses his grip and falls/flies off when the wave reaches him
because:
(A)He is the smallest person
(B)The wave amplitude doubles by reflection
(C)He can only hold on with one hand
(D)The wave speed increases down the chain
[16] A standing wave on a string is called "standing" because:
(A) it has nodes where the string doesn't move
(B) it has no velocity because its wavenumber is zero
(C) it can't move because the ends of the string are tied down
(D) it really is moving, but too fast to see
[17] A jet engine produces 130 dB noise at 10 feet, which is why airport
staff wear those big orange ear muffs. If the approach to the airport
brings the jets 1000 feet above your house, what is the noise intensity
in your front yard?
(A) 130 dB (a rock concert) (B) 110 dB (a day care)
(C) 90 dB (a party) (D) 70 dB (a conversation) (E) 50 dB (a murmur)
[18] In my grandmother's day, the baby in its mother's womb was thought
to come alive (the quickening) around the 15th week of gestation when the
mother could feel it kick. Today, a "Doppler stethoscope" can detect the
baby's heart beating at 6 or 7 weeks gestation, by bouncing ultrasound
waves off the beating heart. If the ultrasound has a frequency of 1.00
Mhz, and the heart contracts at a maximum speed of 1.00 m/s, what is the
beat frequency heard by the receiver?
(A)300 Hz (B) 675 Hz (C) 1480 Hz (D) 998 kHz (E) 1.001MHz
[19] I once heard a Christmas concert performed on handbells. Handbells
have a very pure tone, that can be approximated as a sine wave. To my
dismay and annoyance, one of the bells appeared to not work. I could see
the bellplayer shake the bell, but I never heard the note. I was sitting
under a balcony which created a closed air space at the back of the music
hall. If we can assume that the five walls around me (balcony, floor,
left,right and back wall) formed a closed tube, then I must have been
sitting at a nodal point in a standing wave. If we assume that there
existed only one node, and the back wall was 1.5 meters behind me, what
was the frequency of the handbell that I couldn't hear?
(A) 57 Hz (B) 89 Hz (C) 114 Hz (D) 179 Hz (E) 228 Hz
[20] Two right triangular glass prisms are used in periscopes in
submarines (or windows on tanks). Their purpose is to reflect the light
down into the submarine. Which sentence describes best how they work.
(A) The light reflects at 45 degrees because it is a mirror -plated
surface.
(B) The light reflects at 45 degrees because it is at Brewster's angle
(C) The light reflects at 45 degrees because it is less than the
critical angle.
(D) The light reflects at 45 degrees because it is more than the
critical angle.
[21] If I hold a shiny dinner spoon up to my eye I see an upright
magnified eyeball. As I move the spoon farther away, I see it become
cloudy and then I reappear small and upside down. The best description of
these views of myself are:
(A) A virtual image that becomes a real image as I move the focal point.
(B) A virtual image that goes from magnified to reduced as I move the
focal point.
(C) A real image that becomes a virtual image as I move the focal point.
(D) A real image that goes from magnified to reduced as I move the focal
point.
[22] At night many people see rings (called entopic halos) surrounding
bright outdoor street lights in otherwise dark surroundings. The rings
are the first of the side maxima in diffraction patterns produced by
structures that are thought to be within the cornea of the observer's
eye. (The central maxima of such patterns overlap the lamp.) Would a
particular ring become smaller or larger if the lamp were switched from
blue to red light? If a lamp emits white light, is blue or red on the
outside edge of the ring?
(A)larger, blue (B) smaller, blue (C)larger, red (D)smaller, red
[23] I don't know it for a fact, but it seems that most women I know have
bigger eyes than men. It is a fact, however, that stars come in different
colors, with Betegeuse being red and Rigel being blue. The ancient Arabs,
who gave them those names, had a "eye-chart" test that required the
observer to see both stars of a close binary system. Who would have the
best eyesight, for which type on a pair of similar stars?
(A) women, Betelgeuse (B)men, Betelgeuse (C) women, Rigel (D)men, Rigel
[24] In the following figure, ship A sends a laser pulse to an oncoming
ship B, while scout ship C races away. The indicated speeds of the ships
are all measured from the same reference frame. Rank the ships according
to the speed of the pulse as measured from each ship, greatest first.
0.5c C <======= 0.4c A ==========> <======== 0.3c B
(A) a,b,c (B) c,b,a (C) a,c,b (D) b,a,c (E) none of the above
[25] A metal plate is illuminated with light of a certain frequency.
Which of the following determine whether or not electrons are emitted:
(A) the light intensity;
(B) the length of exposure time to light;
(C) the thermal conductivity of the plate;
(D) the material of the plate.
(E) the area of the plate;
[26] Which has the greatest effect on the tunnelling transmission
coefficient T of a potential energy barrier:
(A)Raising the barrier height by 1%
(B)Lowering the particle energy by 1%
(C)Lengthening the tunnelling barrier by 1%
(D)Increasing the mass by 1%
(E) All have identical effect.
Please answer 3 of the following 6 problems:
[1P] Can an incoming intercontinental ballistic missile be destroyed by
an intense laser beam? A beam of intensity 10^8 W/m^2 would probably burn
into and destroy a hardened (non-spinning) missile in 1s. (a) If the
laser had 5.0 MW power, 1500 nm wavelength (an infrared YAG laser), and a
4.0 m beam diameter (a very powerful laser indeed, similar to the one
deployed at the Starfire Optical range in New Mexico), would it destroy a
missile at a distance of 3000 km? (b) If the wavelength could be changed,
what maximum value would work? (Use the circular aperature diffraction
formula to determine the area of the central disk.)
[2P] (a) Find the angle theta between adjacent nearest-neighbor bonds in
the diamond lattice. Recall that each carbon atom is bonded to four of
its nearest neighbors. The four neighbors form a regular tetrahedron: a
three-sided pyramid whose sides and base are equilateral triangles. (b)
Find the bond length, given that the atoms at the corners of the
tetrahedron are 318 pm apart.
[3P] A potassium chloride crystal has an energy band gap of 7.5 eV above
the topmost occupied band, which is full. (a) Is this crystal opaque or
transparent to light of wavelength 137 nm? (b) We doped this crystal with
an impurity that formed an energy band within the band gap, making it
opaque to wavelengths 400 nm and shorter. What is the impurity band
energy level? (And what color is the crystal?)
[4P] A 6.00g charcoal sample from an ancient fire pit has a 14C activity
of 69.5 disintegrations per minute. A living tree has a 14C activity of
15.5 disintegrations per minute per 1.00 g. The half-life of 14C is 5730
y. (a) How old is the charcoal sample? (b) What is the precision of your
measurement? (c) What assumptions did you make in part that might affect
the accuracy of your measurement? (d) Explain the difference between
precision and accuracy.
[5P] An early proposal for a hydrogen bomb had a spherical fission shell
surrounding a fusion core. The fusion fuel is deuterium, 2H. The high
temperature and particle density needed for fusion are provided by an
atomic bomb "trigger," which involves a 235U or 239Pu fission fuel that
is arranged to impress an imploding, compressive shock wave on the
deuterium. The operative fusion reaction is:
5 2H ==> 3He + 4He + 1H +2n.
(a) Calculate Q for the fusion reaction. (b) Calculate the TNT rating of
the fusion part of the bomb if it contains 500 kg of deuterium, 30% of
which undergoes fusion.
[6P] Will the universe continue to expand forever? To attack this
question, make the (reasonable?) assumption that the recessional speed,
v, of a glaxy a distance r from us is determined only by the matter that
lies inside a sphere of radius r centered on us. If the total mass
inside this sphere is M, the escape speed v_e from the sphere is given
by:
v_e = (2 GM/r)1/2 .
(a) Show that to prevent unlimited expansion, the average density, r,
inside the sphere must be at least equal to:
r = 3H2 / 8 p G
(b) Evaluate this "critical density" numerically; express your answer in
terms of hydrogen atoms per cubic meter. Measurements of the actual
density of space are difficult and are complicated by the presence of
"dark" matter. (c) What are the two types of "dark" matter?
Solve 3 of the additional following 6 problems:
[7P] An illuminated slide of my son's first birthday party is held 44 cm
from a projection screen. How far from the slide must a lens of focal
length 11 cm be placed in order to form an image of the proud father's
face on the screen?
[8P] A guitar string 1.0 meter long is oscillating as a 3 node standing
wave with an amplitude of 5 mm. The wave speed is 100 m/s. (a) What is
the frequency? (b) Write equations for 2 waves that, when combined, will
result in this standing wave.
[9P](a) Can a person, in principle, within a human lifetime, travel from
the Earth to the center of our galaxy, 5,000 ly away? Explain using
either time-dilation or length-contraction arguments. (b) What constant
speed would be needed to make the trip in 30 y of proper time?
[10P] What are the wavelengths for: (a) a photon with an energy of 10.0
keV, (b) an electron with an energy of 10.0 keV, (c) a proton with an
energy of 10.0 keV?
[11P] Laser light of wavelength 550 nm passes through a double-slit
arrangement at the front of a lecture room, reflects off a mirror 15.0 m
away at the back of the room, and then produces an interference pattern
on a screen at the front of the room. The distance between adjacent
bright fringes is 10.0 cm. (a) What is the slit separation? (b)What
happens to the pattern when the lecturer places a thin cellophane sheet
over one slit, thereby increasing by 2.50 the number of wavelengths along
the path that includes the cellophane?
[12P] A particle in the solar system is under the combined influence of
the Sun's gravitational attraction and the radiation force due to the
Sun's rays. Assume that the particle is a black sphere of density 1000
kg/m3 and that all the incident light is absorbed. (a) Show that, if its
radius is less than some critical radius, R, the particle will be blown
out of the solar system. (b) Calculate R.