Official Physics GRE Exam #4

An engine absorbs heat at a temperature of 727 C and exhausts heat at a temperature of 527 C. If the engine operates at maximum possible efficiency, for 2000 joules of heat input the amount of work the engine performs is most nearly (A) 400 J (B) 1450 J (C) 1600 J (D) 2000 J (E) 2760 J

(2000 J)(1 - 800/1000) = (2000 J)(1/5) = 400 J. Choice (A).

A system is known to be in the normalized state described by the wave function phi(theta, phi) = (1/sqrt(30))(5Y3,4 + Y3,6 - 2Y0,6), where the Yl,m(theta, phi) are the spherical harmonics. The probability of finding the system in a state with azimuthal orbital quantum number m = 3 is (A) 0 (B) 1/15 (C) 1/6 (D) 1/3 (E) 13/15

(5^2 + 1^2)/30 = 26/30 = 13/15. Choice (E).

In the circuit above, the resistances are given in ohms and the battery is assumed ideal with emf equal to 3.0 V. - The resistor that dissipates the most power is (A) R1 (B) R2 (C) R3 (D) R4 (E) R5 - The voltage across resistor R4 is (A) 0.4 V (B) 0.6 V (C) 1.2 V (D) 1.5 V (E) 3.0 V

- R1 gets the most resistance and has the most voltage gain, which implies resistor R1 dissipates the greatest power. Choice (A). - Breaking up the resistances and voltages between series and parallel, we get a voltage of 0.4 V across R4. Choice (A).

Isotherms and coexistence curves are shown in the pV diagram above for a liquid-gas system. The dashed lines are the boundaries of the labeled regions. - Which numbered curve is the critical isotherm? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 - In which region are the liquid and the vapor in equilbrium with each other? (A) A (B) B (C) C (D) D (E) E

- The isotherm is represented by the tangential curve to the region labeled B. Choice (B). - Equilibrium occurs in the proximity of the liquid region and is choice (B).

A cylinder with moment of inertia 4 kg*m^2 about a fixed axis initially about a fixed axis initially rotates at 80 radians per second about this axis. A constant torque is applied to slow it down to 40 radians per second. - The kinetic energy lost by the cylinder is (A) 80 J (B) 800 J (C) 4000 J (D) 9600 J (E) 19,200 J - If the cylinder takes 10 seconds to reach 40 radians per second, the magnitude of the applied torque is (A) 80 N*m (B) 40 N*m (C) 32 N*m (D) 16 N*m (E) 8 N*m

- dKE = (1/2)I(w,f^2 - w,i^2) = (1/2)(4 kg*m^2)(80^2 - 40^2) = 9600 J. Choice (D). - a = (80 - 40)/10 = 4 rad/s^2. N = Ia = (4 kg*m^2)(4 rad/s^2) = 16 N*m. Choice (D).

Solid argon is held together by which of the following bonding mechanisms? (A) Ionic bond only (B) Covalent bond only (C) Partly covalent and partly ionic bond (D) Metallic bond (E) van der Waals bond

Argon is a noble gas and it is held together by van der Waals bond in its solid state. Choice (E).

A radioactive nucleus decays, with the activity shown in the graph above. The half-life of the nucleus is (A) 2 min (B) 7 min (C) 11 min (D) 18 min (E) 23 min

By observing the graph, it can be seen that the half-life is ostensibly 7 minutes. Choice (B).

Two horizontal scintillation counters are located near the Earth's surface. One is 3.0 meters directly above the other. Of the following, which is the largest scintilllator resolving time that can be used to distinguish downward-going relativistic muons form upward-going relativistic muons using the relative time of the scinllator signals? (A) 1 picosecond (B) 1 nanosecond (C) 1 microsecond (D) 1 millisecond (E) 1 second

Close to the speed of light which gives us (3.0 m)/(1E-9 s) = 3.0E9 m/s. Choice (B).

Given that the binding energy of the hydrogen atom ground state is E0 = 13.6 eV, the binding energy of the n = 2 state of positronism (positron-electron system) is (A) 8E0 (B) 4E0 (C) E0 (D) E0/4 (E) E0/8

For some reason, the binding energy of positronium is one-half the binding energy of the hydrogen atom. So (13.6 eV)/(2*2^2) = 1.7 eV. Choice (E).

Light of wavelength 5200 angstroms is incident normally on a transmission diffraction grating with 2000 lines per centimeter. The first-order diffraction maximum is at an angle, with respect to the incident beamn, that is most nearly (A) 3 degrees (B) 6 (C) 9 (D) 12 (E) 15

Do (1E-2/2000)/(5.2E-7) gives us 6 degrees. Choice (B).

The state of a quantum mechanical system is described by a wave function phi. Consider two physical observables that have discrete eigenvalues: observable A with eigenvalues {a}, and observable B with eigenvalues {B}. Under what circumstances can all wave functions be expanded in a set of basis states, each of which is a simultaneous eigenfunction of both A and B? (A) Only if the values {a} and {B} are nondegenerate (B) Only if A and B commute (C) Only if A commutes with the Hamiltonian of the system (D) Only if B commutes with the Hamiltonian of the system (E) Under all circumstances

Eigenvalues like to commute. Choice (B).

A conducting cavity is driven an an electromagnetic resonator. If perfect conductivity is assumed, the transverse and normal field components must obey which of the the following conditions at the inner cavity walls? (A) E,n = 0, B,n = 0 (B) E,n = 0, B,t = 0 (C) E,t = 0, B,t = 0 (D) E,t = 0, B,n = 0 (E) None of the above

Electrical is transverse and Magnetic is normal, so to get perfect conduction we will need vector components in synchronicity with those components. Choice (D).

A plane-polarized electromagnetic wave is incident normally on a flat, perfectly conducting surface. Upon reflection at the surface, which of the following is true? (A) Both the electric vector and magnetic vector are reversed. (B) Neither the electric vector nor the magnetic vector is reversed. (C) The electric vector is reversed; the magnetic vector is not. (D) The magnetic vector is reversed; the electric vector is not. (E) The directions of the electric and magnetic vectors are interchanged.

Similar to question #34, where we have a perfect conductor and we have the electric vector direction reversed and the magnetic vector not. Choice (C).

Listed below are Maxwell's equations of electromagnetism. If magnetic monopoles exist, which of these equations would be INCORRECT? I. Curl H = J + dD/dt II. Curl E = -dB/dt III. div D = p IV. div B = 0 (A) I only (B) I and II (C) I and III (D) II and IV (E) III and IV

If the magnetic fields exist in the equation, then there will be a change in those equations which give us Curl E = -dB/dt and div B = 0 and is representative of choice (D).

In experiments located deep underground, the two types of cosmic rays that most commonly reach the experimental apparatus are (A) alpha particles and neutrons (B) protons and electrons (C) iron nuclei and carbon nuclei (D) muons and neutrinos (E) positrons and electrons

Important and comes from astronomy, where most of the cosmic rays typically come from the particles that reach the apparatus are muons and neutrinos. Choice (D).

In a double-slit interference experiment, d is the distance between the centers of the slits and w is the width of each slit, as shown in the figure above. For incident plane waves, an interference maximum on a distant screen will be "missing" when (A) d = sqrt(2)w (B) d = sqrt(3)w (C) 2d = w (D) 2d = 3w (E) 3d = 2w

Let d be bigger than w and have integer coefficients, so we get choice (D).

The total energy of a blackbody radiation source is collected for one minute and used to heat water. The temperature of the water increases from 20.0 C to 20.5 C. If the absolute temperature of the blackbody were doubled and the experiment repeated, which of the following statements would be most nearly correct? (A) The temperature of the water would increases from 20 C to a final temperature of 21 C. (B) The temperature of the water would increase from 20 C to a final temperature of 24 C. (C) The temperature of the water would increase from 20 C to a final temperature of 28 C. (D) The temperature of the water would increase from 20 C to a final temperature of 36 C. (E) The water would boil within the one-minute time period.

Let's cube the double, so we get 2^4 = 16(the difference) = 16(0.5 C) = 8.0 C. Choice (C).

What is F(R)/F(2R)? (A) 32 (B) 8 (C) 4 (D) 2 (E) 1

Simple, the ratio becomes F(R)/F(2R) = (2/1)^2 = 4. Choice (C).

A simple telescope consists of two convex lenses, the objective and the eyepiece, which have a common focal point P, as shown in the figure above. If the focal length of the objective is 1.0 meter and the angular magnification of the telescope is 10, what is the optical path length between objective and eyepiece? (A) 0.1 m (B) 0.9 m (C) 1.0 m (D) 1.1 m (E) 10 m

It is a telescope man, it is obvious that we want to magnify, so we are left with choices (D) and (E). But choice (D) seems to be the best choice due to the ratio of the magnification being 10 and giving significant contraction by a factor of 10. Hence 1 + 0.1 = 1.1. Choice (D).

A particle of mass m on the Earth's surface is confined to move on the parabolic curve y = ax^2, where y is up. Which of the following is a Lagrangian for the particle?

L = 91/2)my*^2(1 + 1/(4ay)) - mgy. Choice (A) because there is parabolic umph given to a particle.

The capacitor in the circuit shown above is initially charged. After closing the switch, how much time elapses until one-half of the capacitor's initial stored energy is dissipated? (A) RC (B) RC/2 (C) RC/4 (D) 2RCln(2) (E) RCln(2)/2

Look at the repetitive 2's in the numerator and the denominator and we see that the time elapsed becomes RCln(2)/2. Choice (E).

An attractive, one-dimensional square well has depth V0 as shown above. Which of the following best shows a possible wave function for a bound state?

Look for the function which exhibits the most sinusoidal behavior and you get choice (B) with an odd sine wave in between the values of x1 and x2.

A classical model of a diatomic molecule is a springy dumbbell, as shown above, where the dumbbell is free to rotate about axes perpendicular to the spring. In the limit of high temperature, what is the specific heat per mole at constant volume? (A) (3/2)R (B) (5/2)R (C) (7/2)R (D) (9/2)R (E) (11/2)R

Remember: The specific heat per mole at constant volume is (7/2)R. Choice (C).

The ratio of the energies of the K characteristic x-rays of carbon (Z = 6) to those of magnesium (Z = 12) is most nearly (A) 1/4 (B) 1/2 (C) 1 (D) 2 (E) 4

The energy of an atom is e(Z/n)^2. So the ratio of the atomic masses gives us (6/12)^2 = (1/2)^2 = 1/4. Choice (A).

The magnitude of the force F on an object can be determined by measuring both the mass m of an object and the magnitude of its acceleration a, where F = ma. Assume that these measurements are uncorrelated and normally distributed. If the standard deviations of the measurements of the mass and the acceleration are sigma,m and sigma,a, respectively, then sigma,F/F is what?

The Pythagorean looking answer is the correct one. Choice (C).

Two wedges, each of mass m, are placed next to each other on a flat floor. A cube of mass M is balanced on the wedges as shown above. Assume no friction between the cube and the wedhes, but a coefficient of static friction u < 1 between the wedges and the floor. What is the largest M that can be balanced as shown without motion of the wedges?

The answer is 2um/(1 - u), where as the coefficient of friction increases, we should get more mass M necessary to keep up the block. Choice (D).

Which of the following is most nearly the mass of the Earth? (The radius of the Earth is about 6.4E6 meters.)

The mass of the Earth is 5.98E24 kg, remember it. Choice (A).

A cylindrical tube of mass M can slide on a horizontal wire. Two identical pendulums, each of mass m and length l, hang from the ends of the tube, as shown above. For small oscillations of the pendulums in the plane of the paper, the eigenfrequencies of the normal modes of oscillation of this system are 0, sqrt[g(M + 2m)/(lM)], and what?

The limiting and limited period given by sqrt(g/l) is in fact the last eigenvalue. Choice (A).

The longest wavelength x-ray that can undergo Bragg diffraction in a crystal for a given family of planes of spacing d is (A) d/4 (B) d/2 (C) d (D) 2d (E) 4d

The longest wavelength for an x-ray in this crystal is 2d, double the spacing. Choice (D).

Tau leptons are observed to have an average half-life of dt1 in the frame @1 in which the leptons are at rest. In an inertial frame S2, which is moving at a speed v12 relative to S1, the leptons are observed to have an average half-life of dt2. In another inertial reference frame S3, which is moving at a speed v13 relative to S1 and v23 relative to S2, the leptons have an observed half-life of dt3. Which of the following is a correct relationship among two of the half-lives, dt1, dt2, and dt3?

The only choice with the "skipped" feature is choice (B) with dt1 = dt3*sqrt(1 - (v13)^2/c^2).

In a 3S state of the helium atom, the possible values of the total electronic angular momentum quantum number are (A) 0 only (B) 1 only (C) 0 and 1 only (D) 0, 1/2, and 1 (E) 0, 1/2, 1

The only value for the total electronic angular momentum is 1. Choice (B).

Suppose there is a very small shaft in the Earth such that the point mass can be placed at a radius of R/2. What is F(R)/F(R/2)?

The outside radius gravitation gives us: F(r) = Gm1m2R^2/r. So the ratio becomes F(R)/F(R/2) = 2/1 = 2. Choice (C).

The outputs of two electrical oscillators are compared on an oscilloscope screen. The oscilloscope spot is initially at the center of the screen. Oscillator Y is connected to the vertical terminals of the oscilloscope and oscillator X to the horizontal terminals. Which of the following patterns could appear on the oscilloscope screen, if the frequency of oscillator Y is twice that of oscillator X?

The ratio between the two is choice (A) with an upside-down V-shape fashion.

Two large conducting plates form a wedge of angle a as shown in the diagram above. The plates are insulated from each other; one has a potential V0 and the other is grounded. assuming that the plates are large enough so that the potential difference between them is independent of the cylindrical coordinates z and p, the potential anywhere between the plates as a function of the angle phi is (A) V0/a (B) V0*phi/a (C) V0*a/phi (D) V0*phi^2/a (E) V0*a/phi^2

The ratio of the arc length for phi to alpha is the determining factor for the rescaling of the voltage V0. Choice (B).

A pi0 meson (rest-mass energy 135 MeV) is moving with velocity 0.8ck in the laboratory rest frame when it decays into two photons, y1 and y2. In the pi0 rest frame, y1 is emitted forwatd and y2 is emitted backward relative to the pi0 direction of flight. The velocity of y2 in the laboratory rest frame is (A) -1.0ck (B) -0.2k (C) +0.8ck (D) +1.0ck (E) +1.8ck

The relative velocity should be -1.0ck. Choice (A).

A rigid cylinder rolls at constant speed without slipping on top of a horizontal plane surface. The acceleration of a point on the circumference of the cylinder at the moment when the point touches the plane is (A) directed forward (B) directed backward (C) directed up (D) directed down (E) zero

The rolling of the cylinder causes the "centripetal" acceleration to be pointed upwards. Choice (C).

The Fermi temperature of Cu is about 80,000 K. Which of the following is most nearly equal to the average speed of a conduction electron in Cu? (A) 2E-2 m/s (B) 2 m/s (C) 2E2 m/s (D) 2E4 m/s (E) 2E6 m/s

The speed of electrons is generally on the order of magnitude of 6, which gives us 2E6 m/s as the best estimate. Choice (E).

A coaxial cable having radii a, b, and c carries equal and opposite currents of magnitude i on the inner and outer conductors. What is the magnitude of the magnetic induction at point P outside of the cable at a distance r from the axis? (A) 0 (B) uir/(2*pi*a^2) (C) ui/(2*pi*r) (D) (ui/2*pi*r)[(c^2 - r^2)/(c^2 - b^2)] (E) (ui/2*pi*r)[(r^2 - b^2)/(c^2 - b^2)]

There is a cancellation of the currents and magnetic fields on the outside of the axis, which causes there to be zero current on the outside of the wire. Choice (A).

A soap film with index of refraction greater than air is formed on a circular wire frame that is held in a vertical plane. The film is viewed by reflected light from a white-light source. Bands of color are observed at the lower parts of the soap film, but the area near the top appears black. A correct explanation for this phenomenon would involve which of the following? I. The top of the soap film absorbs all of the light incident on it; none is transmitted. II. The thickness of the top part of the soap film has become much less than a wavelength of visible light. III. There is a phase change of 180 degrees for all wavelengths of light reflected from the front surface of the soap film. IV. There is no phase change for any wavelength of light reflected from the back surface of the soap film. (A) I only (B) II and III only (C) III and IV only (D) I, II, and III (E) II, III, and IV

There is no perfect light absorber known in existence and hence only the three last selections remain. Choice (E).

If n is an integer ranging from 1 to infinity, w is an angular frequency, and t is time, then the Fourier series for a square wave, as shown above, is given by what?

V(t) = (4/pi)Csigma[sin[(2n + 1)wt]/(2n + 1), which gives 0's and 1's when we want them on the graph. Choice (B).

A ball is dropped from a height h. As it bounces off the floor, its speed is 80 percent of what it was just before it hit the floor. The ball will then rise to a height of most nearly (A) 0.94h (B) 0.80h (C) 0.75h (D) 0.64h (E) 0.50h

This is really as simple as it may seem. (0.8)^2 = 0.64. So the maximum height of the ball is 0.64h. Choice (D).

In transmitting high frequency signals on a coaxial cable, it is important that the cable be terminated at an end with its characteristic impedance in order to avoid (A) leakage of the signal out of the cable (B) overheating of the cable (C) reflection of signals from the terminated end of the cable (D) attenuation of the signal propagating in the cable (E) production of image currents in the outer conductor

We do not want the reflection of signals so we terminate the end of the cable. Choice (C).

If a freely moving electron is localized in space to within d(x0), its wave function can be described by a wave packet phi(x, t) = indeintegral(e^[i(kx - wt)]f(k)dk, where f(k) is peaked around a central value k0. Which of the following is most nearly the width of the peak in k?

We get an inverse relationship based on units, so we have dk = 1/dx0, which is choice (B).

Two positive charges of q and 2q coulombs are located on the x-axis at x = 0.5a and 1.5a, respectively, as shown above. There is an infinite, grounded conducting plane at x = 0. What is the magnitude of the net force on the charge q?

[7/4*pi*e](q/a)^2/2. Choice (E).

If dL/dq = 0 , where L is the Lagrangian for a conservative system without constraints and q,n is a generalized coordinate, then what is the generalized momentum p,n?

constant. Choice (B).

The wave function of a particle is e^[i(kx - wt], where x is the distance, t is time, and k and w are positive real numbers. The x-component of the momentum of the particle is (A) 0 (B) hw (C) hk (D) hw/c (E) hk/w

h = [J*s] and k = [1/m] and so we get the following: hk = [J*s]/[m] = [kg*m^2/s]*[1/m] = [kg*m/s]. Choice (C).

A solid cone hangs from a frictionless pivot at the origin ), as shown above. If i, j, and k are unit vectors, and a, b, and c are positive constants, which of the following forces F applied to the rim of the cone at a point P results in a torque t on the cone with a negative component t,z? (A) F = ak, P is (0, b, -c) (B) F = -ak, P is (0, -b, -c) (C) F = aj, P is (-b 0, -c) (D) F = aj, P is (b, 0, -c) (E) F = -ak, P is (-b, 0, -c)

r x F = (-b, 0, -c) x (0, a, 0) = -abk. Choice (C).