Conquering Physics GRE, Lecture notes of Physics

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TABLE OF INFORMATION

Rest mass of the electron me = 9.11 × 10 −^31 kg Magnitude of the electron charge e = 1.60 × 10 −^19 C Avogadro’s number NA = 6.02 × 1023 Universal gas constant R = 8.31 J/(mol · K) Boltzmann’s constant k = 1.38 × 10 −^23 J/K Speed of light c = 3.00 × 108 m/s Planck’s constant h = 6.63 × 10 −^34 J · s = 4.14 × 10 −^15 eV · s ℏ = h/ 2 π hc = 1240 eV · nm Vacuum permittivity  0 = 8.85 × 10 −^12 C^2 /(N · m 2 ) Vacuum permeability μ 0 = 4 π × 10 −^7 T · m/A Universal gravitational constant G = 6.67 × 10 −^11 m 3 /(kg · s^2 ) Acceleration due to gravity g = 9.80 m/s^2 1 atmosphere pressure 1 atm = 1.0 × 10 5 N/m 2 = 1.0 × 10 5 Pa 1 angstrom 1 Å = 1 × 10 −^10 m = 0.1 nm

Prefixes for Powers of 10

10 −^15 femto f

10 −^12 pico p

10 −^9 nano n

10 −^6 micro μ

10 −^3 milli m

10 −^2 centi c

103 kilo k

106 mega M

109 giga G

1012 tera T

1015 peta P

Rotational inertia about center of mass

Rod

M 2

Disk

MR^2

Sphere

MR^2

Time — 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 large gravitational forces. Consider a centrifuge consisting of an arm of length 4 meters, rotating about a fixed pivot at con- stant speed. What must this speed be to simulate a gravitational acceleration of 9g? (

|g| =

(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 v collides with a heavier block of mass 4m, initially at rest. If the colli- sion is perfectly elastic, what is the velocity of the heavier block after the collision? (A) 4v (B) (1/4)v (C) v (D) (5/2)v (E) (2/5)v
3. An LC circuit, consisting of a solenoid and a parallel- plate capacitor, has resonant frequency ω. If the linear dimensions of all circuit elements are doubled, the new resonant frequency is (A)

2 ω (B) 2ω (C) ω (D) ω/ 2 (E) ω/

4. A point dipole with dipole moment p = p z ˆ is placed at the center of a thin spherical conducting shell of radius R. What is the electric field outside the shell? (A)

4 π 0

p r^2 R r ˆ (B) 0 (C)

4 π 0

3( p · ˆ r )ˆ r − p r^3 (D) −

4 π 0

p r^2 R r ˆ

(E) −

4 π 0

3( p · ˆ r ) r ˆ − p r^3

5. The ground state energy of helium is 79 eV. If the ground state wavefunction of helium were a simple product of 1s wavefunctions,  100 ( r 1 ) 100 ( r 2 ), the pre- dicted ground state energy would be 108 eV. What is the MAIN factor that accounts for this discrepancy? (A) Electron–electron Coulomb repulsion (B) Nonzero orbital angular momentum in the ground state (C) Spin–spin coupling between the orbital electrons (D) Spin–spin coupling between the nucleons (E) None of these
6. The energy of gamma rays from a transition of a nucleus from the first excited state to its ground state is mea- sured. Which of the following is true of the measure- ment? (A) Gamma rays from this transition are part of a con- tinuum of gamma rays from the de-excitation of low-lying states. (B) The measured mean energy must correspond to the energy of a vibrational state of the nucleus. (C) The measured width of the spectral peak must be ℏ/(2τ ), where τ is the lifetime of the excited state. (D) The measured mean energy is greater than the true transition energy. (E) The measured mean energy is less than the true transition energy.

7. A system of electrons is in a box of fixed volume. If the number of electrons in the box is doubled, the Fermi energy is multiplied by a factor of (A) 2−^1 /^2 (B) 2^1 /^2 (C) 2^2 /^3 (D) 2 (E) 2^3 /^2
8. A gas of electrons is confined to a two-dimensional sur- face at z = 0 but is otherwise free to move in the x- and y-directions. An external magnetic field is applied so that the electrons feel a harmonic oscillator potential, U = 12 mω^2 (x^2 + y^2 ). The temperature of the system is well above the Fermi temperature. What is the specific heat per particle of the electron gas? (A) 12 k (B) k (C) 32 k (D) 2k (E) 52 k
9. A particle with mass m and angular momentum l moves in a constant central potential U(r) = −k/r, with k > 0. What, if any, is the radius of its stable circular orbit? (A) The particle has no allowed stable circular orbit. (B) l^2 mk

(C) l^2 2 mk

(D) 2 l^2 mk

(E) 2 l^2 3 mk

X

h R

10. Two identical disks shown in the figure above, each of thickness h, radius R, and mass M, are rigidly attached at a point on their edges. What is the moment of inertia of the pair of disks about an axis X, perpendicular to the plane of the disks, which passes through the point where the disks are connected? (A) MR^2 (B) 32 MR^2 (C) 3MR^2 (D) 6MR^2 (E) 32 MRh
11. A distant galaxy is located at redshift 2. What is the observed wavelength of the 21 cm hyperfine transition line of hydrogen originating from the galaxy? (A) 7 cm (B) 10.5 cm (C) 21 cm (D) 42 cm (E) 63 cm

19. Monochromatic blue light of wavelength 450 nm is shined on a slit of width a. A diffraction pattern is observed on a screen 10 m away. What must a be such that the width of the central diffraction maximum is 100 times the width of the slit? (A) 45 nm (B) 450 nm (C) 0.045 mm (D) 0.21 mm (E) 0.30 mm

a

m

g

b

Questions 20 and 21 refer to a particle of mass m, con- fined to the surface of a torus with central radius a and cross-sectional radius b, oriented such that the Earth’s gravitational field points perpendicular to the plane of the circle of radius a. Letting φ and θ be the angular coordinates on the circles of radii a and b, respectively, a Lagrangian for this system is

L =

m(a + b cos θ)^2 φ˙^2 +

mb^2 θ˙^2 − mgb sin θ.

20. What is the conjugate momentum to φ?

(A) 12 m φ˙(a + b cos θ)^2 (B) m φ˙(a + b cos θ)^2 (C) 12 mb^2 θ˙ (D) mb^2 θ˙ (E) mgb cos θ

21. Which of the following quantities represents the total energy? (A) L (B) L + mgb sin θ (C) L − mgb sin θ (D) L + 2 mgb sin θ (E) L − 2 mgb sin θ
22. A resistor with resistance R and an inductor with induc- tance L are in series with a voltage source. For t < 0, the voltage is 0. For t > 0, the voltage source is V. What time 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)

2 L

R

(E) 0

I

i

23. A straight wire carrying current I passes through the center of a circular wire carrying current i. If the circu- lar loop of wire has radius R, what is the tension on the circular wire due to the field produced by the straight wire? (A) μ 0 iI 2 πR^2 (B) μ 0 I^2 2 πR (C) μ 0 i^2 2 πR (D) μ 0 iI 2 πR (E) 0

h (^) g

r

θ

24. A uniform cylinder of height h and radius r is placed on a flat surface and tipped at an angle θ from the vertical. Find θ 0 such that, when the cylinder is released from θ > θ 0 , it falls over. (A) arctan(2r/h) (B) arctan(r/h) (C) arctan(r/ 2 h) (D) arccos(2r/h) (E) arccos(r/h)
25. Consider a beam of muons with energy 3 GeV. The muon’s mass is approximately 100 MeV/c^2 , and its life- time at rest is 2 × 10 −^6 s. What is the muon lifetime measured by an experimenter in the lab? (A) 67 ns (B) 20 μs (C) 60 μs (D) 20 ms (E) 60 ms
26. Graphene, a two-dimensional allotrope of carbon, dis- plays unusual electronic properties. In particular, the dispersion relation for conduction electrons in graphene is (A) ω ∝

|k| (B) ω ∝ |k| (C) ω ∝ |k|^2 (D) ω ∝ |k|^3 (E) ω ∝ |k|^4

27. An electron placed in a one-dimensional harmonic oscillator potential V = 12 kx^2 is subject to a uniform electric field E = E 0 x ˆ. For small E 0 , the lowest-order, nonzero correction to the ground state energy is (A) independent of E 0 (B) proportional to E 0 (C) proportional to E^20 (D) proportional to E^30 (E) proportional to E^40

5 cm

2 cm

A B

28. In the optical arrangement shown above, converging lenses A and B both have focal length 5 cm. An object is placed 2 cm to the left of lens A. Where is the image of the object located? (A) 5 cm to the right of B (B) 6.25 cm 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 star of mass m orbits a galaxy of mass M in a circu- lar orbit. The MOND theory postulates that, at small accelerations, Newton’s Second Law is replaced by the force law F = ma^2 /a 0 , where a 0 is a constant with dimensions of acceleration. Assuming the MOND force law and Newton’s Law of Gravity, what is the relation between the velocity v of the star and the radius of its orbit r? (A) v is independent of r (B) v is proportional to r−^1 /^2 (C) v is proportional to r^1 /^4 (D) v is proportional to r−^1 (E) v is proportional to r^2
30. Which values of spin quantum numbers are NOT pos- sible for a system consisting of a spin-1 particle and a spin-2 particle? (A) s = 3, m (^) s = 3 (B) s = 1, ms = 0 (C) s = 2, ms = 1 (D) s = 2, ms = 0 (E) s = 0, ms = 0

38. An electron in a cyclotron moves in a circular orbit at a fixed radius in the presence of a constant magnetic field B. If the strength of the magnetic field is tripled, by what factor must the electron’s momentum change to keep it orbiting at the same radius? (A)

(B) 3

(C) 1/

(D) 1/ 3

(E) 3/ 2

a

b I I

39. Two circular loops of wire of radii a and b are oriented concentrically in the same plane, and they each carry a current I circulating in opposite directions, as shown in the figure above. What is the magnetic field at the center of the loops? (A) μ 20 I

a −^ 1 b

, pointed out of the page (B) μ 20 I

a −^ 1 b

, pointed into the page (C) μ 40 I

a −^

1 b

, pointed into the page (D) μ 20 I^1 a , pointed out of the page (E) 0

40. Which of the following is true about the total orbital angular momentum operator, L^2 , of a particle subjected to an arbitrary force? I. Always commutes with L (^) x, L (^) y, L (^) z II. Always commutes with the total angular momen- tum J^2 III. Always commutes with the Hamiltonian (A) I only (B) II only (C) III only (D) I and II (E) I, II, and III
41. A quantum system has a Hamiltonian given by

H =

a 0 0 0 0 −ib 0 ib 0

where a, b, c are real positive constants. What are the possible results of a measurement of the energy of the system? (A) b, ±a (B) a, ±b (C) a, b, a + b (D) a, ±

ab (E) a, ±b^2

42. If magnetic monopoles existed, which of the follow- ing expressions would be proportional to the “magnetic charge” of the monopole? You may assume that there are no other sources of electric or magnetic fields present. (A)

(∇ · E ) dV (B)

(∇ · B ) dV (C)

| E |^2 dV (D)

| B |^2 dV (E)

( E · B ) dV

E

v R

L

43. A beam of nonrelativistic protons (mass m, charge q) of velocity v enters a region of length L with an electric field E perpendicular to the direction of the beam. At the end of the region of length L is a circular target of radius R. Assuming that the diameter of the beam is much smaller than R, what is the minimum electric field E needed to deflect all protons before they strike the target?

(A) mLv^2 2 qR^2 (B) 2 mLv^2 qR^2 (C) mRv^2 q^2 L^2 (D) 2 mRv^2 qL^2 (E) 4 mLv^2 qR^2

44. Put the following in chronological order, starting with the earliest. I. Epoch of reionization II. Nucleosynthesis III. Inflation (A) I, II, III (B) I, III, II (C) II, I, III (D) III, I, II (E) III, II, I
45. For a monoatomic ideal gas, which of the following is constant during adiabatic changes of state? (A) PV^1 /^2 (B) PV (C) PV^5 /^3 (D) PV^7 /^5 (E) PV^9 /^7

R

m

g

M

46. A string of length L and negligible mass is completely wound around a solid cylinder of uniform density, of mass M and radius R, and it has a small weight of mass m attached to its end. If the weight is released from rest under the influence of gravity, what is its velocity when the string is entirely unwound?

(A)

4 mgL M + 2 m

(B)

2 mgL − MR^2 2 m (C)

2 gL

(D)

2(m + M)gL m

(E)

2 mgL − 2 MR^2 m

47. An object is placed at rest in a potential field U(x, y, z) = x + y^2 − cos z. What is the force on the object? (A) F (x, y, z) = −ˆ x − 2 y y ˆ − sin zˆ z (B) F (x, y, z) = x x ˆ + 2 y y ˆ − cos z z ˆ (C) F (x, y, z) = −xˆ x − 2 y y ˆ + cos zˆ z (D) F (x, y, z) = −ˆ x − 2 y y ˆ + cos z z ˆ (E) F (x, y, z) = ˆ x + 2 y y ˆ + sin z z ˆ
48. Consider a system with three energy levels −, 0, , and degeneracies d(−) = 2, d(0) = 1, d() = 3. What is the energy of the system as T → ∞? (A) / 5 (B) / 6 (C) 5/ 6 (D) 0 (E) 
49. In process 1, a monoatomic ideal gas is heated from temperature T to temperature 2T reversibly and at con- stant volume. In process 2, a monoatomic ideal gas freely expands from V to 2V. Which is the correct relationship between the change in entropy S 1 in process 1 and the change in entropy S 2 in process 2? (A) 0 < S 1 < S 2 (B) 0 < S 1 = S 2 (C) 0 = S 1 < S 2 (D) 0 < S 2 < S 1 (E) S 1 = S 2 < 0
50. An electromagnetic wave propagates in vacuum with electric field E 0 cos(kx − ωt)ˆ z. What is the average mag- nitude of the Poynting vector in SI units, where the average is taken over one period of oscillation?

(A)

4 E^20

cμ 0 (B) 0 (C)

E^20

cμ 0 (D)

E^20

2 cμ 0

(E) −

E^20

2 cμ 0

Support wire

Pivot

d

v

α

58. A rod of length d and mass M is attached to a pivot and suspended at an angle α from the vertical using a sup- port wire, as shown in the diagram. A lump of clay of mass m is fired at the end of the rod with a velocity v. Just before the clay makes contact with the rod, the wire is cut. Assuming the clay and rod stick together after collision, what is the angular velocity in radians of the rod–clay system? (You may treat the lump of clay as a point mass.)

(A) mv cos α (M + m)d (B) 3 mv sin α (M + m)d (C) 3 mv cos α (M + 3 m)d (D) 3 mv (M + 3 m)d (E) 3 mv Md

B R

L

ω

59. A square loop of wire of side length L, containing a load resistor R, is oriented perpendicular to the xy-plane and rotates about the z-axis at angular frequency ω in the presence of a uniform magnetic field B = B 0 x ˆ, as shown in the diagram. If L = 10 cm, B 0 = 2 tesla, and R = 100.0 , what must ω be so that the average power dissipated in the resistor is 0.5 W? (A) 25 rad/s (B) 50 rad/s (C) 314 rad/s (D) 354 rad/s (E) 500 rad/s
60. In calculating the entropy of a microcanonical ensem- ble, the inverse temperature β = 1 /kT can be viewed as a Lagrange multiplier enforcing the constraint of fixed total energy. Similarly, the chemical potential μ is related to the Lagrange multiplier for (A) fermion number (B) particle number (C) pressure (D) volume (E) magnetization

61. A spin-1/2 particle interacts with a magnetic field B = Bˆ z through a Hamiltonian H = (−μB gB/ 2 ℏ)σz, where μB is the Bohr magneton and g is the particle’s gyromag- netic ratio. Consider a system of these spin-1/2 particles in equilibrium at temperature T. Let A be the ratio of the number of spin-up particles to spin-down particles. If the strength of the magnetic field is doubled, the new ratio of spin-up to spin-down particles is (A) A−^2 (B) A (C) A^2 (D) e A (E) A exp(μB gB/ℏkT)
62. Which of the following is equivalent to ∇^2 (1/r)?

(A) − 4 πδ^3 ( r ) (B) 4πδ^3 ( r ) (C) 0 (D) 4π (E) − 4 π

63. The mass of the proton is 1.67 × 10 −^27 kg. Which of the following is closest to the Compton wavelength of the proton? (A) 10−^15 m (B) 10−^13 m (C) 10−^12 m (D) 10−^10 m (E) 10−^9 m Questions 64 and 65 refer to the following scenario. A K^0 of mass mK and energy E in the lab frame decays to a π+^ and a π−, both of mass mπ. The π+^ is observed to be emitted parallel to the K^0 momentum.
64. What is the speed of the π+^ in the K^0 rest frame?

(A) (1 − 4 m^2 K /m^2 π )^1 /^2 c (B) (1 − 4 m^2 π /m^2 K )^1 /^2 c (C) (1 − m^2 K /m^2 π )^1 /^2 c (D) (1 − m^2 π /m^2 K )^1 /^2 c (E) 2(m^2 π /m^2 K )^1 /^2 c

65. What must be the initial K^0 energy such that the π−^ is stationary in the lab frame?

(A) m^2 π c^2 2 mK (B) mK c^2 2 (C) mπ c^2 2 (D) (m^2 K + m^2 π )c^2 2 mπ (E) m^2 K c^2 2 mπ

66. A clarinet can be treated as a half-open pipe, where sounds are produced by standing pressure waves. For a clarinet of length 0.6 m, which of the following is a possible wavelength of a standing wave? (A) 0.3 m (B) 0.6 m (C) 0.8 m (D) 1.2 m (E) 1.5 m
67. A sphere has a polarization of P ( r ) = Cr^2 ˆ r. What is the electric field inside the sphere? (You may find the following fact useful: ∇ · (v(r) r ˆ) = (^) r^12 dr^ d (r^2 v(r)).)

(A) − 4 Cr^2  0 r ˆ

(B) 2 Cr^2  0 ˆ r

(C) − Cr^2  0 ˆ r

(D) Cr^2 4 π 0 ˆ r (E) 0

68. Suppose an electromagnetic plane wave propagating in vacuum in the +ˆ z -direction has a polarization with the electric field in the +ˆ x -direction immediately before it strikes a perfect conductor at normal incidence. What are the directions of the E and B vectors of the transmit- ted wave? (A) E in +ˆ x -direction & B in +ˆ y -direction (B) E in −ˆ x -direction & B in +ˆ y -direction (C) E in +ˆ x -direction & B in −ˆ y -direction (D) E in −ˆ x -direction & B in −ˆ y -direction (E) There is no transmitted wave in a perfect conductor.

75. The  is a spin-3/2 bound state of three spin-1/2 quarks. The spin part of the wavefunction of the state with m =
    • 3 /2 is |〉 = |↑ ↑ ↑〉. What is the spin part of the wavefunction with definite spin m = − 1 /2? (A) |↑↓↓〉 (B) √^13 (|↑↓↓〉 + |↓↑↓〉 − |↓↓↑〉) (C) √^13 (− |↑↓↓〉 + |↓↑↓〉 − |↓↓↑〉) (D) √^13 (|↑↓↓〉 + |↓↑↓〉 + |↓↓↑〉) (E) |↓↓↓〉
76. What is true of the electromagnetic field at a p-n junc- tion at equilibrium with zero bias voltage applied? (A) The electric field points toward the p-type semicon- ductor. (B) The electric field points toward the n-type semicon- ductor. (C) The electric field is parallel to the interface between the p-type and n-type semiconductors. (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. In an inertial frame S, two events E 1 and E 2 occur at (t, x, y, z) = (3, 4, 1, 1) and (1, 3, 0, 1), respectively (in units where c = 1). In another inertial frame S′, which of the following could an observer measure as the spacetime 4-vector between E 1 and E 2? (A) (1, 0.5, 1, 1) (B) (2, 1, 0, 0) (C) (3, 2,

(D) (2, 0,

(E) None of these

78. A dark matter experiment takes data for a time T and observes no events. What is the 90% confidence level upper limit that one can 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) ln 0. (C) −(1/T) ln 0. (D) (1/T) ln 0. (E) 0
79. If the proton were a spin-0 particle, which of the follow- ing features of the hydrogen energy spectrum would be absent? (A) Lyman series (B) Balmer series (C) 21 cm hyperfine transition (D) Lamb shift (E) fine-structure splitting of the 2p state
80. An electron neutrino emitted from the Sun may be detected as a tau neutrino on Earth because: (A) Conservation of lepton number does not apply to tau neutrinos. (B) Electron neutrinos from the Sun can annihilate and be reemitted as a pair of tau neutrinos. (C) Electron neutrinos interact with the Earth’s mag- netic field. (D) A freely propagating neutrino is a superposition of electron and tau neutrinos. (E) Electron neutrinos decay faster than tau neutrinos.
81. A pair of electrons is trapped in a “quantum dot.” A magnetic field is applied along the z-direction so that the singlet state has energy −, and the triplet state has energies −/2, −, and − 3 /2 for spins +ℏ, 0, and −ℏ along the z-axis, respectively. What is the prob- ability of finding the electrons in the triplet state, at temperature T? (A) 0 (B) 1 (C)

2 + e/^2 kT^ + e−/^2 kT

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

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

m

m

y 1

y 2

82. The diagram above illustrates a system consisting of a block of mass m hanging from a spring of spring con- stant k, with another block of mass m hanging from the first block by another spring of spring constant k. What is the total energy of this system? (A) 12 m( y˙ 12 + ˙y 22 ) + 12 k(y^21 + (y 2 − y 1 )^2 ) − mg(y 1 + y 2 ) (B) 12 m( y˙ 12 + ˙y 22 ) + 12 k(y^21 + (y 2 − y 1 )^2 ) + mg(y 1 + y 2 ) (C) 12 m( y˙ 12 + ˙y 22 ) − 12 k(y^21 + (y 2 − y 1 )^2 ) + mg(y 1 + y 2 ) (D) 12 m( y˙ 12 + ˙y 22 ) − 12 k(y^21 + (y 2 − y 1 )^2 ) − mg(y 1 + y 2 ) (E) − 12 m( y˙ 12 + ˙y 22 ) − 12 k(y^21 + (y 2 − y 1 )^2 ) − mg(y 1 + y 2 )
83. A particle of mass m is in the ground state of an infinite square well of size a, with energy E. The well suddenly expands to size 2a. What is E′/E, where E′^ is the expec- tation value of the energy of the particle after this sudden expansion? (A) 0 (B) 1 (C) 1/

(D) 1/

(E) 1/

84. A particle of mass m and energy E is incident from the left on a delta-function barrier, V(x) = αδ(x) with α > 0. Which of the following gives the coefficient of reflection for the system? (A) α^2 (B) α^2 E (C) α ℏ

m 2 E (D)

1 + 2 ℏ^2 E/mα^2 (E)

1 + mα^2 / 2 ℏ^2 E

85. Which of the following is NOT true about the 2s → 1 s transition in the hydrogen atom? (A) The dominant decay mode is two-photon emission. (B) It violates l = ±1. (C) It violates m = ±1 or 0. (D) It cannot occur in the electric dipole approximation. (E) None of these.
86. Measurements of the electric dipole moment of the neu- tron provide sensitive tests of fundamental physics. If the neutron were found to 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

0.01 0.1 1.0 10. Energy [MeV]

Cross section [barn]

10 −

100

102

104

Total

c

a (^) b

87. The figure above shows the total cross section for pho- ton scattering on a Pb atom as well as the cross sections for several individual process. Why does curve b drop quickly near 1 MeV? (A) Penetration depth of low-energy photons is small. (B) Interactions with electrons become significant. (C) 1.022 MeV threshold for pair production. (D) Pb has no absorption lines below 1 MeV. (E) Conservation of angular momentum.

R

r

95. A toroidal solenoid of radius R and cross-sectional radius r  R has N winds and carries current R. The vol- ume enclosed by the torus is 2π^2 Rr^2. What is the energy stored in the toroidal solenoid? (A) 0 (B) μ 0 NI^2 r^2 4 πR^3

(C) μ 0 N^2 I^2 r^2 4 πR^3

(D) μ 0 NI^2 r^2 4 R

(E) μ 0 N^2 I^2 r^2 4 R

96. Which of the following is true about a longitudinally polarized wave in three dimensions? I. There are two linearly independent polarization vectors. II. The polarization vector(s) is/are perpendicular to the wavevector. III. The polarization vector(s) is/are parallel to the wavevector. (A) III only (B) II only (C) I only (D) I and II (E) I and III
97. Deep water waves obey the dispersion relation ω = A

k, where A is a constant. What is the correct rela- tionship between phase velocity and group velocity for deep water waves? (A) vphase = 12 vgroup (B) vphase = vgroup (C) vphase = 2 vgroup (D) vphasevgroup = A^4 t^2 (E) none of these

98. When light of 5000 Å is shined on a thin film of oil (n = 1.5) that sits on top of a medium with n = 2.0, the inten- sity of reflected light is minimized. What is the thickness of the oil? (A) 4 × 10 −^8 m (B) 8.33 × 10 −^8 m (C) 1.67 × 10 −^7 m (D) 1.25 × 10 −^7 m (E) 5.0 × 10 −^7 m
99. Suppose a particle has a normalized wavefunction ψ(x) given by

ψ(x) =

3(1 − x), 0 < x < 1, 0, otherwise. What is the expectation value of the position of this particle? (A) 0 (B) 1 (C) 1/ 12 (D) 1/ 4 (E) 1/ 2

100. What are the energy levels of a quantized system con- sisting of a massless rigid rod of length a connecting two masses m, where n is a non-negative integer?

(A) ℏ^2 n(n + 1) ma^2 (B) ℏ^2 n(n + 1) 2 ma^2 (C) ℏ^2 n 2 ma^2 (D) ℏ^2 n ma^2 (E) ℏ^2 (n + 1) ma^2

TABLE OF INFORMATION

Rest mass of the electron me = 9.11 × 10 −^31 kg Magnitude of the electron charge e = 1.60 × 10 −^19 C Avogadro’s number NA = 6.02 × 1023 Universal gas constant R = 8.31 J/(mol · K) Boltzmann’s constant k = 1.38 × 10 −^23 J/K Speed of light c = 3.00 × 108 m/s Planck’s constant h = 6.63 × 10 −^34 J · s = 4.14 × 10 −^15 eV · s ℏ = h/ 2 π hc = 1240 eV · nm Vacuum permittivity  0 = 8.85 × 10 −^12 C^2 /(N · m 2 ) Vacuum permeability μ 0 = 4 π × 10 −^7 T · m/A Universal gravitational constant G = 6.67 × 10 −^11 m 3 /(kg · s 2 ) Acceleration due to gravity g = 9.80 m/s 2 1 atmosphere pressure 1 atm = 1.0 × 105 N/m 2 = 1.0 × 10 5 Pa 1 angstrom 1 Å = 1 × 10 −^10 m = 0.1 nm

Prefixes for Powers of 10

10 −^15 femto f

10 −^12 pico p

10 −^9 nano n

10 −^6 micro μ

10 −^3 milli m

10 −^2 centi c

103 kilo k

106 mega M

109 giga G

1012 tera T

1015 peta P

Rotational inertia about center of mass

Rod

M 2

Disk

MR^2

Sphere

MR^2