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Physics GRE Sample Exam 1

Version 1.0.0: June 25, 2011

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TABLE OF INFORMATION

Rest mass of the electron me = 9.11 1031 kilogram = 9.11 1028 gramMagnitude of the electron charge e = 1.60 1019 coulomb = 4.80 1010 statcoulomb (esu)

Avogadros number NA = 6.02 1023

per moleUniversal gas constant R = 8.31 joules/mole K

Boltzmanns constant k = 1.38 1023 joule/K = 1.38 1016 erg/KSpeed of light c = 3.00 108 m/s = 3.00 1010 cm/s

Plancks constant h = 6.63 1034 joule second = 4.14 1015 eV second = h/2

Vacuum permittivity 0 = 8.85 1012 coulomb2/(newton meter2)Vacuum permeability 0 = 4 107 weber/(ampere meter)

Universal gravitational constant G = 6.67 1011 meter3/(kilogram second2)Acceleration due to gravity g = 9.80 m/s2 = 980 cm/s2

1 atmosphere pressure 1 atm = 1.0 105 newtons/meter2 = 1.0 105 pascals (Pa)1 angstrom 1 A = 1 1010 meter

1 weber/m2 = 1 tesla = 104 gauss

Moments of inertia about center of mass

Rod1

12M l2

Disc

1

2 M R2

Sphere2

5M R2

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PHYSICS TESTTime 170 minutes

100 questions

Directions: Each of the questions or incomplete statements below is followed by five

suggested answers or completions. Select the one that is best in each case and then fill in

the corresponding space on the answer sheet.

1. A centrifuge can be used to simulate largegravitational forces. Consider a centrifuge

consisting of an arm of length 4 meters, ro-tating about a fixed pivot at constant speed.What must this speed be to simulate a grav-itational acceleration of 9g?

(A) 2

g m/s

(B) 3

g m/s

(C) 6

g m/s

(D) 18

g m/s

(E) 36

g m/s

2. A block of mass m moving with velocity vcollides with a heavier block of mass 3m,initially at rest. If the blocks stick togetherafter collision, they move with velocity V.If the collision is perfectly elastic and thelighter block bounces back in the oppositedirection, the heavier block moves with ve-locity V. What is V/V?

(A) 1/4

(B) 1/2

(C) 1

(D) 3/2

(E) 2

3. An LC circuit, consisting of a solenoid anda parallel-plate capacitor, has resonant fre-

quency . If the linear dimensions of all cir-cuit elements are doubled, the new resonantfrequency is:

(A)

2

(B) 2

(C)

(D) /2

(E) /

2

4. A point dipole with dipole moment p = pzis placed at the center of a thin sphericalconducting shell of radius R. What is theelectric field outside the shell?

(A)1

40

p

r2Rr

(B) 0

(C)1

40

3(p r)r pr3

(D)

1

40

p

r2Rr

(E) 140

3(p r)r pr3

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5. The ionization energy of helium is 79 eV.If the ground-state wavefunction of heliumwere a simple product of 1s wavefunctions,100(r1)100(r2), the predicted ionizationenergy would be 108 eV. What is the MAIN

factor that accounts for this discrepancy?

(A) electron-electron Coulomb repulsion

(B) Pauli exclusion principle

(C) spin-spin coupling between the orbitalelectrons

(D) spin-spin coupling between the nucle-ons

(E) none of these

6. The energy of gamma rays from a transitionof a nucleus from the first excited state toits ground state is measured. Which of thefollowing is true of the measurement?

(A) Gamma rays from this transition arepart of a continuum of gamma raysfrom the de-excitation of low-lyingstates.

(B) The measured mean energy must cor-respond to the energy of a vibrationalstate of the nucleus.

(C) The measured width of the spectralpeak must be /(2), where is thelifetime of the excited state.

(D) The measured mean energy is greaterthan the true transition energy.

(E) The measured mean energy is less thanthe true transition energy.

Questions 7-8 refer to a gas of electrons,which is confined to a 2-dimensional surfaceat z = 0 but is otherwise free to move in thex and y directions. The temperature of thesystem is well above the Fermi temperature.

7. What is the specific heat per particle of theelectron gas?

(A) 1/2k

(B) k

(C) 3/2k

(D) 2k

(E) 5/2k

8. An external magnetic field is applied so that

the electrons feel a harmonic oscillator po-tential, U = 12m

2(x2 + y2). What is thespecific heat per particle of the electron gas?

(A) 1/2k

(B) k

(C) 3/2k

(D) 2k

(E) 5/2k

9. A particle with mass m and angular momen-

tum l moves in a constant central potentialU(r) = k/r. What, if any, is the radius ofits stable circular orbit?

(A) The particle has no allowed stable cir-cular orbit

(B) l2/(mk)

(C) l2/(2mk)

(D) 2l2/(mk)

(E) 2l2/(3mk)

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10. A disk (shown edge-on in the figure) hasthickness h, density (r) = Ar3, mass M,and radius R. What is its moment of iner-tia about axis X, which is perpendicular tothe axis of the disk and passes through the

edge of the disk?

(A) MR2

(B) MR2/2

(C) 12M R2/7

(D) 3M R2/2

(E) 5M R2/7

11. What is the correct relationship betweenphase velocity, vphase, and group veloc-

ity, vgroup, for a quantum-mechanical wavepacket?

(A) vphase = vgroup

(B) vphase = 2vgroup

(C) vphase =12vgroup

(D) vphasevgroup = c2

(E) none of these

12. A free particle incident from x = en-counters a potential barrier V(x) = A(x)at x = 0. The correct boundary conditionsfor the wavefunction near x = 0 are:

(A) (0+) = (0

)

d

dx|x=0+

d

dx|x=0

=2mA

2(0)

(B) (0+) = (0)

d

dx|x=0+

d

dx|x=0

=2mA

2

(C) (0+) = (0)

d

dx|x=0+

d

dx|x=0

= 2mA2

(0)

(D) (0+) (0) = 2mA2

d

dx|x=0+ =

d

dx|x=0

(E) (0+) = (0)

d

dx|x=0+ =

d

dx|x=0

13. What is the numerical value of the Planckmass (not the reduced Planck mass)?

(A) 1.22 1054 kg(B) 6.63 1034 kg(C) 1.06 1034 kg(D) 4.34 109 kg(E) 2.18 108 kg

14. A spaceship traveling at 0.6c toward aplanet transmits a signal at 1 GHz to theplanets inhabitants. What frequency is thesignal when it is received on the planet?

(A) 1GHz

(B) 2 GHz

(C) 2.5 GHz

(D) 4 GHz

(E) 8 GHz

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15. A point charge of mass m and charge q isheld a distance d above an infinite conduct-ing plate. When the charge is released, itwill be attracted towards the plate. Whatis its speed when it reaches a distance d/2

above the plate?(A) q/

8m0d

(B) q/

4m0d

(C) 2q/

m0d

(D) 4q/

m0d

(E) q2/

16m0d2

16. An initially uncharged 1-F parallel-platecapacitor is charged with a constant currentof 1 A. What is the displacement current in

the gap between the plates while the capac-itor is being charged?

(A) 1 A

(B) 2 A

(C) 4 A

(D) 1/2 A

(E) 1/4 A

17. A particle in a 1-dimensional infinite square

well between x = 0 and x = L is subject tothe following perturbation

V(x) =

V0, x < L/20, otherwise.

What is the leading-order shift in the energyof the first excited state?

(A) V0(B) V0

(C) V0/4

(D) 0

(E) V0/2

18. Which of the following is NOT true aboutthe isothermal expansion phase of a Carnotcycle?

(A) The free energy of the gas increases.

(B) The entropy of the gas increases.(C) The isothermal expansion phase is re-

versible.

(D) The expansion takes place at the tem-perature of the hot reservoir.

(E) The gas does work on its surroundings.

19. Monochromatic blue light of wavelength 450nm is shined on a slit of width a. A diffrac-tion pattern is observed on a screen 10 maway. What must a be such that the widthof the central diffraction maximum is 100times the width of the slit?

(A) 45 nm

(B) 450 nm

(C) 0.045 mm

(D) 0.21 mm

(E) 0.30 mm

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Questions 20-21 refer to a particle of massm, confined to the surface of a torus withinner radius a and cross-sectional radius b,oriented such that the Earths gravitationalfield points perpendicular to the plane of thecircle of radius a.

20. Letting and be the angular coordinateson the circles of radii a and b, respectively,what is a possible Lagrangian for this sys-tem?

(A) 12m(a22 + b22)mgb sin

(B) 12m(a22 + b22)mgb cos

(C) 12m((a + b cos )22 + b22)mgb sin

(D) 12m((a + b sin )22 + b22)mgb sin

(E) 12m((a + b cos )22 + b22)

mgb cos

21. Which of the following is a conserved quan-tity for this system?

I. m(a + b cos )2

II. 12m[(a + b cos )22 + b22] + mgb sin

III. mb2

(A) I only

(B) II only

(C) III only(D) I and II

(E) I and III

22. A resistor with resistance R and an induc-tor with inductance L are in series with avoltage source. For t < 0, the voltage is 0.For t > 0, the voltage source is V. At whattime t does it take for the voltage across the

inductor to drop to half of its initial level?

(A)L ln 2

R

(B)L

R

(C)L

R ln 2

(D)2L

R(E) 0

I

i

23. A straight wire carrying current I passesthrough the center of a circular wire car-rying current i. If the circular loop of wirehas radius R, what is the tension on the cir-cular wire due to the field produced by thestraight wire?

(A) 0iI/(2R2)

(B) 0I2/(2R)

(C) 0i2/(2R)

(D) 0iI/(2R)

(E) 0

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24. A uniform cylinder of height h and radius ris placed on a flat surface and tipped at anangle from the vertical. Find 0 such that

when the cylinder is released from > 0, itfalls over.

(A) arctan(2r/h)

(B) arctan(r/h)

(C) arctan(r/2h)

(D) arccos(2r/h)

(E) arccos(r/h)

25. Consider a beam of muons produced in

the upper atmosphere (100 km above theEarths surface) with energy 3 GeV. Themuons mass is approximately 100 MeV, andits lifetime at rest is 2 106 s. The frac-tion of muons detected at ground level ismost nearly

(A) e0.5

(B) e1

(C) e5

(D) e10

(E) e15

26. Graphene, a 2-dimensional allotrope of car-bon, displays unusual electronic properties.In particular, the dispersion relation forelectrons in graphene is:

(A) |k|

(B) |k|(C) |k|2(D) |k|3(E) |k|4

27. Consider an electron subject to a Coulombpotential V(r) = e240 1r and a uniformelectric field E = E0z. For small E0,the leading-order nonzero correction to the

n = 2 states is:

(A) independent of E0

(B) proportional to E0

(C) proportional to E20

(D) proportional to E30

(E) proportional to E40

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5cm

2cm

BA

28. In the optical arrangement shown above,converging lenses A and B both have focallength 5 cm. An object is placed 2 cm tothe left of lens A. Where is the image of theobject located?

(A) 5 cm to the right of B

(B) 6.25cm to the right B

(C) 12.5 cm to the left of B

(D) 12.5 cm to the right of B

(E) No image is formed.

29. A square picture frame is made of four iden-tical rods of mass M, length L, and uniformmass density. The frame is hung from the

wall by two massless wires, also of of lengthL, attached to a nail, as shown in the figure.What is the frames moment of inertia withrespect to rotations about the axis throughthe nail?

(A) 14ML2

(B)83 +

3

M L2

(C)163 + 2

3

ML2

(D)

193 + 2

3

ML2

(E)416 + 33ML2

30. Which values of spin quantum numbers areNOT possible for system consisting of spin-1particle and a spin-2 particle?

(A) l = 3, m = 3

(B) l = 1, m = 0

(C) l = 2, m = 1

(D) l = 2, m = 0

(E) l = 0, m = 0

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31. The radial wavefunction of the 2p state ofhydrogen is

R21(r) =124

a5/20 r exp(r/2a),

where a0 is the Bohr radius. What is theexpectation value of r in this state?

(A) 5a0

(B) 2a0

(C) a0

(D) a0/2

(E) a0/5

32. The device commonly used to measure

the energies of particles produced in high-energy collisions at particle accelerators iscalled:

(A) veto

(B) return yoke

(C) calorimeter

(D) tracker

(E) beam dump

33. A musician tuning a violin to a tuning fork

at 440 Hz hears a beat frequency of 3 Hz.What is the frequency of the note producedby the violin?

(A) 428 Hz

(B) 434 Hz

(C) 437 Hz

(D) 443 Hz

(E) It is impossible to tell from the giveninformation

34. The USS Enterprise, moving at speed 0.6cwith respect to a nearby planet, fires a pho-ton torpedo of speed c at a retreating Romu-lan warship, initially 300 km away. Accord-ing to the Enterprises clock, the torpedo

made contact with the warship 10 secondsafter firing. How fast was the warship trav-eling, in the frame of the planet?

(A) 0.975c

(B) 0.9975c

(C) 0.99975c

(D) 0.999975c

(E) c

35. An ice skater is spinning with arms extendedat an angular velocity of 5.0 radians/sec.After drawing her arms in, her new angularvelocity is 8.0 radians/sec. If the skatersmoment of inertia with arms extended wasI, her moment of inertia with arms drawnin is:

(A) I

(B) 3I

(C) (8/5)I

(D) (5/8)I

(E)

5/8I

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36. Suppose that a particle in one-dimensionalsystem has a Lagrangian L with a potentialthat is constant in time and such that

L

t= 0

Lx

= 0.

Which of the following must be true?

I. Energy is conserved

II. Linear momentum is conserved

III. The potential is nonzero

IV. The Euler-Lagrange equations are notsatisfied.

(A) I. only

(B) II only

(C) I and II

(D) II and III

(E) III and IV

37. A beam of particles with luminosity1022 cm2s1 is incident upon a target withscattering cross section 1020 cm2. Assum-

ing a detector has an efficiency of 0.5 for de-tecting products of the scattering process,how many events will the detector see if theexperiment runs for 1 day?

(A) 4.32 106 events(B) 8.64 106 events(C) 4.20 102 events(D) 4.0 106 events(E) 7.2

104 events

38. An electron in a cyclotron moves in a circu-lar orbit at a fixed radius in the presence ofa constant magnetic field B. If the strengthof the magnetic field is tripled, by what fac-tor must the electrons momentum change

to keep it orbiting at the same radius?(A)

3

(B) 3

(C) 1/

3

(D) 1/3

(E) 3/2

39. Two circular loops (radius b) of current,with centers at (0, 0, a/2) and (0, 0,a/2),are oriented parallel to the xy-plane. One

loop carries a current I in the directionand the other carries current I in the direction. What is the magnitude of themagnetic field on the z-axis for z a, b?

(A) 30Iab/(4z3)

(B) 0Iab2/(4z4)

(C) 30Ia2b/(8z4)

(D) 30Iab2/(2z4)

(E) 0

40. Which of the following is true about the to-tal orbital angular momentum operator, L2,of a particle subjected to an arbitrary force?

I. Always commutes with Lx, Ly, Lz

II. Always commutes with the total spinJ2

III. Always commutes with the Hamilto-nian

(A) I only(B) II only

(C) III only

(D) I and II

(E) I, II, and III

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41. A quantum system has a Hamiltonian givenby

H =

a 0 00 0 ib

0 ib 0

,

where a,b,c are real positive constants.What are the possible results of a measure-ment of the energy of the system?

(A) b, a(B) a, b(C) a, b, a + b

(D) a, ab(E) a, b2.

42. If magnetic monopoles were discovered,which of the following would no longer bea globally well-defined quantity?

(A) B(B) The magnetic vector potential A(r)

(C) A magnetic dipole moment

(D) The stress-energy tensor

(E) E

Ev R

, ,

43. A beam of nonrelativistic protons (mass m,charge q) and neutrons of velocity v entersa region of length L with an electric field Eperpendicular to the direction of the beam.At the end of the region of length L, thebeam strikes a circular target of radius R.Assuming that the diameter of the beam ismuch smaller than R, what is the minimumelectric field needed to deflect all protonsbefore they strike the target?

(A) E = mLv2/(2qR2)(B) E = 2mLv2/(qR2)

(C) E = mRv2/(2qL2)

(D) E = 2mRv2/(qL2)

(E) E = 4mLv2/(qR2)

44. Put the following in chronological order,starting with the earliest.

I. Epoch of reionization

II. Nucleosynthesis

III. Big bang

IV. Lepton epoch

(A) I, II, III, IV

(B) III, I, II, IV

(C) III, II, IV, I

(D) III, I, IV, II

(E) III, IV, II, I

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45. For a monoatomic ideal gas, which of the fol-lowing is constant during adiabatic changesof state?

(A) P V1/2

(B) P V(C) P V5/3

(D) P V7/5

(E) P V9/7

R

M

g

46. A string of length L and negligible mass iscompletely wound around a solid cylinder ofuniform density, of mass M and radius R,and it has a small weight of mass m attachedto its end. If the weight is released from restunder the influence of gravity, what is its ve-locity when the string is entirely unwound?

(A)

2mgLMR2m

(B)

2mgLMR22m

(C)

2gL

(D) 2mgL(2/3)MR2m

(E)

2mgL2MR2m

47. A particle orbits in a central potential andhas an energy E such that Vmin < E < 0,where Vmin is the minimum of the effectivepotential. What conic section correspondsto the shape of the orbit?

(A) circular

(B) elliptical

(C) parabolic

(D) hyperbolic

(E) A body with this energy does not orbit

48. Consider a system with three energy lev-els , 0, , and degeneracies d() = 2,d(0) = 1, d() = 3. What is the energy ofthe system as T?(A) /5

(B) /6

(C) 5/6

(D) 0

(E)

49. The heat capacity per particle at constantvolume of a relativistic ideal gas is:

(A) 3k/2

(B) k/2

(C) k

(D) 2k

(E) 3k

50. An electromagnetic wave propagates in vac-uum with electric field E0 cos(kx t)z.What is the average magnitude of thePoynting vector, where the average is takenover one period of oscillation?

(A) 4E20/(c0)

(B) 0

(C) E20/(c0)

(D) E20/(2c0)

(E) E20/(2c0)

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51. An observation of the reaction e+e would violate which of the following conser-vation laws?

(A) lepton number

(B) photon number

(C) angular momentum

(D) energy-momentum

(E) baryon number

52. The nucleus can be considered as a degener-ate Fermi gas. A neutron scattering experi-ment determines that the Fermi momentumof nucleons in the carbon nucleus is about40 MeV/c. Which of the following is ap-proximately an absolute lower bound on the

nuclear radius?

(A) 976 fm

(B) 87.4 fm

(C) 1.23 fm

(D) 0.132 fm

(E) 0.00510 fm

53. Which of the following does NOT obeyBose-Einstein statistics?

(A) neutrinos(B) Cooper pairs

(C) photons

(D) 4He atoms

(E) pions

54. The observation of a sharp line of gammarays of energy 511 keV from the center ofour galaxy is most naturally explained bywhich of the following processes?

(A) hydrogen hyperfine transitions(B) Hawking radiation

(C) ammonia maser transitions

(D) electron-positron annihilation

(E) supernovae

cannon

enemy100m

300m

T

55. A soldier can fire a cannon horizontally fromthe top of a 100 m cliff. He wants to hitan enemy encampment at 300 m horizontaldistance from the cliff. What must the ini-tial velocity of his cannonball be in order tostrike the encampment, neglecting air resis-tance?

(A) 22.4 m/s

(B) 47.4 m/s

(C) 67.1 m/s

(D) 94.9 m/s

(E) 134.2 m/s

56. In empty space, a spaceship of mass M canaccelerate by expelling matter at velocity v0relative to the spaceship from its exhaust.How much matter must be exhausted in or-

der to achieve a final velocity v?

(A) M(1 exp(vf/v0))(B) M(1 exp(vf/2v0))(C) Mexp(vf/2v0)

(D) Mexp(vf/v0)

(E) M(1 exp(2vf/v0))

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57. Consider the circuit shown in the diagram.When switch S is open, the current throughthe 10 k resistor is I1. After switch S isclosed, the current through the same resistoris I2. What is I2/I1?

(A) 3/5(B) 11/6

(C) 1

(D) 4/5

(E) 1/6

58. A rod of length L and mass M is attachedto a pivot and suspended at an angle fromthe vertical using a support wire, as shownin the diagram. A lump of clay of mass m isfired at the end of the rod with a velocity v.Just before the clay makes contact with therod, the wire is cut. Assuming the clay androd stick together after collision, and thatthe rod does not rotate past vertical, whatis the maximum angle the rod attains? Youmay treat the lump of clay as a point mass.

(A) cos1

cos 3(M + m)mv2 cos2

(M + 2m)(M + 3m)gL

(B) cos1

cos 3m2v2 cos2

(M + 2m)(M + 3m)gL

(C) cos1

cos 3m2v2 cos2

M(M + 2m)gL

(D) cos1

cos 3m2v2

(M + 2m)(M + 3m)gL

(E) cos1

cos 3m2v2 cos2

(M + m)(M + 2m)gL

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59. A square loop of wire of side length L,containing a load resistor R, is orientedperpendicular to the xy-plane and rotatesabout the z-axis at angular frequency inthe presence of a uniform magnetic field

B = B0x, as shown in the diagram. IfL = 10 cm, B0 = 2 tesla, and R = 100.0 ,what must be so that the average powerdissipated in the resistor is 1.0 W?

(A) 25 rad/sec

(B) 50 rad/sec

(C) 314 rad/sec

(D) 500 rad/sec

(E) 707 rad/sec

60. In calculating the entropy of a micro-canonical ensemble, the inverse temperature = 1/kT can be viewed as a Lagrangemultiplier enforcing the constraint of fixedtotal energy. Similarly, the chemical poten-tial is related to the Lagrange multiplierfor:

(A) fermion number

(B) particle number

(C) pressure(D) volume

(E) magnetization

61. A spin-1/2 particle interacts with a mag-netic field B = Bz through a HamiltonianH = (BgB/2)z, where B is the Bohrmagneton and g is the particles gyromag-netic ratio. Consider a system of these spin-

1/2 particles in equilibrium at temperatureT. Let A be the ratio of the number of spin-up particles to spin-down particles. If thestrength of the magnetic field is doubled, thenew ratio of spin-up to spin-down particlesis:

(A) A2

(B) A

(C) A2

(D) eA

(E) A exp(BgB/kT)

62. Evaluate 2(1/r).

(A) 43(r)(B) 43(r)

(C) 0

(D) 4

(E) 4

63. Which of the following is closest to theCompton wavelength of the proton?

(A) 1015 m

(B) 1013 m

(C) 1012 m

(D) 1010 m

(E) 109 m

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Questions 64-65 refer to the following sce-nario. A K0 of mass mK and energy E inthe lab frame decays to a + and a , bothof mass m. The

is observed to be emit-ted antiparallel to the K0 momentum.

64. What is the speed of the + in the labframe?

(A)

1 m2Kc4E2

1/2c +

1 4m2

m2K

1/2c

(B)

1m

2Kc

4

E2

1/2+

14m

2

m2K

1/2

1+

1m

2Kc4

E2

1/214m

2

m2K

1/2 c

(C)

1 m2Kc4E21/2

c

1 4m2m2K

1/2c

(D)1+

m2Kc4

E21/2

+1

m2Kc4

E21/2

1+

1+

m2Kc4

E2

1/21m

2Kc4

E2

1/2 c

(E)

1 4m2m2K

1/2c

65. What must be the initial K0 energy suchthat the is stationary in the lab frame?

(A) m2c

2

2mK

(B) mKc2

2

(C) mc2

2

(D)(m2K+m

2)c

2

2m

(E)m2Kc

2

2m

66. A clarinet can be treated as a half-openpipe, where sounds are produced by stand-ing pressure waves. For a clarinet of length0.6 m, which of the following is a possiblewavelength of a standing wave?

(A) 0.3 m(B) 0.6 m

(C) 0.8 m

(D) 1.2 m

(E) 1.5 m

d

a

, ,

67. A conducting sphere of radius a is placed adistance d above a large conducting plate,as shown in the diagram. The two conduc-tors are connected by a wire with a volt-age source between them. What is the ca-pacitance of this arrangement, assuming theplate is large enough that it can be treatedas infinite?

(A) 20a

(B) 20a(2d+a)

d

(C) 20d

(D) 20(2da)a(da)

(E) 20(d2a)a

d

68. Suppose an electromagnetic plane wave

propagating in vacuum in the +z-directionhas a polarization with the electric field inthe +x-direction immediately before it is re-flected at normal incidence off a perfect con-ductor. What are the directions of the Eand B vectors of the reflected wave?

(A) E in +x-direction & B in +y-direction

(B) E in x-direction & B in +y-direction(C) E in +x-direction & B in y-direction(D) E in

x-direction & B in

y-direction

(E) Electromagnetic plane waves do not re-flect off perfect conductors

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69. What is the value of the following commu-tator?

[[[Lx, Ly] , Lx] , Lx] .

(A) i3Lz(B) i3Lz

(C) i3Ly(D) i3Ly

(E) i3Lx

70. The vibrational frequency of diatomic oxy-gen is approximately 5 1013 Hz. The tem-perature at which the vibrational modes ofO2 will begin to be excited is closest to:

(A) 20 K

(B) 200 K

(C) 2000 K

(D) 20000 K

(E) 2 105 K

71. Which of the following does NOT representa possible observable, written in the positionbasis, for a free particle in three dimensions?

(A) i(B) x2/y

(C) x2/y2

(D) x2y2z2

(E) xyz

72. The BCS theory of superconductivity ex-plains the superconducting properties ofmetals at low temperature by supposingthat a macroscopic number of conductionelectrons all lie in the same ground state.

Why does this not violate the Pauli exclu-sion principle?

(A) BCS theory is incorrect

(B) Cooper pairs behave as bosons

(C) spin-spin coupling prevents electronsfrom being in the same state

(D) the Pauli exclusion principle does notapply to systems at low temperature

(E) electrons are not fermions

73. A hoop of radius R rotates at constant an-gular velocity . A small bead of mass m isattached to the hoop, with a frictional forceon the bead proportional to the differencein velocity between the bead and edge ofthe hoop, F = k(R R), where is theangular velocity of the bead. If the beadbegins at angular velocity 0, which of thefollowing describes its subsequent motion?

(A) (t) = 0ekt/m

(B) (t) = 0ekt/m

(C) (t) = 0emt/k(D) (t) = (1 ekt/m)(E) (t) = ( 0)ekt/m

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74. Consider a cylinder of radius R, mass M,length z, and density (r) = Ar that rollswithout slipping down an inclined plane ofheight h at an angle . What is the velocityof the cylinder at the bottom of the inclined

plane?(A)

3gh

(B)

2gh

(C)

gh

(D) 2

gh/3

(E)

5gh/2

75. The is a spin-3/2 bound state of threespin-1/2 quarks. The spin part of the wave-function of the state with m = +3/2 is

| = |. What is the spin part of thewavefunction with definite spin m = 1/2?(A) |(B) 1

3(| + ||)

(C) 13

(|+ ||)(D) 1

3(| + | + |)

(E) |76. What is true of the electromagnetic field at

a p-n junction at equilibrium with zero biasvoltage applied?

(A) The electric field points toward the p-type semiconductor

(B) The electric field points toward the n-type semiconductor

(C) The electric field is parallel to the in-terface between the p-type and n-typesemiconductors

(D) There is no electromagnetic field(E) There is no electric field, but there is

a magnetic field pointing toward the n-type semiconductor

77. An event E occurs at (2, 1, 1, 0) in a refer-ence frame S (in units where c = 1). Atwhat coordinates in another frame S couldan observer measure E?

(A) (1, 0.5, 1, 1)

(B) (2, 1, 0, 0)

(C) (3, 2,

3, 0)

(D) (2, 0,

3, 0)

(E) None of these

78. A dark matter experiment takes data for atime T and observes no events. What is the90% confidence level upper limit that onecan place on the event rate in the detector?

(A) One cannot place a limit at the 90%confidence level for this experiment

(B) (1/T) l n 0.9(C) (1/T) l n 0.1(D) (1/T) l n 0.9

(E) 0

79. An approximate Hamiltonian for the hydro-gen atom is

H0 =p2

2me e2

40

1

r

.

The true Hamiltonian contains all of the fol-lowing corrections to H0. Which correction,taken by itself, results in the smallest changeto the ground-state energy of H0?

(A) spin-orbit coupling between the protonspin and the electrons orbital angularmomentum

(B) first-order relativistic correction to thekinetic term

(C) spin-spin coupling between the electronand proton spins

(D) Lamb shift

(E) replacing electron mass with reducedmass of the electron-proton system

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80. An electron neutrino emitted from the sunmay be detected as a tau neutrino on Earthbecause:

(A) neutrinos are so difficult to detect thatflavor misidentification is common

(B) electron neutrinos from the sun can an-nihilate and be reemitted as a pair oftau neutrinos

(C) electron neutrinos interact with theearths magnetic field

(D) neutrino flavor eigenstates are not masseigenstates

(E) scattering on 3He in the sun causes fla-vor transmutation

81. A pair of electrons is trapped in a quan-tum dot. A magnetic field is applied alongthe z-direction so that the singlet state hasenergy , and the triplet state has energies/2, , and 3/2 for spins +, 0, and along the z-axis, respectively. What isthe probability of finding the electrons inthe triplet state, at temperature T?

(A) 0

(B) 1

(C) 22 + e/2kT + e/2kT

(D)e/2kT + e/2kT

2 + e/2kT + e/2kT

(E)1 + e/2kT + e/2kT

2 + e/2kT + e/2kT

m

m

k

k

y1

y2

, ,

82. The diagram above illustrates a system con-sisting of a block of mass m hanging froma spring of spring constant k, with anotherblock of mass m hanging from the firstblock by another spring of spring constantk. What is a possible Lagrangian for thissystem?

(A) 12m( y12 + y2

2) 12k(y21 + (y2 y1)2)+ mg(y1 + y2)

(B) 12m( y12 + y2

2)

12k(y

21 + (y2

y1)

2)

mg(y1 + y2)(C) 12m( y1

2 + y22) + 12k(y

21 + (y2 y1)2)

mg(y1 + y2)(D) 12m( y1

2 + y22) + 12k(y

21 + (y2 y1)2)

+ mg(y1 + y2)

(E) 12m( y12 + y22) + 12k(y21 + (y2 y1)2)+ mg(y1 + y2)

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Questions 83-84 refer to a particle of massm in the ground state of an infinite squarewell of size a, with energy E.

83. The well suddenly expands to size 2a. Whatis E/E, where E is the expectation value ofthe energy of the particle after this suddenexpansion?

(A) 0

(B) 1

(C) 1/

2

(D) 1/2

(E) 1/4

84. The same experiment is repeated, but the

well is now expanded slowly to size 2a.What is E/E?

(A) 0

(B) 1

(C) 1/

2

(D) 1/2

(E) 1/4

85. Which of the following is NOT true aboutthe 2s

1s transition in the hydrogen

atom?

(A) The dominant decay mode is two-photon emission

(B) It violates the selection rule l = 1(C) It violates the selection rule m = 1

or 0

(D) It cannot occur in the electric dipoleapproximation

(E) none of these

86. Measurements of the electric dipole momentof the neutron provide sensitive tests of fun-damental physics. If the neutron were foundto have a nonzero electric dipole moment,one could directly conclude that which of

the following symmetries is violated?

I. Parity

II. Charge conjugation

III. Time reversal

(A) I

(B) II

(C) III

(D) I and II

(E) I and III

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0.01 0.1 1.0 10.0Energy [MeV]

Crosss

ection[barn]

10-2

100

102

104

total

ca b

T 11

87. The figure above shows the total cross sec-tion for photon scattering on a Pb atom aswell as the cross sections for several indi-

vidual process. Identify the processes corre-sponding to the partial cross sections a, b,and c.

(A) a = Compton scattering; b = photo-electric effect; c = pair production

(B) a = photoelectric effect; b = pair pro-duction; c = Compton scattering

(C) a = photoelectric effect; b = pair pro-duction; c = Compton scattering

(D) a = photon absorption; b = Comptonscattering; c = pair production

(E) a = Compton scattering; b = photonabsorption; c = pair production

88. The following are the first three normalizedLegendre polynomials:

P0(x) =1

2

P1(x) =3

2 x

P2(x) =

5

8(3x2 1).

What is the value of the following integral?

0

9

2cos4 sin 3cos2 sin + 1

2sin

d.

(A) 1(B)

16/5

(C) 0

(D) 4/5

(E) 5/8

89. Suppose that the magnetic field in a regionof space is given by B = B0(x+2xz). Whichof the following could be the vector poten-tial?

(A) B0(xy + x2z)

(B) B0(xy + x2

z)(C) B0(x2y + yz)(D) B0(y

2x + zy)

(E) B0(x2y + yz)

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a

Q

90. Consider a charge configuration consistingof a ball of charge Q surrounded by a thinconducting shell of radius a. The conduc-tor initially has no net charge, but is thenconnected to ground. What is the change inenergy of the configuration?

(A) Q2/(40a)(B) Q2/(40a)

(C) Q2/(820a2)

(D) Q2/(80a)(E) Q2/(80a)

91. A recent experiment used a measurementof the Lamb shift in muonic hydrogen, anexotic atom made up of a muon orbiting aproton, to infer which of the following prop-erties of the proton?

(A) electric dipole moment

(B) spin

(C) charge radius

(D) mass

(E) magnetic moment

92. In tabletop atomic spectroscopy experi-ments using free nuclei, the difference be-tween the frequencies of emitted and ab-sorbed photons driven by the same elec-tronic transition is due to:

(A) measurement error

(B) nuclear recoil

(C) gravitational redshift

(D) time dilation

(E) none of these

93. A sequence of NAND gates can create whichof the following effective logic gates:

(A) AND

(B) OR

(C) NOT

(D) XOR

(E) all of the above

94. A particle moving in one dimension has thefollowing Lagrangian

L =1

2Aq2 Bq2.

What is the equation of motion of the par-ticle?

(A) q = (2B/A)q

(B) q = (2B/A)q(C) q = (2B/A)q

(D) q = (2A/B)q

(E) q = (2B/A)q

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95. What is the energy stored in a toroidalsolenoid with N winds, carrying current I,of radius R, and enclosing a volume V? Youmay assume that the radius r of an individ-ual wire wind is much less than the toroid

radius R.(A) 0

(B) (0N I2r2)/(4R3)

(C) (0N I2V)/(82R2)

(D) (0N2I2r2)/(4R3)

(E) (0N2I2V)/(82R2)

96. Which of the following is true about a lon-gitudinally polarized wave in 3 dimensions?

I. There are two linearly independent po-larization vectors

II. The polarization vector(s) is/are per-pendicular to the wave vector

III. The polarization vector(s) is/are par-allel to the wave vector

(A) III only

(B) II only

(C) I only

(D) I and II

(E) I and III

97. Deep water waves obey the dispersion re-lation = A

k, where A is a constant.

What is the correct relationship betweenphase velocity and group velocity for deepwater waves?

(A) vphase =12vgroup

(B) vphase = vgroup

(C) vphase = 2vgroup

(D) vphasevgroup = A4t2

(E) none of these

98. When light of 5000 A is shined on a thin filmof oil (n = 1.5) that sits on top of a mediumwith n = 2.0, the intensity of reflected lightis minimized. What is the thickness of theoil?

(A) 4 108 m(B) 8.33 108 m(C) 1.67 107 m(D) 1.25 107 m(E) 5.0 107 m

99. Suppose a particle has a wavefunction (x)given by

(x) =

3(1 x) : 0 < x < 1

0 : otherwiseWhat is the expectation value of the posi-tion of this particle?

(A) 0

(B) 1

(C) 1/12

(D) 1/4

(E) 1/2

100. What are the energy levels of a quantizedsystem consisting of a rigid rod of lengtha connecting two masses of mass m, forn N?(A) 2n(n + 1)/(ma2)

(B) 2n(n + 1)/(2ma2)

(C) 2n/(2ma2)

(D) 2n/(ma2)

(E) 2(n + 1)/(ma2)

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