Official GRE Physics Exam #2

Two long, identical bar magnets are placed under a horizontal piece of paper, as shown in the figure above. The paper is covered with iron filings. When the two north poles are a small distance apart and touching the paper, the iron filings move into a pattern that shows the magnetic field lines. Which of the following best illustrates the pattern that results?

The two bar magnets will create a field which is similar to that of two positive charges in proximity of each other with their electric field. Choice (B).

In the circuit shown above, the switch S is closed at t = 0. Which of the following best represents the voltage across the inductor, as seen on an oscilloscope?

The voltage quickly drops and we get choice (D).

In the hydrogen spectrum, the ratio of the wavelengths for Lyman-a radiation (n = 2 to n = 1) to Balmer-a radiation (n = 3 to n = 2) is (A) 5/48 (B) 5/27 (C) 1/3 (D) 3 (E) 27/5

The wavelengths are less in the Lyman-alpha radiation and in the Balmer-alpha radiation is bigger. Look for the cubed value and get 3^3 = 27 and we spot 5/27. Choice (B).

A constant amount of an ideal gas undergoes the cyclic process ABCA in the PV diagram shown above. The path BC is isothermal. The work done by the gas during one complete cycle, beginning and ending at A, is most nearly (A) 600 kJ (B) 300 kJ (C) 0 (D) -300 kJ (E) -600 kJ

The work done by the gas during this process is 300 kJ (1/2 * 300 kJ * 2 m^3) = 300 kJ by the area of the approximated triangle. Choice (B).

Which of the following is the orbital angular momentum eigenfunction Ym,l(theta, phi) in a state for which the operators L^2 and L,z have eigenvalues 6h^2 and -h, respectively?

The z-component and the bottom give us Y-1,2(theta, phi), which is choice (B).

The cylinder shown above, with mass M and radius R, has a radially dependent density. The cylinder starts from rest and rolls without slipping down an inclined plane of height H. At the bottom of the its translational speed is (8gH/7)^0.5. Which of the following is the rotational inertia of the cylinder? (A) (1/2)MR^2 (B) (3/4)MR^2 (C) (7/8)MR^2 (D) MR^2 (E) (7/4)MR^2

This one is tricky and can fool you easily like it did to me. So the translational and rotational inertia of an object is given by (1/2)MR^2 intrinsically, not determinately. Choice (A).

Five classes of students measure the height of a building. Each class uses a different method and each measures the height many different times. The data for each class are plotted below. Which class made the most precise measurement?

This question is kindergarten. The most precise measurement is given by the smallest deviation from the measured results. Choice (A).

Suppose that a system in quantum state i has energy E,i. In thermal equilibrium, the expression zeta(E,i*e^(-E,i/kT)/zeta(e^(-E,i/kT) represents what? (A) The average energy of the system (B)The partition function (C) Unity (D) The probability to find the system with energy E,i. (E) The entropy of the system

This question is trippy, but it is the average energy of a system. Choice (A), not the probability.

In an inertial reference frame S, two events occur on the x-axis separated in time by Dt and in space by Dx. In another inertial reference frame S¢, moving in the x-direction relative to S, the two events could occur at the same time under which, if any, of the following conditions? (A) For any values of Dx and Dt (B) Only if ΩDx /DtΩ< c (C) Only if ΩDx /DtΩ> c (D) Only if ΩDx /DtΩ= c (E) Under no condition

Timelike events occur when the light acts as x/t = c. Choice (D).

A sample of radioactive nuclei of a certain element can decay only by g -emission and b -emission. If the half-life for g -emission is 24 minutes and that for b -emission is 36 minutes, the half-life for the sample is (A) 30 minutes (B) 24 minutes (C) 20.8 minutes (D) 14.4 minutes (E) 6 minutes

[1/(24 min) + 1/(36 min)]^-1 = [5/(72 min)]^-1 = 14.4 min. Choice (D).

The components of the orbital angular momentum operator L = (Lx , Ly , Lz) satisfy the following commutation relations. [Lx , Ly] = i = Lz , [Ly , Lz] = i = Lx , [Lz , Lx] = i = Ly . What is the value of the commutator [LxLy , Lz] ? (A) 2i = Lx L y (B) i = L L x y 2 2 e + j (C) -i = L L x y 2 2 e + j (D) i = L L x y 2 2 e - j (E) -i = L L x y 2 2 e - j

[L,x*L,y, L,z] = L,x*L,y*L,z - L,z*L,x*L,y = ih(L,x^2 - L,y^2), where only the x- and y-terms are involved due to the L,z at the end of the commutator. Choice (D)>

The raising and lowering operators for the quantum harmonic oscillator satisfy aT*n> = sqrt(n + 1)(n + 1)>, a(n> = sqrt(n)*(n - 1)> for energy eigenstates n> with energy E,n. Which of the following gives the first-order shift in the n = 2 energy level due to the perturbation delta(H) = V(a + a^t)^2, where V is a constant? (A) 0 (B) V (C) sqrt(2)*V (D) 2*sqrt(2)*V (E) 5V

1^2 + 2^2 = 5. So the potential V becomes 5V. Choice (E).

A nonrelativistic particle with a charge twice that of an electron moves through a uniform magnetic field. The field has a strength of p 4 tesla and is perpendicular to the velocity of the particle. What is the particle's mass if it has a cyclotron frequency of 1,600 hertz? (A) 2.5 ¥ 10-23 kg (B) 1.2 ¥ 10-22 kg (C) 3.3 ¥ 10-22 kg (D) 5.0 ¥ 10-21 kg (E) 7.5 ¥ 10-21 kg

2*pi*f = qB/r ==> f = (q*B)/(2*pi*r) = (3.20E-19 C * pi/4)/(2*pi*

The coefficient of static friction between a small coin and the surface of a turntable is 0.30. The turntable rotates at 33.3 revolutions per minute. What is the maximum distance from the center of the turntable at which the coin will not slide? (A) 0.024 m (B) 0.048 m (C) 0.121 m (D) 0.242 m (E) 0.484 m

33.3 rev/min = 33.3*2*pi/60 rad/s = 1.11*pi rad/s. u(v^2)/r = g (masses cancel out). r = u*(v^2)/g = [(0.30)(1.11*pi)^2]/(9.8 m/s^2) = 0.30*12.3/9.8 = 0.30 *1.25 = 0.3 m, which is closest to choice (D).

An ideal monatomic gas expands quasi-statically to twice its volume. If the process is isothermal, the work done by the gas is Wi . If the process is adiabatic, the work done by the gas is Wa . Which of the following is true? (A) Wi = Wa (B) 0 = Wi < Wa (C) 0 < Wi < Wa (D) 0 = Wa < Wi (E) 0 < Wa < Wi

A adiabatic expansion is more ideal in nature and allows for no heat to be taken or given to the environment, whereas an isothermal process does the opposite. Choice (E).

Three wire loops and an observer are positioned as shown in the figure above. From the observer's point of view, a current I flows counterclockwise in the middle loop, which is moving towards the observer with a velocity u . Loops A and B are stationary. This same observer would notice that (A) clockwise currents are induced in loops A and B (B) counterclockwise currents are induced in loops A and B (C) a clockwise current is induced in loop A, but a counterclockwise current is induced in loop B (D) a counterclockwise current is induced in loop A, but a clockwise current is induced in loop B (E) a counterclockwise current is induced in loop A, but no current is induced in loop B

A reversed current is induced in the front loop, while the back loop gets an induced current in the same direction as the velocity v of the loops. Choice (C).

A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of the oscillator is (A) (1/2)kT (B) kT (C) (3/2)kT (D) 3kT (E) 6kT

A three-dimensional oscillator at thermal equilibrium has kT average total energy for each degree of freedom, which is represented by choice (D).

Five positive charges of magnitude q are arranged symmetrically around the circumference of a circle of radius r. What is the magnitude of the electric field at the center of the circle? ( ) k = 1 4p 0 (A) 0 (B) kq r2 (C) 5 2 kq r (D) ( / ) cos / kq r2 a f 2 5 p (E) ( / ) cos / 5 25 2 kq r a f

Anytime all similar charges are arranged in circle and are symmetrically distributed, the net electric field becomes zero due to total cancellation. Choice (A).

A free particle with initial kinetic energy E and de Broglie wavelength l enters a region in which it has potential energy V. What is the particle's new de Broglie wavelength? (A) l (1 + E/V) (B) l (1 - V/E) (C) l (1 - E/V)^-1 (D) l (1 + V/E)^1/2 (E) l (1 - V/E)^-1/2

As the potential energy increases, then them more the wavelength should increase as well. Choice (E) is the only choice which is capable of this.

A particle of mass m is moving along the x-axis with speed u when it collides with a particle of mass 2m initially at rest. After the collision, the first particle has come to rest, and the second particle has split into two equal-mass pieces that move at equal angles q > 0 with the x-axis, as shown in the figure above. Which of the following statements correctly describes the speeds of the two pieces? (A) Each piece moves with speed u. (B) One of the pieces moves with speed u, the other moves with speed less than u. (C) Each piece moves with speed u/2. (D) One of the pieces moves with speed u/2, the other moves with speed greater than u/2. (E) Each piece moves with speed greater than u/2.

Each piece must move with a speed greater than v/2 because momentum is conserved in the direction of motion and the leg of the right triangle is v/2, so the velocity vector should be greater than v/2. Choice (E).

Which of the following best represents the temperature dependence of the resistivity of an undoped semiconductor?

By process of elimination, it becomes apparent that the resistivity does not increase with increasing temperature and does not figit. Choice (B).

Which of the following functions could represent the radial wave function for an electron in an atom? (r is the distance of the electron from the nucleus; A and b are constants.) I. A e- b r II. A sin(br) III. A/r (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III

Choice (A) gives the common form of the wave function for radial wave functions of an electron in an atom as from quantum mechanics. Choice (A).

One end of a Nichrome wire of length 2L and cross-sectional area A is attached to an end of another Nichrome wire of length L and cross-sectional area 2A. If the free end of the longer wire is at an electric potential of 8.0 volts, and the free end of the shorter wires is at an electric potential of 1.0 V, the potential at the junction of the two wires is most nearly equal to (A) 2.4 V (B) 3.3 V (C) 4.5 V (D) 5.7 V (E) 6.6 V

Choice (A), lucky guess.

A Michelson interferometer is configured as a wavemeter, as shown in the figure above, so that a ratio of fringe counts may be used to compare the wavelengths through distance d, 100,000 intergerence fringes pass across the detector for green light and 85,865 fringes pass across the detector for red (lambda = 632.82 nm) light. The wavelength of the green laser light is what?

Choice (B), we get (85,865/100,000)(632.82 nm) = 543.37 nm (about 5/6 of 632.82 nm).

A photon strikes an electron of mass m that is initially at rest, creating an electron-positron pair. The photon is destroyed and the positron and two electrons move off at equal speeds along the initial direction of the photon. The energy of the photon was (A) mc^2 (B) 2mc^2 (C) 3mc^2 (D) 4mc^2 (E) 5mc^2

Choice (D), double the double.

The muon decays with a characteristic lifetime of about 10^-6 second into an electron, a muon neutrino, and an electron antineutrino. The muon is forbidden from decaying into an electron and just a single neutrino by the law of conservation of (A) charge (B) mass (C) energy and momentum (D) baryon number (E) lepton number

Common Question: Muons do not like to decay into an electron and a single neutrino because the lepton number is not conserved. Choice (E).

The eigenvalues of a Hermitian operator are always (A) real (B) imaginary (C) degenerate (D) linear (E) positive

Definition: The eigenvalues of a Hermitian operator are always real. Choice (A)

The 238U nucleus has a binding energy of about 7.6 MeV per nucleon. If the nucleus were to fission into two equal fragments, each would have a kinetic energy of just over 100 MeV. From this, it can be concluded that (A) 238U cannot fission spontaneously (B) 238U has a large neutron excess (C) nuclei near A = 120 have masses greater than half that of 238U (D) nuclei near A = 120 must be bound by about 6.7 MeV/nucleon (E) nuclei near A = 120 must be bound by about 8.5 MeV/nucleon

Definition: When A = 120, then the nuclei must be bound at about 8.5 MeV/nucleon. Choice (E).

Two small equal masses m are connected by an ideal massless spring that has equilibrium length l0 and force constant k, as shown in the figure above. The system is free to move without friction in the plane of the page. If p1 and p2 represent the magnitudes of the momenta of the two masses, a Hamiltonian for this system is?

Do not include the 2 to the spring internal energy and we get (1/2){p1^2/m + p2^2/m + k(l - l0)^2}. Choice (E).

Consider the quasi-static adiabatic expansion of an ideal gas from an initial state i to a final state f. Which of the following statements is NOT true? (A) No heat flows into or out of the gas. (B) The entropy of state i equals the entropy of state f. (C) The change of internal energy of the gas is - z PdV. (D) The mechanical work done by the gas is PdV z . (E) The temperature of the gas remains constant.

During a quasi-"static" expansion, we should expect the temperature of the gas to remain constant throughout its expansion. Choice (E).

Positronium is an atom formed by an electron and a positron (antielectron). It is similar to the hydrogen atom, with the positron replacing the proton. If a positronium atom makes a transition from the state with n = 3 to a state with n = 1, the energy of the photon emitted in this transition is closest to (A) 6.0 eV (B) 6.8 eV (C) 12.2 eV (D) 13.6 eV (E) 24.2 eV

Positronium has a base ground energy of 6.8 eV and the transition gives us: 6.8 eV(1/(1^2) - 1/(3^2)) = 6.8 eV(1 - 1/9) = 6.0 eV. Choice (A).

The ground state electron configuration for phosphorus, which has 15 electrons, is (A) 1s2 2s2 2p6 3s1 3p4 (B) 1s2 2s2 2p6 3s2 3p3 (C) 1s2 2s2 2p6 3s2 3d3 (D) 1s2 2s2 2p6 3s1 3d4 (E) 1s2 2s2 2p6 3p2 3d3

Simple chemistry and a bit of counting. Choice (B).

The state ww w w = ++ − 1 6 1 2 1 3 112 is a linear combination of three orthonormal eigenstates of the operator Ô corresponding to eigenvalues -1, 1, and 2. What is the expectation value of Ô for this state? (A) 2/3 (B) sqrt(7/6 (C) 1 (D) 4/3 (E) (sqrt(3) + 2sqrt(2) - 1)/sqrt(6)

Since the function O corresponds to all eigenvalues, then the expectation value of O for this state is going to be 1. Choice (C).

A spherical, concave mirror is shown in the figure above. The focal point F and the location of the object O are indicated. At what point will the image be located? (A) I (B) II (C) III (D) IV (E) V

Since the object is within the focal point, then the image will appear behind the lenses, which is only the last choice. Choice (E).

In a Maxwell-Boltzmann system with two states of energies and 2, respectively, and a degeneracy of 2 for each state, the partition function is (A) e- /kT (B) 2e-2 /kT (C) 2e-3 /kT (D) e- /kT + e-2 /kT (E) 2[e - /kT + e-2 /kT ]

Since there are two degenerate states and the partition functions are given as the sum of the exponentials, then the partition function is choice (E).

An electromagnetic plane wave, propagating in vacuum, has an electric field given by ( ) 0 E E kx t = - cos w and is normally incident on a perfect conductor at x = 0, as shown in the figure above. Immediately to the left of the conductor, the total electric field E and the total magnetic field B are given by which of the following? E B (A) 0 0 (B) 0 2 cos E t w 0 (C) 0 ( ) 0 2 cos Ec t w (D) 0 2 cos E t w ( ) 0 2 cos Ec t w (E) 0 2 cos E t w ( ) 0 2 sin Ec t

Since we have the incident wave perpendicular to the surface, the magnetic field is existent, whereas the electric field is not existent at the surface. Choice (C).

Two spherical, nonconducting, and very thin shells of uniformly distributed positive charge Q and radius d are located a distance 10d from each other. A positive point charge q is placed inside one of the shells at a distance d/2 from the center, on the line connecting the centers of the two shells, as shown in the figure above. What is the net force on charge q?

Take the difference, 2*10d - d = 19d and take the square and we get 360qQ/(361*pi*e*d^2) to the left. Choice (E).

A sealed and thermally insulated container of total volume V is divided into two equal volumes by an impermeable wall. The left half of the container is initially occupied by n moles of an ideal gas at temperature T. Which of the following gives the change in entropy of the system when the wall is suddenly removed and the gas expands to fill the entire volume? (A) 2nR ln2 (B) nR ln2 (C) 1 2 nR ln2 (D) -nR ln2 (E) -2nR ln2

Take the number of moles and the universal gas constant and the change in entropy formula and we get nR*ln(2) which is choice (B).

An ensemble of systems is in thermal equilibrium with a reservoir for which kT = 0.025 eV. State A has an energy that is 0.1 eV above that of state B. If it is assumed the systems obey Maxwell-Boltzmann statistics and that the degeneracies of the two states are the same, then the ratio of the number of systems in state A to the number in state B is (A) e^+4 (B) e^+0.25 (C) 1 (D) e^-0.25 (E) e^-4

Take the ratio between the energy for state A and state B and we get (0.1 eV)/(0.025 eV) = 4. So the decay gives us e^-4. Choice (E).

C kN hv kT A hv kT hv kT = F H I K - 3 1 2 2 e (e / / ) Einstein's formula for the molar heat capacity C of solids is given above. At high temperatures, C approaches which of the following? (A) 0 (B) 3kN hv kT A F H I K (C) 3kN hv A (D) 3kNA (E) N hv

Taylor expanding this polynomial gives us that 3kNhv is the limit at high temperatures. Choice (D).

Seven pennies are arranged in a hexagonal, planar pattern so as to touch each neighbor, as shown in the figure above. Each penny is a uniform disk of mass m and radius r. What is the moment of inertia of the system of seven pennies about an axis that passes through the center of the central penny and is normal to the plane of the pennies? (A) (7/2) mr2 (B) (13/2) mr2 (C) (29/2) mr2 (D) (49/2) mr2 (E) (55/2) mr2

The 6 outer pennies cause a moment of inertia of (1/2)mr^2 each and the pennies altogether create a total of (7^2)mr^2/2 moments of inertia. So (6/2)mr^2 + (49/2)mr^2 = (55/2)mr^2. Choice (E).

The Lagrangian for a mechanical system is L = aq*^2 + bq^4, where q is a generalized coordinate and a and b are constants. The equation of motion for this system is what?

The Euler-Lagrange Equation is (d/dt)(dL/dx*) - (dL/dx) = 0. (d/dt)(dL/dq*) = 2aq** and dL/dq = 4bq^3. q** = (2b/a)q^3. Choice (D).

For an inductor and capacitor connected in series, the equation describing the motion of charge is L d Q dt C Q 2 2 1 + = 0, where L is the inductance, C is the capacitance, and Q is the charge. An analogous equation can be written for a simple harmonic oscillator with position x, mass m, and spring constant k. Which of the following correctly lists the mechanical analogs of L, C, and Q ?

The L, C, and Q terms become mass (m), 1/k, and displacement (x). Choice (B).

Maxwell's equations can be written in the form shown below. If magnetic charge exists and if it is conserved, which of these equations will have to be changed? I. ∇ = E r/ o II. ∇ = B 0 III. ∇× = − ∂ ∂ E B t IV. ∇× = + ∂ ∂ B J E µ µ o oo t (A) I only (B) II only (C) III only (D) I and IV (E) II and III

The Maxwell's Equations involving the divergence and time derivative of the magnetic field vector will be changed. Choice (E).

A positive charge Q is located at a distance L above an infinite grounded conducting plane, as shown in the figure above. What is the total charge induced on the plane? (A) 2Q (B) Q (C) 0 (D) -Q (E) -2Q

The positive charge induces a negative coulomb charge on the plate which is -Q. Choice (D).

A uniformly charged sphere of total charge Q expands and contracts between radii R1 and R2 at a frequency f. The total power radiated by the sphere is (A) proportional to Q (B) proportional to f^2 (C) proportional to f^4 (D) proportional to (R2/R1) (E) zero

The power is radiated is not proportional to the frequency at all. Choice (E), zero.

The conventional unit cell of a body-centered cubic Bravais lattice is shown in the figure above. The conventional cell has volume a3 . What is the volume of the primitive unit cell? (A) a3/8 (B) a3/4 (C) a3/2 (D) a3 (E) 2a3

The primitative unit cell is actually smaller than the conventional cell, with a volume of (a^3)/2. Choice (C).

The matrix shown above transforms the components of a vector in one coordinate frame S to the components of the same vector in a second coordinate frame S'. This matrix represents a rotation of the reference frame S by

The rotation is 60 degrees about the counterclockwise rotation of the z-axis. Choice (E).

An energy-level diagram of the n = 1 and n = 2 levels of atomic hydrogen (including the effects of spin-orbit coupling and relativity) is shown in the figure above. Three transitions are labeled A, B, and C. Three transitions are labeled A, B, and C. Which of the transitions will be possible electric-dipole transitions? (A) B only (B) C only (C) A and C only (D) B and C only (E) A, B, and C

The slanted lines B and C are each characteristic of an electric-dipole transition, Choice (D).

A particle is constrained to move in a circle with a 10-meter radius. At one instant, the particle's speed is 10 meters per second and is increasing at a rate of 10 meters per second squared. The angle between the particle's velocity and acceleration vectors is (A) 0∞ (B) 30∞ (C) 45∞ (D) 60∞ (E) 90∞

theta = tan^-1[w^2/g] = tan^-1[(v/R)^2/g] = tan^-1{[(10 m/s)/(10 m)]^2/(9.8 m/s^2)} = tan^-1(1) = 45 degrees. Choice (C).

An AC circuit consists of the elements shown above, with R = 10,000 ohms, L = 25 millihenries, and C an adjustable capacitance. The AC voltage generator supplies a signal with an amplitude of 40 volts and angular frequency of 1,000 radians per second. For what value of C is the amplitude of the current maximized? (A) 4 nF (B) 40 nF (C) 4 mF (D) 40 mF (E) 400 mF

w = sqrt^-1(LC) ==> C = (w^2)L = (1000^2)(2.5E-2) = 2.5e4

A tube of water is traveling at (1/2)c relative to the lab frame when a beam of light traveling in the same direction as the tube enters it. What is the speed of light in the water relative to the lab frame? (The index of refraction of water is 4/3.) (A) (1/2)c (B) (2/3)c (C) (5/6)c (D) (10/11)c (E) c

y = sqrt(1 - 0.5^2)^-0.5 = sqrt(0.75) = (c/2)(4/3) = (2/3)c. Choice (D).

An 8-centimeter-diameter by 8-centimeter-long NaI(Tl) detector detects gamma rays of a specific energy from a point source of radioactivity. When the source is placed just next to the detector at the center of the circular face, 50 percent of all emitted gamma rays at that energy are detected. If the detector is moved to 1 meter away, the fraction of detected gamma rays drops to (A) 10-4 (B) 2 ¥ 10-4 (C) 4 ¥ 10-4 (D) 8p ¥ 10-4 (E) 16p ¥ 10-4

{[(8E-2 m)/(1 m)]^2} (50%) = 4(50%)/625 = 200/625 % - 8/25 % = 0.32% = 3.2E-3, which is somewhat close to choice (C).

A satellite of mass m orbits a planet of mass M in a circular orbit of radius R. The time required for one revolution is (A) independent of M (B) proportional to m (C) linear in R (D) proportional to R3/2 (E) proportional to R2

Kepler's Third Law of Planetary Motion ==> R^3 = T^2, which implies the time for one revolution is proportional to R^(3/2). Choice (D).

The solution to the Schrodinger equation for the ground state of hydrogen is phi0 = sqrt^-1(pi*a0^3)*e^(-r/a0), where a0 is the Bohr radius and r is the distance from the origin. Which of the following is the most probable value for r? (A) 0 (B) a0/2 (C) a0 (D) 2a0 (E) infinity

Look at the denominator and we get r = a0, so the most probable value for r is r = a0. Choice (C).

When 4 7Be transforms into 3 7Li, it does so by (A) emitting an alpha particle only (B) emitting an electron only (C) emitting a neutron only (D) emitting a positron only (E) electron capture by the nucleus with emission of a neutrino

We decay from emitting a neutrino and the electron as well in this case. Choice (E).

A thin uniform rod of mass M and length L is positioned vertically above an anchored frictionless pivot point, as shown above, and then allowed to fall to the ground. With what speed does the free end of the rod strike the ground? (A) sqrt(1/3 gL (B) sqrt(gL (C) sqrt(3gL (D) sqrt(12gL (E) sqrt(12 gL

When it comes to rods, we have a moment of inertia at the edges as (1/3)Mr^2, which gives us ample speed at the end of the fall, which is choice (C).

An object is located 40 centimeters from the first of two thin converging lenses of focal lengths 20 centimeters and 10 centimeters, respectively, as shown in the figure above. The lenses are separated by 30 centimeters. The final image formed by the two-lens system is located (A) 5.0 cm to the right of the second lens (B) 13.3 cm to the right of the second lens (C) infinitely far to the right of the second lens (D) 13.3 cm to the left of the second lens (E) 100 cm to the left of the second lens

When two converging lenses are put together they only make the image smaller and smaller, so the image put to the right of the second lens should be significantly smaller than the focal lengths and the distance between the lenses. Choice (A).

If the absolute temperature of a blackbody is increased by a factor of 3, the energy radiated per second per unit area does which of the following? (A) Decreases by a factor of 81. (B) Decreases by a factor of 9. (C) Increases by a factor of 9. (D) Increases by a factor of 27. (E) Increases by a factor of 81.

Wien's Law gives us, the 3^4 = 81 as the factor of energy radiated per second per unit area due to the temperature tripling. Choice (E).

Let a> represent the state of an electron with spin up and B> the state of an electron with spin down. Valid spin eigenfunctions for a triplet state (3S) of a two-electron atom include which of the following? I. a>1*a>2 II. [sqrt^-1(2)(a>1*b>2 - a>2*b>1) III. [sqrt^-1(2)(a>1*b>2 - a>2*b>1) (A) I only (B) II only (C) III only (D) I and III (E) II and III

You get a positive superposition state for the spin functions of the material, so we get choice (D) which is selection I and III.

The state of a spin-1/2 particle can be represented using the eigenstates up> and down> of the S,z operator. S,z up> = (1/2)h*up> and S,z down> = (-1/2)h*down> Given the Pauli matrix sigma,x = (0 1/1 0), which of the following is an eigenstate of S,x with eigenvalue (-1/2)h?

You must take the difference between the two and normalize. Hence we get 1/sqrt(2)*(up> - down>). Choice (C).

A gaseous mixture of O2 (molecular mass 32 u) and N2 (molecular mass 28 u) is maintained at constant temperature. What is the ratio u u rms rms ( ) ( ) N O 2 2 of the root-mean-square speeds of the molecules? (A) 7 8 (B) 7 8 (C) 8 7 (D) 8 7 2 F H I K (E) ln 8 7 F H I

sqrt(32/28) = sqrt(8/7). Choice (C).

Two identical springs with spring constant k are connected to identical masses of mass M, as shown in the figures above. The ratio of the period for the springs connected in parallel (Figure 1) to the period for the springs connected in series (Figure 2) is (A) 1/2 (B) 1/sqrt(2) (C) 1 (D) sqrt(2) (E) 2

sqrt[(2k)/(k/2)] = sqrt(4) = 2. The series is stiffer and will vibrate quicker and have less period than figure 2. Choice (A).

A stream of water of density r, cross-sectional area A, and speed u strikes a wall that is perpendicular to the direction of the stream, as shown in the figure above. The water then flows sideways across the wall. The force exerted by the stream on the wall is (A) ru2A (B) ruA/2 (C) rghA (D) u2A/r (E) u2A/2r

Recall from Physics GRE Practice Textbook that p*v^2*A. Choice (A).

The primary source of the Sun's energy is a series of thermonuclear reactions in which the energy produced is c^2 times the mass difference between (A) two hydrogen atoms and one helium atom (B) four hydrogen atoms and one helium atom (C) six hydrogen atoms and two helium atoms (D) three helium atoms and one carbon atom (E) two hydrogen atoms plus two helium atoms and one carbon atom

Remember this and remember it well that there are 4 hydrogen atoms and 1 helium atom in the system of thermonuclear reactions. Choice (B).

Which two of the following circuits are high-pass filters? (A) I and II (B) I and III (C) I and IV (D) II and III (E) II and IV

Resist and oscillate and capacitate and resist are what gives us high-pass filters. Choice (D).

In the production of X rays, the term "bremsstrahlung" refers to which of the following? (A) The cut-off wavelength, lmin , of the X-ray tube (B) The discrete X-ray lines emitted when an electron in an outer orbit fills a vacancy in an inner orbit of the atoms in the target metal of the X-ray tube (C) The discrete X-ray lines absorbed when an electron in an inner orbit fills a vacancy in an outer orbit of the atoms in the target metal of the X-ray tube (D) The smooth, continuous X-ray spectra produced by high-energy blackbody radiation from the X-ray tube (E) The smooth, continuous X-ray spectra produced by rapidly decelerating electrons in the target metal of the X-ray tube

"Bremstrahlung" tends to involve a metal and the smooth, continuous X-ray spectra produced by decelerating electrons. Choice (E).

The energy eigenstates for a particle of mass m in a box of length L have wave functions f p n( ) / sin( / ) x L nxL = 2 and energies E n mL n = 222 2 p = / , 2 where n = 123 , , ,.... At time t = 0, the particle is in a state described as follows. ( ) 123 1 0 [ 2 3] 14 Y t == + + fff Which of the following is a possible result of a measurement of energy for the state Y ? (A) 2E1 (B) 5E1 (C) 7E1 (D) 9E1 (E) 14E1

(1/14)(1 + 2*2^2 + 3*3^2) = (1/14)(1 + 8 + 27) = 36/14. Choice (D).

A child is standing on the edge of a merry-go-round that has the shape of a solid disk, as shown in the figure above. The mass of the child is 40 kg. The merry-go-round has a mass of 200 kg and a radius of 2.5 m, and it is rotating with an angular velocity of w = 2.0 rad/s. The child then walks slowly toward the center of the-merry-go-round. What will be the final angular velocity of the merry-go-round when the child reaches the center? (The size of the child can be neglected.) (A) 2.0 rad/s (B) 2.2 rad/s (C) 2.4 rad/s (D) 2.6 rad/s (E) 2.8 rad/s

(2.5 rad/s)(240/200) = 2.8 rad/s. Choice (E).

The distribution of relative intensity I ( ) l of blackbody radiation from a solid object versus the wavelength l is shown in the figure above. If the Wien displacement law constant is 2.9 ¥ 10-3 mK, what is the approximate temperature of the object? (A) 10 K (B) 50 K (C) 250 K (D) 1,500 K (E) 6,250 K

(2.9E-3 m*K)/(2E-6 m) = 1.5E3 K. Choice (D).

A balloon is to be filled with helium and used to suspend a mass of 300 kilograms in air. If the mass of the balloon is neglected, which of the following gives the approximate volume of helium required? (The density of air is 1.29 kilograms per cubic meter and the density of helium is 0.18 kilogram per cubic meter.) (A) 50 m3 (B) 95 m3 (C) 135 m3 (D) 270 m3 (E) 540 m3

(300 kg)[(1.29)/(1.47)] = 270 m^3. Choice (D).

At 20∞C, a pipe open at both ends resonates at a frequency of 440 hertz. At what frequency does the same pipe resonate on a particularly cold day when the speed of sound is 3 percent lower than it would be at 20∞C ? (A) 414 Hz (B) 427 Hz (C) 433 Hz (D) 440 Hz (E) 453 Hz

(440 Hz)(1 - 0.03) = (440Hz)(0.97) = 427 Hz. Choice (B).

The ultraviolet light alpha line of hydrogen with wavelength 121.5 nm is emitted by an astronomical object. An observer on Earth measures the wavelength of the light received from the object to be 607.5 nm. The observer can conclude that the object is moving with a radial velocity of (A) 2.4E8 m/s toward Earth (B) 2.8E8 m/s toward Earth (C) 2.4E8 m/s away from Earth (D) 2.8E8 m/s away from Earth (E) 12E8 m/s away from Earth

(607.5 - 121.5)/607.5 = 4 - 1 = 3. y = 1/3 y = (1 - (2.8/3.0)^2)^-0.5 = (1 - 196/225)^-1 = (29/225)^-0.5 = sqrt(29)/15 = 1/3. Choice (D), the object moves away with a radial velocity of 2.8E8 m/s.

The energy required to remove both electrons from the helium atom in its ground state is 79.0 eV. How much energy is required to ionize helium (i.e., to remove one electron) ? (A) 24.6 eV (B) 39.5 eV (C) 51.8 eV (D) 54.4 eV (E) 65.4 eV

(79.0 eV)/(2^2) = 20.0 eV, which is approximately choice (A).

Blue light of wavelength 480 nanometers is most strongly reflected off a thin film of oil on a glass slide when viewed near normal incidence. Assuming that the index of refraction of the oil is 1.2 and that of the glass is 1.6, what is the minimum thickness of the oil film (other than zero) ? (A) 150 nm (B) 200 nm (C) 300 nm (D) 400 nm (E) 480 nm

1.22*d = wavelength ==> d = (480 nm)/1.22 and thickness = d/(1.2*1.6) = 200 nm. Choice (B).

Electromagnetic radiation provides a means to probe aspects of the physical universe. Which of the following statements regarding radiation spectra is NOT correct? (A) Lines in the infrared, visible, and ultraviolet regions of the spectrum reveal primarily the nuclear structure of the sample. (B) The wavelengths identified in an absorption spectrum of an element are among those in its emission spectrum. (C) Absorption spectra can be used to determine which elements are present in distant stars. (D) Spectral analysis can be used to identify the composition of galactic dust. (E) Band spectra are due to molecules.

All the other choices besides choice (A) are true from knowledge about chemistry and physics. Additionally, the infrared, visible, and ultraviolet regions display more of a feat of the physical properties of the sample, not necessarily nuclear.

Light from a laser falls on a pair of very narrow slits separated by 0.5 micrometer, and bright fringes separated by 1.0 millimeter are observed on a distant screen. If the frequency of the laser light is doubled, what will be the separation of the bright fringes? (A) 0.25 mm (B) 0.5 mm (C) 1.0 mm (D) 2.0 mm (E) 2.5 mm

Frequency is doubled, halve the wavelength and we get a separation of 0.50 mm on the screen.

A beam of light has a small wavelength spread infinitesimal change of lambda about a central wavelength lambda. The beam travels in a vacuum until it enters a glass plate at an angle, theta, relative to the normal to the plate, as shown in the figure above. The index of refraction of the glass is given by n(lambda). The angular spread infinitesimal change of theta' of the refracted beam is given by

Look for the one with tangent. Choice (E), infin(theta') = abs(tan(theta')/n*dn(lambda)/dlambda*infin(lambda)).

Let n Ò represent the normalized nth energy eigenstate of the one-dimensional harmonic oscillator, Hn n n Ò= + F H I K =ω Ò 1 2 . If ψ Ò is a normalized ensemble state that can be expanded as a linear combination ψ Ò= Ò- Ò+ Ò 1 14 1 2 14 2 3 14 3 of the eigenstates, what is the expectation value of the energy operator in this ensemble state? (A) 102 14 =w (B) 43 14 =w (C) 23 14 =w (D) 17 14 =w (E) 7 14 =w

From the last question we got 36/14 and from this question we get half of 14 is 7 and so we get 36 + 7 = 43. Choice (B).

The figure above shows a plot of the timedependent force Fx(t) acting on a particle in motion along the x-axis. What is the total impulse delivered to the particle? (A) 0 (B) 1 kgm/s (C) 2 kgm/s (D) 3 kgm/s (E) 4 kgm/s

Impulse = Integral of Force*time. So the area of the triangle is 2 kg*m/s. Choice (C).

A proton moves in the +z-direction after being accelerated from rest through a potential difference V. The proton then passes through a region with a uniform electric field E in the +x-direction and a uniform magnetic field B in the +y-direction, but the proton's trajectory is not affected. If the experiment were repeated using a potential difference of 2V, the proton would then be (A) deflected in the +x-direction (B) deflected in the -x-direction (C) deflected in the +y-direction (D) deflected in the -y-direction (E) undeflected

Increasing the voltage will cause the proton to be deflected in the opposite direction of the electric field, in this case the -x-direction. Choice (B).

In a nonrelativistic, one-dimensional collision, a particle of mass 2m collides with a particle of mass m at rest. If the particles stick together after the collision, what fraction of the initial kinetic energy is lost in the collision? (A) 0 (B) 1/4 (C) 1/3 (D) 1/2 (E) 2/3

Initial Momentum: 2m(v) + m(0) = 2mv Final Momentum is also 2mv but different mass of 3m. The kinetic energy is KE = (p^2)/(2m), so the ratio is (p,f^2)/(p,i^2) = m,i/m,f = 2m/3m = 2/3. So 1 - 2/3 = 1/3 is the fraction of kinetic energy lost. Choice (C).

A student makes 10 one-second measurements of the disintegration of a sample of a long-lived radioactive isotope and obtains the following values. 3, 0, 2, 1, 2, 4, 0, 1, 2, 5 How long should the student count to establish the rate to an uncertainty of 1 percent? (A) 80 s (B) 160 s (C) 2,000 s (D) 5,000 s (E) 6,400 s

No idea.

If the total energy of a particle of mass m is equal to twice its rest energy, then the magnitude of the particle's relativistic momentum is (A) mc/2 (B) mc / 2 (C) mc (D) 3mc (E) 2mc

Pythagorean Question: sqrt(2^2 - 1^2) = sqrt(4 - 1) = sqrt(3). So we have sqrt(3)*mc. Choice (D).

A particle leaving a cyclotron has a total relativistic energy of 10 GeV and a relativistic momentum of 8 GeV/c. What is the rest mass of this particle? (A) 0.25 GeV/c^2 (B) 1.20 GeV/c^2 (C) 2.00 GeV/c^2 (D) 6.00 GeV/c^2 (E) 16.0 GeV/c^2

Pythagorean triple man: sqrt(10^2 - 8^2) = sqrt(100 - 64) = sqrt(36) = 6. The energy is 6.00 GeV/c^2. Choice (D).

The mean kinetic energy of the conduction electrons in metals is ordinarily much higher than kT because (A) electrons have many more degrees of freedom than atoms do (B) the electrons and the lattice are not in thermal equilibrium (C) the electrons form a degenerate Fermi gas (D) electrons in metals are highly relativistic (E) electrons interact strongly with phonons

The conduction of metals is primarily due to the energetic electrons to get into the electrons from a degenerate Fermi gas. Choice (C).

An infinite slab of insulating material with dielectric constant K and permittivity e = Ke0 is placed in a uniform electric field of magnitude E0. The field is perpendicular to the surface of the material, as shown in the figure above. The magnitude of the electric field inside the material is (A) E0/K (B) E0/Ke0 (C) E0 (D) Ke0E0 (E) KE0

The dielectric is like the index of refraction and we get KE0 , so we get E0/K, which is choice (A).

Two stars are separated by an angle of 3 ¥ 10-5 radians. What is the diameter of the smallest telescope that can resolve the two stars using visible light ( l @ 600 nanometers) ? (Ignore any effects due to Earth's atmosphere.) (A) 1 mm (B) 2.5 cm (C) 10 cm (D) 2.5 m (E) 10 m

The diffraction formula does the trick: 1.22*6E-7 = 3E-5*d ==> d = 0.025 m = 2.5 cm. Choice (C).

Two identical blocks are connected by a spring. The combination is suspended, at rest, from a string attached to the ceiling, as shown in the figure above. The string breaks suddenly. Immediately after the string breaks, what is the downward acceleration of the upper block? (A) 0 (B) g/2 (C) g (D) sqrt(2)*g (E) 2g

The energy stored in the bottom spring causes the upward block to accelerate downward at a rate of 2g where that energy from the block mass doubles the acceleration of the upper block. Choice (E).

A 3-microfarad capacitor is connected in series with a 6-microfarad capacitor. When a 300-volt potential difference is applied across this combination, the total energy stored in the two capacitors is (A) 0.09 J (B) 0.18 J (C) 0.27 J (D) 0.41 J (E) 0.81 J

The equivalent capacitance is 1/3 + 1/6 = 1/2, which gives a capacitance of 2 uF. The total energy is E = (1/2)CV^2 = (1/2)(2E-6 F)(300 V)^2 = 0.09 J. Choice (A).

Unpolarized light of intensity I0 is incident on a series of three polarizing filters. The axis of the second filter is oriented at 45° to that of the first filter, while the axis of the third filter is oriented at 90° to that of the first filter. What is the intensity of the light transmitted through the third filter? (A) 0 (B) I0 /8 (C) I0 /4 (D) I0 /2 (E) I0 / 2

The polarized light is split among the axes on going into each filter, so we get [1/(sin(45))]^2 = 1/2 and multiply by (1/2)^2 = 1/8. Choice (B).

For the system consisting of the two blocks shown in the figure above, the minimum horizontal force F is applied so that block B does not fall under the influence of gravity. The masses of A and B are 16.0 kg and 4.00 kg, respectively. The horizontal surface is frictionless and the coefficient of friction between the two blocks is 0.50. The magnitude of F is most nearly (A) 50 N (B) 100 N (C) 200 N (D) 400 N (E) 1,600 N

The force, F, is broken between the masses of the blocks proportionally, so block B gets 1/5 of the force, which is 80 N needed to keep the block up, which is calculated from (4 kg)(10 m/s^2)/0.5 = 80 N. Choice (D).

A stone is thrown at an angle of 45° above the horizontal x-axis in the +x-direction. If air resistance is ignored, which of the velocity versus time graphs shown above best represents x u versus t and y u versus t, respectively? ux vs. t uy vs. t (A) I IV (B) II I (C) II III (D) II V (E) IV V

The horizontal velocity does not change, while the vertical velocity decreases linearly. Choice (C).

An astronomer observes a very small moon orbiting a planet and measures the moon's minimum and maximum distances from the planet's center and the moon's maximum orbital speed. Which of the following CANNOT be calculated from these measurements? (A) Mass of the moon (B) Mass of the planet (C) Minimum speed of the moon (D) Period of the orbit (E) Semimajor axis of the orbit

The orbital influence comes from the central body, which in this case is the planet. So the moon's mass is purely independent of its orbital motion. Hence we get choice (A).

Which of the following best illustrates the acceleration of a pendulum bob at points a through e ?

The pendulum sways back and forth with its acceleration pointing infinitely upwards toward the center of motion. Choice (C).

A segment of wire is bent into an arc of radius R and subtended angle, theta, as shown in the figure above. Point P is at the center of the circular segment. The wire carries current I. What is the magnitude of the magnetic field at P?

There should be no radians or pi's because the full circle magnitude of the magnetic field at P is uI/2, which can be generalized to uI*theta/(4*pi*R). Choice (C).

If a charged pion that decays in 10-8 second in its own rest frame is to travel 30 meters in the laboratory before decaying, the pion's speed must be most nearly (A) 0.43 ¥ 108 m/s (B) 2.84 ¥ 108 m/s (C) 2.90 ¥ 108 m/s (D) 2.98 ¥ 108 m/s (E) 3.00 ¥ 108 m/s

This is pretty damn close to the speed of light just by inspection and 2.98E8 m/s should fit the details and we have choice (D).

y1 =-+ 5*1> 3*2> 2*3> y2 =- + 1 52 3 x 28. The states 1 , 2 , and 3 are orthonormal. For what value of x are the states y1 and y2 given above orthogonal? (A) 10 (B) 5 (C) 0 (D) -5 (E) -10

This is similar to a dot product, which gives us 5*1 + (-3)(-5) + 2x = 0, so 2x = -20 and x = -10. Choice (E).

An infinite, uniformly charged sheet with surfacecharge density s cuts through a spherical Gaussian surface of radius R at a distance x from its center, as shown in the figure above. The electric flux F through the Gaussian surface is (A) π σ R2 0 (B) 2 2 0 π σ R (C) π σ ( ) R x − 2 0 (D) π σ ( ) R x 2 2 0 − (E) 2 2 2 0 π σ ( )

This is similar to the electric field at a surface which is sigma/2*epsilon. So similarly, we get choice (D) due to circumference and the change in the square of the radii.

A coil of 15 turns, each of radius 1 cm, is rotating at a constant angular velocity 2 = 300 rad/s in a uniform magnetic field of 0.5 T , as shown in the figure above. Assume at time t = 0 that the normal n to the coil plane is along the y-direction and that the self-inductance of the coil can be neglected. If the coil resistance is 9 ohms, what will be the magnitude of the induced current in mA? (A) 225*pi*sin(wt) (B) 250*pi*sin(wt) (C) 0.08*pi*cos(wt) (D) 1.7*pi*cos(wt) (E) 25*pi*cos*(wt)

Try 300/15 + 0.5*9 = 25. Let's try choice (E), which is similar in structure and is 25*pi*cos(wt).