Physics GRE Sample Exam #1
A wire loop that encloses an area of 10 cm^2 has a resistance of 5 ohms. The loop is placed in a magnetic field of 0.5 T with its plane perpendicular to the field. The loop is suddenly removed from the field. How much charge flows past a given point in the wire? (A) 10^-4 C (B) 10^-3 C (C) 10^-2 C (D) 10^-1 C (E) 1 C
(0.5 T)(1E-3 m^2)/(5 ohms) = 1E-4 C. Choice (A).
During a hurricane, a 1,200 Hz warning siren on the town hall sounds. The wind is blowing at 55 m/s in a direction from the siren toward a person 1 km away. With what frequency does the sound wave reach the person? (The speed of sound in air is 330 m/s.) (A) 1,000 Hz (B) 1,030 Hz (C) 1,200 Hz (D) 1,400 Hz (E) 1,440 Hz
(1,200 Hz)(55 m/s + 330 m/s)/(330 m/s) = (1,200 Hz)(385/330) = (1,200 Hz)(7/6) = 1,400 Hz. Got ya, the frequency of sound waves in air do not change, but the perceived frequency does. Choice (C)
A uniform thin film of soapy water with index of refraction n = 1.33 is viewed in air via reflected light. The film appears dark for long wavelengths and first appears bright for lambda = 540 nm. What is the next shorter wavelength at which the film will appear bright on reflection? (A) 135 nm (B) 180 nm (C) 270 nm (D) 320 nm (E) 405 nm
(1.33 - 1)(540 nm) = (0.33)(540) = 180 nm. Choice (A).
An organ pipe, closed at one end and open at the other, is designed to have a fundamental frequency of C (131 Hz). What is the frequency of th enexy higher armonic for this pipe? (A) 44 Hz (B) 196 Hz (C) 262 Hz (D) 393 Hz (E) 524 Hz
(131 Hz)(2 + 1) = 3(131 Hz) = 393 Hz. The next higher harmonic for this pipe is 393 Hz. Choice (D)
A large, parallel-plate capacitor consists of two square plates that measure 0.5 m on each side. A charging current of 9 A is applied to the capacitor. Which of the following gives the approximate rate of change of the electric field between the plates? (A) 2 V/(m*s) (B) 40 V/(m*s) (C) 1E12 V/(m*s) (D) 4E12 V/(m*s) (E) 2E13 V/(m*s)
([A/d]e0-1 = [18][2.03E10] = 4E11 <-- The 4 is suspicious, let's take it. Choice (D)
A heat pump is to extract heat from an outdoor environment at 7 C and heat the environment indoors to 27 C. For each 15,000 J of heat delivered indoors, the smallest amount of work that must be supplied to the heat pump is approximately (A) 500 J (B) 1,000 J (C) 1,100 J (D) 2,000 J (E) 2,200 J
15,000J(1 - (280K/300K)) = 1,000 J. Choice (B)
An incompressible fluid of density p flows througha horizontal pipe of radius r and then passes through a constriction of radius r/2. If the fluid has pressure P0 and velocity v before the constriction, the pressure in the constriction is (A) P0 - (15/2)pv0^2 (B) P0 - (3/2)pv0^2 (C) Po/4 (D) Po + (3/2)pv0^2 (E) P0 + (15/2)pv0^2
2^4 - 1^4 = 15 and the pressure should be lower than before, so we get choice (A).
A distant galaxy is observed to have its hydrogen-beta line shifted to a wavelength of 580 nm, away from the laboratory value of 434 nm. Which of the following gives the approximate velocity of recession of the distant galaxy? (Note: 580/434 = 4/3) (A) 0.28c (B) 0.53c (C) 0.56c (D) 0.75c (E) 0.86c
4/3 - 1 = 1/3 . So we get about 0.28c for the speed, due to the fairly weak relativistic effects. Choice (A)
Characteristics of the quantum harmonic oscillator include which of the following? I. A spectrum of evenly spaced energy states II. A potential energy function that is linear in the position coordinate III. A ground state that is characterized by zero kinetic energy IV. A nonzero probability of finding the oscillator outside the classical turning points. (A) I only (B) IV only (C) I and IV only (D) II and III only (E) I, II, III, and IV
A spectrum of evenly spaced energy states and a nonzero probability of finding the oscillator outside the classical turning points are characteristic of the quantum harmonic oscillator. Choice (C)
In the Compton effect, a photon with energy E scatters a 90 angle from a stationary electron of mass m. The energy of the scattered photon is (A) E (B) E/2 (C) (E^2)/(mc^2) (D) (E^2)/(E + m*c^2) (E) E*mc^2/(E + mc^2)
Adding the inherent energy E and the mass energy under the energy squared seems reasonable because scattering breaks and takes away energy, so choice (D).
A ball is thrown out of the passenger window of a car moving to the right (ignore air resistance). If the ball is thrown out perpendicular to the velocity of the car, which of the following best depicts the path the ball takes, as viewed from above?
Assuming there is no air resistance, the ball would continue in a straight line. Choice (B)
If the Sun were suddenly replaced by a black hole of the same mass, it would have a Schwarzschild radius of 3,000 m. What effect, if any, would this change have on the orbits of the planets? (A) The planets would move directly toward the Sun. (B) The planets would move in spiral orbits. (C) The planets would oscillate about their former elliptical orbits. (D) The orbits would precess much more rapidly. (E) The orbits would remain unchanged.
At the edge of the Schwarzchild zone, the orbit of the planet does not change because the edge is the outer limit for the planet, which causes it to have typical orbital behavior, choice (E).
The root-mean-square speed of molecules of mass m in an ideal gas at temperature T is (A) 0 (B) sqrt(2kT/m) (C) sqrt(3kT/m) (D) sqrt(8kT/(pi*m)) (E) kT/m
Basic Chemistry, three degrees of freedom, which gives us: sqrt(3kT/m). Choice (C)
Which of the following cannot CANNOT be used as a dopant in germanium to make an n-type semiconductor? (A) As (B) P (C) Sb (D) B (E) N
Boron (B) has three valence electrons where the others have only five, so choice (D).
The energy from electromagnetic waves in equilibrium in a cavity is used to melt ice. If the Kelvin temperature of the cavity is increased by a factor of two, the mass of ice that can be melted by a fixed amount of time is increased by a factor of (A) 2 (B) 4 (C) 8 (D) 16 (E) 32
By Boltzmann law, the flux of radiation is given by F = sigma*T^4, so doubling the temperature will increased the mass of ice melted per unit time by 16 (2^4 = 16). Choice (D).
An electron with total energy E in the region x < 0 is moving in the +x-direction. It encounters a step potential at x = 0. The wave functions for x </ o and x >/ 0 is given in your Quantum Mechanics textbook. Which of the following gives the reflection coefficient for the system? (A) R = 0 (B) R = 1 (C) R = k2/k1 (D) R = [(k1 - k2)/(k1 + k2)]^2 (E) R = 4k1k2/(k1 + k2)^2
Choice (D)
De Broglie hypothesized that the linear momentum and wavelength of a free massive particle are related by which of the following constants? (A) Planck's constant (B) Boltzmann's constant (C) The Rydberg constant (D) The speed of light (E) Avogardo's number
De Broglie wavelength is given by the formulh h/p = h/mc, which contains Planck's constant. Choice (A)
A parallel-plate capacitor has plate separation d. The space between the plates is empty. A battery supplying voltage V0 is connected across the capacitor, resulting in electromagnetic energy U0 stored in the capacitor. A dielectric, of dielectric constant k, is inserted so that it just fills the space between the plates. If the battery is still connected, what are the electric field E and the energy U stored in the dielectric, in terms of V0 and U0?
Dielectric decreases the electric field and the electromagnetic energy does not change due to the dielectric. Choice (D).
A beam of muons travels through the laboratory with speed v = (4/5)c. The lifetime of a muon in its rest frame is t = 2.2E-6 s. The mean distance traveled by the muons in the laboratory frame is (A) 530 m (B) 660 m (C) 880 m (D) 1,100 m (E) 1,500 m
Do not forget about time dilation, relativistically. d = v* t = (1 - 0.8^2)^-0.5 *(2.4E8 m/s)(2.2E-6 s) = 880 m. Choice (C)
A uniform disk with a mass of m and a radius of r rolls without slipping along a horizontal surface and ramp, as shown above. The disk has an initial velocity of v. What is the maximum height h to which the center of mass of the disk rises? (A) h = (v^2)/(2g) (B) h = 3(v^2)/(4g) (C) h = (v^2)/g (D) h = 3(v^2)/(2g) (E) h = 2(v^2)/g
Due to inertia and the recurring 3 in these problems, which can suspect a below ideal height of rise, which is given in choice (B).
Two spaceships approach Earth with equal speeds as measured by an observer on Earth, but from opposite directions. A meterstick on one spaceship is measured to be 60 cm long by an occupant of the other spaceship. What is the speed of each spaceship, as measured by the observer on Earth? (A) 0.4c (B) 0.5c (C) 0.6c (D) 0.7c (E) 0.8c
Einstein's velocity addition rule gives us: (x^2)/(1 - x^2) = 0.6, which gives us 0.6 = 1.6x^2 => x = sqrt(3/8) = sqrt(0.375) = 0.6
Which of the following expressions is proportional to the total energy for the levels of a one-electron Bohr atom? (m is the reduced mass, Z is the number of protons in the nucleus, -e is the charge on the electron, and n is the principal quantum number.) (A) mZe^2/n (B) mZe^2/n^2 (C) mZ^2*e^4/n^2 (D) m^2*Z^2*e^2/n^2 (E) m^2*Z^2*e^4/n^2
Everything is taken to an even power, except for the mass and hence the choice should be choice (C).
Two sinsoidal waveforms of the same frequency are displayed on an oscilloscope screen, as indicated above. The horizontal sweep of the oscilloscope is set to 100 ns/c, and the vertical gains of channels 1 and 2 are each set to 2 V/cm. the zero-voltage level of each channel is given at the right in the figure.
Eyeballing the figure we get about 120 degrees as the phase difference between the two channels. Important for alternating current. Choice (E)
In the figure above, block A has mass m,A = 25 kg and block B has mass m,B = 10 kg. Both blocks move with constant acceleration a = 2 m/s^2 to the right, and the coefficient of static friction between the two blocks is u,s = 0.8. The static frictional force acting between the blocks is (A) 20 N (B) 50 N (C) 78 N (D) 196 N (E) 274 N
F,fB - F,A = (0.8)(10 kg)(9.8 m/s^2) - (25 kg)(2 m/s^2) = 80 N - 50 N = 30 N, closest to choice (A).
Which of the following statements about bosons and/or fermions is true? (A) Bosons have symmetric wave functions and obey the Paul exclusion principle. (B) Bosons have antisymmetric wave functions and do not obey the Pauli exclusion principle. (C) Fermions have symmetric wave functions and obey the Pauli exclusion principle. (D) Fermions have antisymmetric wave functions and obey the Pauli exclusion principle. (E) Bosons and fermions obey the Paul exclusion principle.
Fermions have antisymmetric wave functions and obey the Pauli Exclusion Principles, whereas Bosons have symmetric wave functions. Choice (D).
A massless spring with force constant k launches a ball of mass m. In order for the ball to reach a speed v, by what displacement s should the spring be compressed? (A) s = v*sqrt(k/m) (B) s = v*sqrt(m/k) (C) s = v*sqrt(2k/m) (D) s = v*m/k (E) s = (v^2)*m/(2k)
For the spring, KE = PE. So (1/2)k*s^2 = (1/2)m*v^2, and hence s = v*sqrt(m/k). Choice (B).
Which of the following lasers utilizes transitions that involve the energy levels of free atoms? (A) Diode laser (B) Dye laser (C) Free-electron laser (D) Gas laser (E) Solid-state laser
Gas lasers provide an efficient transition between one state of an atom to another and hence can be used in the transitions for free atoms. Choice (D).
A particle with mass m and charge q, moving with a velocity v, enters a region of uniform magnetic field B< as shown in the figure above. The particle strikes the wall at a distance d from the entrance slit. If the particle's velocity stay the same but its charge-to-mass ratio is doubled, at what distance from the entrance slit will the particle strike the wall? (A) 2d (B) sqrt(2)*d (C) d (D) (1/sqrt(2))d (E) (1/2)d
If the charge-to-mass ratio is doubled, then the distance from the entrance slit where the particle will strike the wall is at (1/2)d. Choice (E).
For the logic circuit shown above, which of the following Boolean statements gives the output E in terms of inputs A, B, C, and D?
Image is on the web. Nots add to the nots and those not do not, but multiply. Choice (C), E = bar(bar(A) + bar(B)) + bar(C*D).
A pair of electric charges of equal magnitude q and opposite sign are separated by a distance l, as shown in the figure above. Which of the following gives the approximate magnitude and direction of the electric field set up by the two charges at a point P on the y-axis, which is located a distance r >> l from the x-axis?
Imagine found on exam, answer: choice (E)
On a frictionless surface, a block of mass M moving at speed v collides elastically with another block of the same mass that is intiailly at rest. After the collision, the first block moves at an angle, theta, to its initia direction and has a speed v/2. The second block's speed after the collision is (A) sqrt(3)*v/4 (B) v/2 (C) sqrt(3)*v/2 (D) sqrt(5)*v/2 (E) v + (v/2)cos(theta)
Initial momentum p,i = Mv Final momentum p,f = - M(v/2) + M(sqrt(5)/2), the speed v= sqrt(5)/2 lies on a diagonal of the right triangle with sqrt(1^2 + (1/2)^2) = sqrt(5)/2. Choice (D)
The discovery of the J/phi particle was especially significant because it provided evidence for which of the following? (A) Parity violation in weak interactions (B) Massive neutrinos (C) Higgs bosons (D) Charmed quarks (E) Strange quarks
Knowledge based question --> We get Charmed quarks
Which of the following is the principal decay mode of the positive mode of the positive muon u+? (A) u+ --> e+ + v,e (B) u+ --> p + v,u (C) u+ --> n + e+ + v,e (D) u+ --> e+ + v,e + /v,u (E) u+ -> pi+ + /v,e + v,u
Look for the three particles which include positron and a particle and an antiparticle, which is only apparent in choice (D).
In static electromagnetism, let E, B, J, and p be the electric field, magnetic field, current density, respectively. Which of the following conditions allows the electric field to be written in the form E = -grad(phi), where phi is the electrostatic potential? (A) div(J) = 0 (B) div(E) = p/e (C) curl(E) = 0 (D) curl(B) = uJ (E)
Naturally, the curl of the electrostatic field is zero. Choice (C).
For an adiabatic process involving an ideal gas having volume V and temperature T, which of the following is constant? (y = Cp/Cv) (A) TV (B) TV^y (C) TV^(y - 1) (D) (T^y)V (E) (T^y)V^-1
Nice mix between thermodynamics and vibrations and waves gives us the constant T*V^(y - 1). Choice (C).
The figure above shows an object O placed at a distance R to the left of a convex spherical mirror that has a radius of curvature R. Point C is the center of curvature R. Point C is the center of curvature of the mirror. The image formed by the mirror is at (A) infinity (B) a distance R to the left of the mirror and inverted (C) a distance R to the right of the mirrot and upright (D) a distance R/3 to the left of the mirrot and inverted (E) a distance R/3 to the right of the mirroe and upright
Obviously to the right of the mirror and the image is smaller due to convexity and apparently the image is upright. Choice (E).
True statements about the absorption and emission of energy by an atom include which of the following? I. An atom can only absorb photons of light that have certain specific energies. II. An atom can emit photons of light of any energy. III. At low temperature, the lines in the absorption spectrum of an atom coincide with the lines in its emission spectrum that represent transitions to the ground state. (A) I only (B) III only (C) I and II only (D) I and III only (E) I, II, and III
Obviously, selection I is correct and selection III is a little trickier to spot, but at low temperature the ground state energy is the limit in the emission spectrum, so selection III is correct as well. Choice (D).
A capacitor containing a switch S, capacitor C, and inductor L. If switch S is closed at time t = 0, which of the following represents the magnetic energy, U, in the inductor as a function of time? (Assume that the capacitor and inductor are ideal.)
Of course, we must start with zero magnetic energy because the current needs to jump up and then the inductor oscillates the circuit as well, so the magnetic energy is at full sinsoidual oscillation. Choice (A), the upside-down cosine curve.
A particle is in an infinite square well potential with walls at x = o and x = L. If the particle is in the state phi(x) = A*sin(3*pi*x/L), where A is a constant, what is the probability that the particle is between x = (1/3)L and x = (2/3)L? (A) 0 (B) 1/3 (C) 1/sqrt(3) (D) 2/3 (E) 1
One-third of the regions => one-third of the chances to pick the particle, which is 1/3. Choice (B).
A resistor in a circuit dissipates energy at a rate of 1 W. If the voltage across the resistor is doubled, what will be the new rate of energy dissipation? (A) 0.25 W (B) 0.5 W (C) 1 W (D) 2 W (E) 4 W
P = (V^2)/R with V' = 2V and so P' = (2V)^2/R = 4(V^2)/R = 4P = 4(1 W) = 4 W. Choice (E)
In the Bohr model of the hydrogen atom, the linear momentum of the electron at radius r n is given by which of the following? (n is the principal quantum number.) (A) nh (B) nrh (C) nh/r (D) (n^2)rh (E) (n^2)h/r
Planck's constant and the radius have units of J*s = N*m *s = kg*m^2/s and m, respectively. So the combination of units, J*s/r would suffice. Remember h/r is proportional to the value of n. Choice (C).
Consider a set of wave functions. Which of the following conditions guarantees that the functions are normalized and mutually orthogonal? (The indices i and j take on the values in the set {1, 2, ...., n}.)
Purely, the definition of orthogonality of functions, which is the integral of the complex conjugate and the function itself gives us the Kronecker Delta function, choice (E).
An electron has total energy equal to four times its rest energy. The momentum of the electron is (A) mc (B) sqrt(2)*m*c (C) sqrt(15)*m*c (D) 4*m*c (E) 2*sqrt(15)*m*c
Pythagorean analogues again gives us sqrt(4^2 - 1^2) = sqrt(16 - 1) = sqrt(15), given by choice (C).
A thermodynamic system, initially at absolute temperature T1, contains a mass m of water with specific heat capacity c. Heat is added until the temperature rises to T2. The change in entropy of the water is (A) 0 (B) T2 - T1 (C) mcT2 (D) mc(T2 - T1) (E) mc*ln(T2/T1)
Q Always think of natural logarithms when it comes to changes in entropy, which is given by choice (E), mc*ln(T2/T1).
A gas at temperature T is composed of molecules of mass m. Which of the following describes how the average time between intermolecular collisions varies with m? (A) It is proportional to 1/m. (B) It is proportional to m^(1/4). (C) It is proportional to sqrt(m). (D) It is proportional to m. (E) It is proportional to m^2.
RMS speed is given by t = sqrt(3kT/m), so the time, t, is reversed and we have it is proportional to sqrt(m). Choice (C).
Heat Q is added to a monatomic ideal gas under conditions of constant volume, resulting in a temperature change T. How much heat will be required to produce the same temperature change, if it is added under conditions of constant pressure? (A) (3/5)Q (B) Q (C) (5/3)Q (D) 2Q (E) (10/3)Q
Remember the combination of the prime numbers 5 and 3 in this scenario because there are three degrees of freedom in the system, which gives us (5/3)Q, which is the energy of the system.
The lifetime for the 2p --> 1s transition in hydrogen is 1.6E-9 s. The natural line width for the radiation emitted during the transition is approximately.
Simply, f = 1/T = 1/1.6E-9 s = 100 MHz, which is the frequency, natural line width for the radiation emitted during the transition. Choice (C)
According to the BCS theory, the attraction is between Cooper pairs in a superconductor is due to (A) the weak nuclear force (B) the strong nuclear force (C) vacuum polarization (D) interactions with the ionic lattice (E) the Casimir effect
Solid-state physics, so let's think of a network where the Cooper pairs interact with the ionic lattice. Choice (D)
A muon can be considered to be a heavy electron with a mass m,u = 207m,e. Imagine replacing the electron in a hydrogen atom with a muon. What are the energy levels E,n for this new form of hydrogen in terms of the binding energy of ordinary hydrogen E,0, the mass of the proton m,p, and the principal quantum number n?
Take the specific and general form in this case, so we get E,n = -E0/n^2[m,e(m,p + m,u)/m,u/(m,p + m,e)], where the m,u, which is the principal decay particle, overlays the m,e. Choice (D).
Two experimental techniques determine the mass of an object to be 11 +/- 1 kg and 10 +/- 2 kg. These two measurements can be combined to give a weighted average. The uncertainty of the weighted average. The uncertainty of the weighted average is equal to which of the following? (A) 1/2 kg (B) 2/sqrt(5) kg (C) 2/sqrt(3) kg (D) 2 kg (E) sqrt(5) kg
Taking the Pythagorean analogue, we get sqrt(1^2 + 2^2) = sqrt(5) and since there are two confidence ranges, then we will have 2/sqrt(5) kg for the uncertainty of the weighted average. Choice (B).
An atom has filled n = 1 and n = 2 levels. How many electrons does the atom have? (A) 2 (B) 4 (C) 6 (D) 8 (E) 10
The Pauli Exclusion Principle gives us the number of electrons as 2*n^2, where n is the energy level of the atom. 2(1^2 + 2^2) = 2(1 + 4) = 2(5) = 10. Choice (E)
A model of an optical fiber is shown in the figure above. The optical fiber has an index of refraction, n, and is surrounded by free space. What angles of incidence, theta, will result in the light staying in the optical fiber? (A) theta > sin^-1(sqrt(n^2 - 1) (B) theta < sin^-1(sqrt(n^2 - 1) (C) theta > sin^-1(sqrt(n^2 + 1) (D) theta < sin^-1(sqrt(n^2 + 1) (E) sin^-1(sqrt(n^2 - 1) < theta < sin^-1(sqrt(n^2 + 1)
The angle should be smaller than the minimum incident angle, so there is Choice (B).
A mass m attached to the end of a massless rod of length L is free to swing below the plane of support, as shown in the figure above. The Hamiltonian for this system is given by H = (p,theta)^2/(2mL^2) + (p,phi)^2/[(2m*L^2*sin^2(theta)] - mgLcos(theta), where theta and phi are defined as shown in the figure. On the basis of Hamilton's equations of motion, the generatlized coordinate or momnetum that is a constant in time is (A) theta (B) phi (C) dtheta (D) p,theta (E) p,phi
The angular momentum in the direction of the spinning motion gives us that p,phi is constant because we have ideally constant velocity and obviously mass. Choice (E).
The normalized ground state wave function of hydrogen is phi(100) = 2*e^(-r/a0)/[((4*pi)^0.5)(a0)^1.5)], where a0 is the Bohr radius. What is the most likely distance that the electron is from the nucleus? (A) 0 (B) a0/2 (C) a0/sqrt(2) (D) a0 (E) 2a0
The denominator of the exponent gives us a radius of r = a0.
A very long, thin, straight wire carries a uniform charge density of lambda per unit length. Which of the following gives the magnitude of the electric field at a radial distance r from the wire?
The electric field about the outside of a wire should decrease as the distance from the wire decreases, which gives us the radius found in the denominator, which is given by choice (A) as (1/2*pi*e)*lambda/r.
Consider a single electron atom with orbita angualr momentum L = sqrt(2)*h. Which of the following gives the possible values of a measurement of L,z, the z-component of L? (A) 0 (B) 0, h (C) 0, h, 2h (D)-h, 0, h (E) -2h, -h, 0, h, 2h
The energies are within the range of the sqrt(2), so choice (D) is the reasonable response.
A particle of mass M decays from rest into two particles. One particle has mass m and the other particle is massless. The momentum of the massless particle is (A) (M^2 - m^2)c/(4M) (B) (M^2 - m^2)c/(2M) (C) (M^2 - m^2)c/M (D) 2(M^2 - m^2)c/M (E) 4(M^2 - m^2)c/M
The energy is split and halved in the experiment, so we get choice (B).
Consider the closed cylindrical Gaussian surface above. Suppose that the net charge enclosed within this surface is +1 x 10^=9 C and the electric flux out through the portion of the surface marked A is -100 N*M^2/C. The flux through the rest of the surface is most nearly given by which of the following? (A) -100 N*M^2/C (B) 0 N*m^2/C (C) 10 N*m^2/C (D) 100 N*m^2/C (E) 200 N*m^2/C
The flux of the surface given as -100 N*m^2/C but eminating on the other sides gives us 200 N*m^2/C to balance the surface. Choice (E).
An infinitely long, straight wire carrying current I1 passes through the center of a circular loop of wire carrying current I2, as shown above. The long wire is perpendicular to the plane of the loop. Which of the following describes the magnetic force on the loop? (A) Outward, along a radius of the loop. (B) Inward, along a radius of the loop (C) Upward, along the axis of the loop. (D) Downward, along the axis of the loop. (E) There is no magnetic force on the loop.
The force from the straight wire and the circular wire cancel out each other.
A quantum mechanical oscillator has an angular frequency w. The Schrodinger equation predicts that the ground state energy of the oscillator will be (A) (-1/2)hw (B) 0 (C) (1/2)hw (D) hw (E) (3/2)hw
The general energy of the quantum mechanical oscillator is (1/2 + n)hw with n = 0 (the value of n at the ground state.)
Consider 1 mole of a real gas that obeys the van der Waals equation of state shown above. If the gas undergoes an isothermal expansion at temperature T0 from volume V1 to volume V2, which of the following gives the work done by the gas?
The integration falls into place and hence we get RT,0 ln[(V2 - b)/(V1 - b)] + a(1/V2 - 1/V1). Choice (D).
In the diamond structure of elemental carbon, the nearest neighbors of each C atom lie at the corners of a (A) square (B) hexagon (C) cube (D) tetrahedron (E) octahedron
The nearest sides of a carbon atom lie on a tetrahedron. Choice (D)
A spin-1/2 particle is in a state described by the spinor X = A(1 + i / 2) where A is a normalization constant. The probability of finding the particle with spin projection S,z = (-1/2)h is (A) 1/6 (B) 1/3 (C) 1/2 (D) 2/3 (E) 1
The negative spin state gives us the bottom entry of the vector with entry over magnitude all squared. So (2^2)/(sqrt(6))^2 = 2/3. Choice (D).
A particle can occupy two possible states with energies E1 and E2, where E2 > E1. At temperature T, the probability of finding the particle in state 2 is given by what?
The partition function for state 2 overlays the sum in the denominator, which gives us e^(-E2/kT)/[(e^(-E1/kT) + e^(-E2/kT)]. Choice (C)
Astronomers observe two separate solar systems, each consisting of a planet orbiting a sun. The two orbits are circular and have the same radius R. It is determined that the planets have angular momenta of the same magnitude L about their suns, and that the orbital periods are in the ratio of three to one, i.e., T1 = 3T2. The ratio m1/m2 of the masses of the two planets is (A) 1 (B) sqrt(3) (C) 2 (D) 3 (E) 9
The period is inversely proportional to the mass of the planets, so we get choice (D).
A mass, m, is attached to a massless spring fixed at one end. The mass is confined to move in a horizontal plane, and its position is given by the polar coordinates r and theta. Both r and theta can vary. If the relaxed length of the spring is s and the force constant is k, what is the Lagrangian, L, for the system?
The polar coordinates do not matter when it comes to this Lagrangian, so we get L = (1/2)m(dr)^2 + (1/2)m(r^2)(dtheta)^2 - (1/2)k(r - s)^2.
Two thin, concentric, spherical conducting shells are arranged as shown in the figure above. The inner shell has radius a, charge +Q, and is at zero electric potential. The outer shell has radius b and charge -Q. If r is the radial distance from the center of the spheres, what is the electric potential in region I (a <r < b) and in region II (r > b)?
The region outside the sphere is ideally constant and inside we have a zeroing electric field between sphere A and B, which gives us region I with Q/(4*pi*e)*(1/r - 1/a) and region II has Q/(4*pi*e)*(1/b - 1/a). Choice (D).
The figure above represents a log-log plot of variable y versus variable x. The origin represents the point x = 1 and y = 1. Which of the following gives the approximate functional relationship between y and x? (A) y = 6*sqrt(x) (B) y = (1/2)x + 6 (C) y = 6x + 0.5 (D) y = (1/6)x^2 (E) y = 6*x^2
The rise (change in y) changes slower than the run (change in x) on a logarithmic scale, which causes the variable y to decrease slower than the variable x, which should indicate a negative concavity, which indicates choice (A) is the winner.
A simple pendulum of length l is suspended from the ceiling of an elevator that is accelerating upward with constant acceleration a. For small oscillations, the period, T, of the pendulum is?
The rising and accelerating elevator decreases the period of the pendulum, which causes the acceleration of the elevator and gravity to add together. So we have T = 2*pi* sqrt[l/(g + a)].
Two nonrelativistic electrons move in circles under the influence of a uniform magnetic field B, as shown in the figure above. The ratio r1/r2 of the orbital radii to 1/3. Which of the following is equal to the ratio v1/v2 of the speeds? (A) 1/9 (B) 1/3 (C) 1 (D) 3 (E) 9
The speeds are directly proportional to the radii of the circles, so we get choice (B).
If the five lenses shown below are made of the same material, which lens has the shortest positive focal length?
The thicker the lenses, there should be less curvature and the smaller the radius of curvature, then the smaller the focal length should be as in choice (E).
Unpolarized light is incident on a pair of ideal linear polarizers whose transmission axes make an angle of 45 degrees with each other. The transmitted light intensity through both polarizers is what percentage of the incident intensity? (A) 100% (B) 75% (C) 50% (D) 25% (E) 0%
The transmitted light is (1 - cos(angle))% = (1 - cos(45)) = 25%. The transmitted light intensity through both polarizers is 25% of the incident intensity. Choice (D).
At the present time, the temperature of the universe (i.e., the microwave radiation background) is about 3 K. When the temperature was 12 K, typical objects in the universe, such as galaxies, were (A) one-quarter as distant as they are today (B) one-half as distant as they are today (C) separated by about the same distances as they are today (D) two times as distant as they are today (E) four times as distant as they are today
The universe was much more condensed and much hotter billions of years ago at the genesis of creation. (3 K)/(12 K) = 1/4, which gives us the galaxies were one-quarter as distant as they are today. Choice (A).
A bar magnet shown in the figure above is moved completely through the loop. Which of the following is a true statement about the direction of the current flow between the two points a and b in the circuit? (A) No current flows between a and b as the magnet passes through the loop. (B) Current flows from a to b as the magnet passes through the loop. (C) Current flows from b to a as the magnet passes through the loop. (D) Current flows from a to b as the magnet enteres the loop and from b to a as the magnet leaves the loop. (E) Current flows from b to a as the magnet enters the loop and from a to b as the magnet leaves the loop.
The vectors in the orthogonal basis of the force, magnetic field, and current indicated by the right-hand rule from b to a as the magnet goes into the loop and vice-versa. Choice (E).
A layer of oil with density 800 kg/m^3 floats on top of a volume of water with density 1,000 kg/m^3. A block floats at the oil-water interfaces with 1/4 of its volume in oil and 3/4 of its volume in water, as shown in the figure above. What is the density of the block? (A) 200 kg/m^3 (B) 50 kg/m^3 (C) 950 kg/m^3 (D) 1,050 kg/m^3 (E) 1,800 kg/m^3
The weighted average is (800 kg/m^3)(1/4) + (1,000 kg/m^3)(3/4) = 950 kg/m^3, so the density of the block is given by the weighted average of the amount of the block submerged at the interface. Choice (C)
A small plane can fly at a speed of 200 k/h in still air. A 30 km/h wind is blowing form west t east. How much time is required for the plane to fly 500 km due north? (A) 50/23 h (B) 50/sqrt(409) h (C) 50/20 h (D) 50/sqrt(391) h (E) 50/17 h
The wind pushes the plane more forward and gives it more thrust. Taking Pythagorean analogues, we get sqrt(20^2 - 3^2) = sqrt(391), which is the inserted into choice (D).
Consider two very long, straight, insulated wires oriented at right angles. The wires carry currents of equal magnitude I In the directions shown in the figure above. What is the net magnetic field at point P?
The wires cross perpendicularly and so their respective magnetic fields go in the + and - directions which cancel each other out. Choice (E)
A spring of force constant k is stretched a certain distance. It takes twice as much work to stretch a second spring by half this distance. The force constant of the second spring is (A) k (B) 2k (C) 4k (D) 8k (E) 16k
The work on a spring is given by W = (1/2)kx^2, so 2W = (1/2)k'(x/2)^2 ==> k' = 8k. Choice (D)
In the AC circuit above V,i is the amplitude of the input voltage and V,a is the amplitude of the output voltage. If the angular frequency w of the input voltage is varied, which of the following gives the ratio V,o/V,i = G as a function of w?
There is "dissipation" going on here and as the angular frequency is varied, we should get exponentially decreasing voltage. Choice (D).
The figure above represents the orbit of a planet around a star, S, and the marks divide the orbit into 14 equal time intervals, t = T/14, where T is the orbital period. If the only force acting on the planet is Newtonian gravitation, then true statement about the situation include which of the following? I. Area A = area B II. The star S is at one focus of an elliptically shaped orbit. III> T^2 = Ca^3, where a is the semimajor axis of the ellipse and C is a constant. (A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III
This is hands-down Kepler's Three Laws of Planetary Motion. Choice (E).
A meter stick with a speed of 0.8c moves past an observer. In the observer's reference frame, how long does it take the stick to pass the observer? (A) 1.6 ns (B) 2.5 ns (C) 4.2 ns (D) 6.9 ns (E) 8.3 ns
Time dilation: y = (1 - 0.8^2)^-0.5 = 5/3. So the new time (t') is t' = (5/3)t. v = x/t => t = x/v = t' = 3x/5v = (3/5)(1 m)/(2.4E8 m/s) = 2.5E-9 s =2.5 ns. Choice (B)
Each of the figures above shows blocks of mass 2m and m acted on by an external horizontal force F. For each figure, which of the following statements about the magnitude of the the force that one block exerts on the other (F12) is correct? (Assume that the surface on which the blocks move is frictionless.) Figure 1 Figure 2 (A) F12 = F/3 F12 = F/3 (B) F12 = F/3 F12 = 2F/3 (C) F12 = 2F/3 F12 = F/3 (D) F12 = 2F/3 F12 = 2F/3 (E) F12 = F F12 = F
To push a smaller block to push a bigger block, the force required should be greater. But to push a bigger block to push a smaller block, the force required should be less. This is given by choice (C).
The circuit shown in the figure above consists of eight resistors, each with resistance R, and a battery with terminal voltage V and negligible internal resistance. What is the current flowing through the battery? (A) (1/3)V/R (B) (1/2)V/R (C) V/R (D) (3/2)V/R (E) 3V/R
Two branches of symmetry and three branches of resistors gives us a reasonable equivalent resistance of (2/3)R and so I = V/R,eq = (3/2)V/R. Choice (D).
Three long, straight wires in the xz-plane, each carrying current I, cross at the origin of coordinates, as shown in the figur e above. Let x, y, and z denote the unit vectors in the x-, y-, and z-directions, respectively. The magnetic field B as a function of x, with y = 0 and z = 0, is
We get the sin(45) for the trigonometric function and the magnetic field becomes uI/(2*pi*x)*(1 + 2*sqrt(2))y. Choice (C)
A 1 kg block attached to a spring vibrates with a frequency of 1 Hz on a frictionless horizontal table. Two springs identical to the original spring are attached in parallel to an 8 kg block placed on the same table. Which of the following gives the frequency of vibration of the 8 kg block? (A) 1/4 Hz (B) 1/2*sqrt(2) Hz (C) 1/2 Hz (D) 1 Hz (E) 2 Hz
We have inversely proportional for the frequency of vibration from the masses, so we get (8 kg/16 kg)(1 Hz) = 1/2 Hz. Choice (C).
A long, straight, hollow cylindrical wire with an inner radius R and an outer radius 2R carries a uniform current density. What graph best represents the magnitude of the magnetic field as a function of the distance from the center of the wire?
We have no magnetic field in the center axis of the wire, but above radius R we get a linearly changing B-field and then exponentially decaying field. Choice (E).
13N --> 13C + e+ + v,e. The nuclear decay above is an example of a process induced by the (A) Mossbauer effect (B) Casimir effect (C) photoelectric effect (D) weak interaction (E) strong interaction
Weak interaction, as in weakly bound together. The lack of tight structure for the nitrogen isotope causes this atom to decay. Choice (C).
The surface of the Sun has a temperature close to 6,000 K and it emits a blackbody (Planck) spectrum that reaches a maximum near 500 nm. For a body with a surface temperature close to 300 K, at what wavelength would the thermal spectrum reach a maximum? (A) 10 um (B) 100 um (C) 10 mm (D) 100 mm (E) 10 m
You love ratios GRE, don't you. (500 nm)(6,000 K/300 K) = 10 um. This is similar to a combined gas law question, but choice (A) is the winner.
The partition function Z in statistical mechanics can be written as Z = sum{e^(-E,r/kT)} where the index r ranges over all possible microstates of a system and E,r is the energy of microstate r. For a single quantum mechanical harmonic oscillator with energies E,n = (n + 1/2)hw, where n =0, 1, 2,..., the partition function Z is given by what?
Z = (e^[hw/(2kT)])/(e^(hw/kT) - 1), which is similar to the Boltzmann distribution. Choice (E).
Let J be a quantum mechanical angular momentum operator. The commutator [J,x*J,y, J,x] is equivalent to which of the following? (A) 0 (B) ihJ,z (C) ihJ,z*J,x (D) -ihJ,x*J,z (E) ihJ,x*J,y
[J,x*J,y, J,x] = J,x*J,y*J,x - J,x*J,x*J,y = J,x(J,y*J,x - J,x*J,y) = J,x[J,y, J,x] = -ihJ,x*J,y. Choice (D).
Which of the following are the eigenvalues of the Hermitian matrix (2 i/-i 2)? (A) 1, 0 (B) 1, 3 (C) 2, 2 (D) i, -i (E) 1 + i, 1 - i
det(A) = (2 - lambda)^2 - 1 = 0 ==> (2 - lambda)^2 = 1, so lambda = 1 and 3. Choice (B).
In an experimental observation of the photoelectric effect, the stopping potential was plotted versus the light frequency, as shown in the figure above. The best straight line was fitted to the experimental points. Which of the following gives the slope of the line? (The work function of the metal is phi.) (A) h/phi (B) h/e (C) e/h (D) e/phi (E) phi/e
h and e have units of J*s and C, respectively. So [h/e] = [J*s/C] = [J/C][s] = [V/Hz], which is given in the graph. Choice (B)
A rod of length L and mass M is placed along the x-axis with one end at the origin, as shown in the figure above. The rod has linear mass density lambda = (2M/L^2)x, where x is the distance from the origin. Which of the following gives the x-coordinate of the rod's center of mass? (A) (1/12)L (B) (1/4)L (C) (1/3)L (D) (1/2)L (E) (2/3)L
lambda = (2M/L^2)x (x*dx) integrated gives us: (2M/3L^2)(x^3) from x = 0 to x = L, so we have (2/3)L. Choice (E).
X-rays of wavelength lambda = 0.250 nm are incident on the face of a crystal at angle, theta, measured from the crystal surface. The smallest angle that yields an intense reflected beam is theta = 14.5 degrees. Which of the following gives the value of the interplanar spacing d? (sin(14.5) = 1/4) (A) 0.125 nm (B) 0.250 nm (C) 0.500 nm (D) 0.625 nm (E) 0.750 nm
m*lambda = d*sin(theta) => d*sin(theta)/lambda = (0.250 nm)/(1/4)/2 = 0.500 nm. The interplanar spacing d is 0.500 nm, choice (C).
What is he canonical equation(s) of motion? (H is the Hamiltonian, q,iare the generalized coordinates, and p,i are the generalized momenta.)
q,i* = dH/dp,i and p,i* = -dH/dq,i.
Consider the Pauli spin matrices sigma,x, sigma,y, and sigma,z and the identity matrix I given above. The commutator [sigma;x, sigma;y] = s,x*s,y - s,y*s,x is equal to which of the following? (A) I (B) 2is,x (C) 2is,y (D) 2is,z (E) 0
s,x*s,y - s,y*s,x = (i 0/0 -i) - (-i 0/0 i) = 2(i 0/0 -i) = 2is,z. Choice (D).
A small particle of mass m is at rest on a horizontal circular platform that is free to rotate about a vertical axis through its center. The particle is located at a radius r from the axis, as shown in the figure above. The platform beings to rotate with constant angular acceleration a. Because of friction between the particle and the platform, the particle remains at rest with respect to the platform. When the platform has reached angular speed w, the angle theta between the static frictional force f,s and the inward radial direction is what?
theta = tan^(-1)(w^2/a). Choice (D).
Sound waves moving at 350 m/s diffract out of a speaker enclosure with an opening that is a long rectangular slit 0.14 m across. At about what frequency will the sound first disappear at an angle of 45 degrees from the normal to the speaker face? (A) 500 Hz (B) 1,750 Hz (C) 2,750 Hz (D) 3,500 Hz (E) 5,000 Hz
v = f*lambda ==> f = v/lambda = (350 m/s)/(0.14 m) = 2,500 Hz. (2,500 Hz)/cos(45) = (2,500 Hz)sqrt(2) = 3,500 Hz. Choice (D)
An object is thrown horizontally from the open window of a building. If the initial speed of the object is 20 m/s and it hits the ground 2.0 s later, from what height was it thrown? (Neglect air resistance and assume the ground is level.) (A) 4.9 m (B) 9.8 m (C) 10.0 m (D) 19.6 m (E) 39.2 m
x = (1/2)gt^2 (1/2)(9.8 m/s^2)(2.0 s)^2 = 19.6 m. Choice (D)
An observer O at rest midway between two courses of light at x = 0 and x = 10 m observes the two sources to flash simultaneously. According to a second observer O', moving at a constant speed parallel to the x-axis, one source of light flashes 13 ns before the other. Which of the following gives the speed of O' relative to O? (A) 0.13c (B) 0.15c (C) 0.36c (D) 0.53c (E) 0.62c
y = (1 - 0.36^2)^(-0.5) = 1.072. t' = (1.072)(13 ns) = 14 ns. (5 m)/(1.4E-8 s) ~= 3E8 m/s. Choice (C).
