shayekh/physics_gre huggingface dataset:
A dielectric of dielectric constant $K$ is placed in contact with a conductor having surface charge density $\sigma$, as shown above. What is the polarization (bound) charge density $\sigma_{p}$ on the surface of the dielectric at the interface between the two materials?
A. $\sigma \frac{K}{1-K}$
B. $\sigma \frac{K}{1+K}$
C. $\sigma K$
D. $\sigma \frac{1+K}{K}$
E. $\sigma \frac{1-K}{K}$


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
The mean kinetic energy of electrons in metals at room temperature is usually many times the thermal energy $k T$. Which of the following can best be used to explain this fact?

A. The energy-time uncertainty relation

B. The Pauli exclusion principle

C. The degeneracy of the energy levels

D. The Born approximation

E. The wave-particle duality


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A: 0
B: 1
C: 0
D: 0
E: 0
If $\psi$ is a normalized solution of the Schrödinger equation and $Q$ is the operator corresponding to a physical observable $x$, the quantity $\psi * Q \psi$ may be integrated in order to obtain the

A. normalization constant for $\psi$

B. spatial overlap of $Q$ with $\psi$

C. mean value of $x$

D. uncertainty in $x$

E. time derivative of $x$


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B: 0
C: 1
D: 0
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Which of the following is an eigenfunction of the linear momentum operator $-i \hbar \frac{\partial}{\partial x}$ with a positive eigenvalue $\hbar k$; i.e., an eigenfunction that describes a particle that is moving in free space in the direction of positive $x$ with a precise value of linear momentum?
A. $\cos k x$
B. $\sin k x$
C. $\mathrm{e}^{-i k x}$
D. $\mathrm{e}^{i k x}$
E. $\mathrm{e}^{-k x}$


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A: 0
B: 0
C: 0
D: 1
E: 0
In an ordinary hologram, coherent monochromatic light produces a 3 -dimensional picture because wave information is recorded for which of the following?

1. Amplitude

II. Phase

III. Wave-front angular frequency

A. I only

B. I and II only

C. 1 and III only

D. II and III only

E. I, II, and III


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A: 0
B: 1
C: 0
D: 0
E: 0
The dispersion law for a certain type of wave motion is $\omega=\left(c^{2} k^{2}+m^{2}\right)^{\frac{1}{2}}$, where $\omega$ is the angular frequency, $k$ is the magnitude of the propagation vector, and $c$ and $m$ are constants. The group velocity of these waves approaches

A. infinity as $k \rightarrow 0$ and zero as $k \rightarrow \infty$

B. infinity as $k \rightarrow 0$ and $c$ as $k \rightarrow \infty$

C. $c$ as $k \rightarrow 0$ and zero as $k \rightarrow \infty$

D. zero as $k \rightarrow 0$ and infinity as $k \rightarrow \infty$

E. zero as $k \rightarrow 0$ and $c$ as $k \rightarrow \infty$


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A: 0
B: 0
C: 0
D: 0
E: 1
A particle of mass $m$ that moves along the $x$-axis has potential energy $V(x)=a+b x^{2}$, where $a$ and $b$ are positive constants. Its initial velocity is $v_{0}$ at $x=0$. It will execute simple harmonic motion with a frequency determined by the value of

A. $b$ alone

B. $b$ and $a$ alone

C. $b$ and $m$ alone

D. $b, a$, and $m$ alone

E. $b, a, m$, and $v_{0}$



Answer scores (1 = correct):
A: 0
B: 0
C: 1
D: 0
E: 0
The equation of motion of a rocket in free space can be written

$$
m rac{d v}{d t}+u rac{d m}{d t}=0
$$

where $m$ is the rocket's mass, $v$ is its velocity, $t$ is time, and $u$ is a constant.

The constant $u$ represents the speed of the

A. rocket at $t=0$

B. rocket after its fuel is spent

C. rocket in its instantaneous rest frame

D. rocket's exhaust in a stationary frame

E. rocket's exhaust relative to the rocket


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The equation of motion of a rocket in free space can be written

$$
m rac{d v}{d t}+u rac{d m}{d t}=0
$$

where $m$ is the rocket's mass, $v$ is its velocity, $t$ is time, and $u$ is a constant.

The equation can be solved to give $v$ as a function of $m$. If the rocket has $m=m_{0}$ and $v=0$ when it starts, what is the solution?

A. $u m_{0} / m$

B. $u \exp \left(m_{0} / m\right)$

C. $u \sin \left(m_{0} / m\right)$

D. $u \tan \left(m_{0} / m\right)$

E. None of the above.


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A point charge $-q$ coulombs is placed at a distance $d$ from a large grounded conducting plane. The surface charge density on the plane a distance $D$ from the point charge is

A. $\frac{q}{4 \pi D}$

B. $\frac{q D^{2}}{2 \pi}$

C. $\frac{q d}{2 \pi D^{2}}$

D. $\frac{g d}{2 \pi D^{3}}$

E. $\frac{q d}{4 \pi \epsilon_{0} D^{2}}$


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B: 0
C: 0
D: 1
E: 0
An alternating current electrical generator has a fixed internal impedance $\boldsymbol{R}_{\boldsymbol{g}}+j X_{g}$ and is used to supply power to a passive load that has an impedance $\boldsymbol{R}_{g}+j X_{\ell}$, where $j=\sqrt{-1}$, $R_{g} \neq 0$, and $X_{g} \neq 0$. For maximum power transfer between the generator and the load, $X_{\ell}$ should be equal to
A. 0
B. $X_{g}$
C. $-X_{g}$
D. $\boldsymbol{R}_{\boldsymbol{g}}$
E. $-\boldsymbol{R}_{\boldsymbol{g}}$


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B: 0
C: 1
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A current $i$ in a circular loop of radius $b$ produces a magnetic field. At a fixed point far from the loop, the strength of the magnetic field is proportional to which of the following combinations of $i$ and $b$ ?
A. $i b$
B. $i b^{2}$
C. $i^{2} b$
D. $*-\frac{i}{b}$
E. $\frac{i}{b^{2}}$


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A: 0
B: 1
C: 0
D: 0
E: 0
For a system in which the number of particles is fixed, the reciprocal of the Kelvin temperature $T$ is given by which of the following derivatives? (Let $P=$ pressure, $V=$ volume, $S=$ entropy, and $U=$ internal energy.)

A. $\left(\frac{\partial P}{\partial V}\right)_{S}$

B. $\left(\frac{\partial P}{\partial S}\right)_{V}$

C. $\left(\frac{\partial S}{\partial P}\right)_{U}$

D. $\left(\frac{\partial V}{\partial P}\right)_{U}$

E. $\left(\frac{\partial S}{\partial U}\right)_{V}$


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A large isolated system of $N$ weakly interacting particles is in thermal equilibrium. Each particle has only 3 possible nondegenerate states of energies $0, \epsilon$, and $3 \epsilon$. When the system is at an absolute temperature $T>\epsilon / k$, where $k$ is Boltzmann's constant, the average energy of each particle is
A. 0
B. $\epsilon$
C. $\frac{4}{3} \epsilon$
D. $2 \epsilon$
E. $3 \epsilon$


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A: 0
B: 0
C: 1
D: 0
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If a newly discovered particle $X$ moves with a speed equal to the speed of light in vacuum, then which of the following must be true?

A. The rest mass of $X$ is zero.

B. The spin of $X$ equals the spin of a photon.

C. The charge of $X$ is carried on its surface.

D. $X$ does not spin.

E. $X$ cannot be detected.



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C: 0
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A car of rest length 5 meters passes through a garage of rest length 4 meters. Due to the relativistic Lorentz contraction, the car is only 3 meters long in the garage's rest frame. There are doors on both ends of the garage, which open automatically when the front of the car reaches them and close automatically when the rear passes them. The opening or closing of each door requires a negligible amount of time.

The velocity of the car in the garage's rest frame is

A. $0.4 c$

B. $0.6 c$

C. $0.8 c$

D. greater than $c$

E. not determinable from the data given


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A: 0
B: 0
C: 1
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A car of rest length 5 meters passes through a garage of rest length 4 meters. Due to the relativistic Lorentz contraction, the car is only 3 meters long in the garage's rest frame. There are doors on both ends of the garage, which open automatically when the front of the car reaches them and close automatically when the rear passes them. The opening or closing of each door requires a negligible amount of time.

The length of the garage in the car's rest frame is

A. $2.4 \mathrm{~m}$

B. $4.0 \mathrm{~m}$

C. $5.0 \mathrm{~m}$

D. $8.3 \mathrm{~m}$

E. not determinable from the data given


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A: 1
B: 0
C: 0
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A car of rest length 5 meters passes through a garage of rest length 4 meters. Due to the relativistic Lorentz contraction, the car is only 3 meters long in the garage's rest frame. There are doors on both ends of the garage, which open automatically when the front of the car reaches them and close automatically when the rear passes them. The opening or closing of each door requires a negligible amount of time.

Which of the following statements is the best response to the question:

"Was the car ever inside a closed garage?"

A. No, because the car is longer than the garage in all reference frames.

B. No, because the Lorentz contraction is not a "real" effect.

C. Yes, because the car is shorter than the garage in all reference frames.

D. Yes, because the answer to the question in the garage's rest frame must apply in all reference frames.

E. There is no unique answer to the question, as the order of door openings and closings depends on the reference frame.

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The measured index of refraction of $x$-rays in rock salt is less than one. This is consistent with the theory of relativity because

A. relativity deals with light waves traveling in a vacuum only

B. $\mathrm{x}$-rays cannot transmit signals

C. $\mathrm{x}$-ray photons have imaginary mass

D. the theory of relativity predates the development of solid-state physics

E. the phase velocity and group velocity are different


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A: 0
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C: 0
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E: 1
It is necessary to coat a glass lens with a nonreflecting layer. If the wavelength of the light in the coating is $\lambda$, the best choice is a layer of material having an index of refraction between those of glass and air and a thickness of
A. $\frac{\lambda}{4}$
B. $\frac{\lambda}{2}$
C. $\frac{\lambda}{\sqrt{2}}$
D. $\lambda$
E. $1.5 \lambda$


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A: 1
B: 0
C: 0
D: 0
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Unpolarized light is incident on two ideal polarizers in series. The polarizers are oriented so that no light emerges through the second polarizer. A third polarizer is now inserted between the first two and its orientation direction is continuously rotated through $180^{\circ}$. The maximum fraction of the incident power transmitted through all three polarizers is
A. zero
B. $\frac{1}{8}$
C. $\frac{1}{2}$
D. $\frac{1}{\sqrt{2}}$
E. 1


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A: 0
B: 1
C: 0
D: 0
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The period of a hypothetical Earth satellite orbiting at sea level would be 80 minutes. In terms of the Earth's radius $\boldsymbol{R}_{e}$, the radius of a synchronous satellite orbit (period 24 hours) is most nearly

A. $3 R$

B.

C. $18 R$

D. $320 R_{e}$

E. $5800 R_{e}$



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B: 1
C: 0
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A hoop of mass $M$ and radius $R$ is at rest at the top of an inclined plane as shown above. The hoop rolls down the plane without slipping. When the hoop reaches the bottom, its angular momentum around its center of mass is

A. $M R \sqrt{g h}$

B. $\frac{1}{2} M R \sqrt{g h}$

C. $M \sqrt{2 g h}$

D. $M g h$

E. $\frac{1}{2} M g h$


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A: 1
B: 0
C: 0
D: 0
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A particle is constrained to move along the $x$-axis under the influence of the net force $\mathbf{F}=-k \mathbf{x}$ with amplitude $A$ and frequency $f$, where $k$ is a positive constant. When $x=A / 2$, the particle's speed is
A. $2 \pi f A$
B. $\sqrt{3} \pi f A$
C. $\sqrt{2} \pi f A$
D. $\pi f A$
E. $\frac{1}{3} \pi f A$


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A: 0
B: 1
C: 0
D: 0
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A system consists of two charged particles of equal mass. Initially the particles are far apart, have zero potential energy, and one particle has nonzero speed. If radiation is neglected, which of the following is true of the total energy of the system?

A. It is zero and remains zero.

B. It is negative and constant.

C. It is positive and constant.

D. It is constant, but the sign cannot be determined unless the initial velocities of both particles are known.

E. It cannot be a constant of the motion because the particles exert force on each other.


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One of Maxwell's equations is $\nabla \cdot \mathbf{B}=\mathbf{0}$. Which of the following sketches shows magnetic field lines that clearly violate this equation within the region bounded by the dashed lines?

A.



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Which of the following electric fields could exist in a finite region of space that contains no charges? (In these expressions, $A$ is a constant, and $\mathbf{i}, \mathbf{j}$, and $\mathbf{k}$ are unit vectors pointing in the $x, y$, and $z$ directions, respectively.)

A. $A(2 x y i-x z k)$

B. $A(-x y \mathbf{j}+x z \mathbf{k})$

C. $A(x z \mathbf{i}+x z \mathbf{j})$

D. $\operatorname{Axyz}(\mathbf{i}+\mathbf{j})$

E. $A x y z 1$



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A small circular wire loop of radius $a$ is located at the center of much larger circular wire loop of radius $b$ as shown above. The larger loop carries an alternating current $I=I_{0} \cos \omega t$, where $I_{0}$ and $\omega$ are constants. The magnetic field generated by the current in the large loop induces in the small loop an emf that is approximately equal to which of the following? (Either use mks units and let $\mu_{0}$ be the permeability of free space, or use Gaussian units and let $\mu_{0}$ be $4 \pi / c^{2}$.)

A. $\left(\frac{\pi \mu_{0} I_{0}}{2}\right) \frac{a^{2}}{b} \omega \cos \omega t$

B. $\left(\frac{\pi \mu_{0} I_{0}}{2}\right) \frac{a^{2}}{b} \omega \sin \omega t$

C. $\left(\frac{\pi \mu_{0} I_{0}}{2}\right) \frac{a}{b^{2}} \omega \sin \omega t$

D. $\left(\frac{\pi \mu_{0} I_{0}}{2}\right) \frac{a}{b^{2}} \cos \omega t$

E. $\left(\frac{\pi \mu_{0} I_{0}}{2}\right) \frac{a}{b} \sin \omega t$


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The emission spectrum of an atomic gas in a magnetic field differs from that of the gas in the absence of a magnetic field. Which of the following is true of the phenomenon?

A. It is called the Stern-Gerlach effect.

B. It is called the Stark effect.

C. It is due primarily to the nuclear magnetic moment of the atoms.

D. The number of emission lines observed for the gas in a magnetic field is always twice the number observed in the absence of a magnetic field.

E. The number of emission lines observed for the gas in a magnetic field is either greater than or equal to the number observed in the absence of a magnetic field.

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A spectral line is produced by a gas that is sufficiently dense that the mean time between atomic collisions is much shorter than the mean lives of the atomic states responsible for the line. Compared with the same line produced by a low-density gas, the line produced by the higher-density gas will appear

A. the same

B. more highly polarized

C. broader

D. shifted toward the blue end of the spectrum

E. split into a doublet


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C: 1
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Sodium has eleven electrons and the sequence in which energy levels fill in atoms is $1 s, 2 s, 2 p$, $3 s, 3 p, 4 s, 3 d$, etc. What is the ground state of sodium in the usual notation ${ }^{2 S+1} L_{J}$ ?
A. ${ }^{1} S_{0}$
B. ${ }^{2} S_{\frac{1}{2}}$
C. ${ }^{1} P_{0}$
D. ${ }^{2} P_{\frac{1}{2}}$
E. ${ }^{3} P_{\frac{1}{2}}$



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A: 0
B: 1
C: 0
D: 0
E: 0
The figure above shows the photon interaction cross sections for lead in the energy range where the Compton, photoelectric, and pair production processes all play a role. What is the correct identification of these cross sections?

A. 1 = photoelectric, 2 = Compton, 3 = pair production

B. $1=$ photoelectric, $2=$ pair production, $3=$ Compton

C. 1 = Compton, $2=$ pair production, $3=$ photoelectric

D. 1 = Compton, $2=$ photoelectric, $3=$ pair production

E. 1 = pair production, $2=$ photoelectric, $3=$ Compton


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A: 0
B: 1
C: 0
D: 0
E: 0
The exponent in Coulomb's inverse square law has been found to differ from two by less than one part in a billion by measuring which of the following?

A. The charge on an oil drop in the Millikan experiment

B. The deflection of an electron beam in an electric field

C. The neutrality of charge of an atom

D. The electric force between two charged objects

E. The electric field inside a charged conducting shell


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B: 0
C: 0
D: 0
E: 1
In a gas of $N$ diatomic molecules, two possible models for a classical description of a diatomic molecule are: [FIGURE]. Which of the following statements about this gas is true? 

A. Model I has a specific heat $c_{v}=rac{3}{2} N k$. 

B. Model II has a smaller specific heat than Model I. 

C. Model I is always correct. 

D. Model II is always correct. 

E. The choice between Models I and II dependes on the temperature.

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Consider a system of $N$ noninteracting particles confined in a volume $V$ at a temperature such that the particles obey classical Boltzmann statistics. If the temperature is lowered to the point at which quantum effects become important, the pressure of the gas may differ depending on whether the particles are fermions or bosons. Let $P_{F}$ be the pressure exerted by the particles if they are fermions, $P_{B}$ be the pressure if they are bosons, and $P_{C}$ be the pressure the particles would exert if quantum effects are ignored. Which of the following is true?

A. $P_{F}=P_{B}=P_{C}$

B. $P_{F}>P_{C}>P_{B}$

C. $P_{F}>P_{B}>P_{C}$

D. $P_{F}<P_{B}<P_{C}$

E. $P_{F}<P_{C}<P_{B}$

Answer scores (1 = correct):
A: 0
B: 1
C: 0
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A system containing two identical particles is described by a wave function of the form

$\psi=\frac{1}{\sqrt{2}}\left[\psi_{\alpha}\left(x_{1}\right) \psi_{\beta}\left(x_{2}\right)+\psi_{\beta}\left(x_{1}\right) \psi_{\alpha}\left(x_{2}\right)\right]$

where $x_{1}$ and $x_{2}$ represent the spatial coordinates of the particles and $\alpha$ and $\beta$ represent all the quantum numbers, including spin, of the states that they occupy. The particles might be

A. electrons

B. positrons

C. protons

D. neutrons

E. deuterons



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The figure above shows one of the possible energy eigenfunctions $\psi(x)$ for a particle bouncing freely back and forth along the $x$-axis between impenetrable walls located at $x=-a$ and $x=+a$. The potential energy equals zero for $|x|<a$. If the energy of the particle is 2 electron volts when it is in the quantum state associated with this eigenfunction, what is its energy when it is in the quantum state of lowest possible energy?
A. $0 \mathrm{eV}$
B. $\frac{1}{\sqrt{2}} \mathrm{eV}$
C. $\frac{1}{2} \mathrm{eV}$
D. $1 \mathrm{eV}$
E. $2 \mathrm{eV}$



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C: 1
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When a narrow beam of monoenergetic electrons impinges on the surface of a single metal crystal at an angle of 30 degrees with the plane of the surface, first-order reflection is observed. If the spacing of the reflecting crystal planes is known from $x$-ray measurements to be 3 angstroms, the speed of the electrons is most nearly

A. $1.4 \times 10^{-4} \mathrm{~m} / \mathrm{s}$

B. $2.4 \mathrm{~m} / \mathrm{s}$

C. $5.0 \times 10^{3} \mathrm{~m} / \mathrm{s}$

D. $2.4 \times 10^{6} \mathrm{~m} / \mathrm{s}$

E. $4.5 \times 10^{9} \mathrm{~m} / \mathrm{s}$


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Which of the following is NOT compatible with the selection rule that controls electric dipole emission of photons by excited states of atoms?

A. $\Delta n$ may have any negative integral value.

B. $\Delta l= \pm 1$

C. $\Delta m_{\ell}=0, \pm 1$

D. $\Delta s= \pm 1$

E. $\Delta j= \pm 1$



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An electric sander has a continuous belt that rubs against a wood surface as shown schematically above. The sander is 100 percent efficient and draws a current of 9 amperes from a 120 -volt line. The belt speed is 10 meters per second. If the sander is pushing against the wood with a normal force of 100 newtons, the coefficient of friction is most nearly
A. 0.02
B. 0.2
C. 0.4
D. 1.1
E. 10



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In the circuit shown above, $R_{2}=3 R_{1}$ and the battery of emf $\varepsilon$ has negligible internal resistance. The resistance of the diode when it allows current to pass through it is also negligible. At time $t=0$, the switch $S$ is closed and the currents and voltages are allowed to reach their asymptotic values. Then at time $t_{1}$, the switch is opened. Which of the following curves most nearly represents the potential at point $A$ as a function of time $t$ ?

A.



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In the cycle shown above, $K L$ and $N M$ represent isotherms, while $K N$ and $L M$ represent reversible adiabats. A system is carried through the Carnot cycle $K L M N$, taking in heat $Q_{2}$ from the hot reservoir $T_{2}$ and releasing heat $Q_{1}$ to the cold reservoir $T_{1}$. All of the following statements are true EXCEPT:

A. $Q_{1} / T_{1}=Q_{2} / T_{2}$

B. The entropy of the hot reservoir decreases.

C. The entropy of the system increases.

D. The work $W$ done is equal to the net heat absorbed, $Q_{2}-Q_{1}$.

E. The efficiency of the cycle is independent of the working substance.

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A particle of mass $M$ is in an infinitely deep square well potential $V$ where

$$
\begin{array}{ll}
V=0 & \text { for }-a \leqq x \leqq a, \text { and } \\
V=\infty & \text { for } x<-a, a<x .
\end{array}
$$

A very small perturbing potential $V^{\prime}$ is superimposed on $V$ such that

$$
\begin{array}{ll}
V^{\prime}=\epsilon\left(\frac{a}{2}-|x|\right) & \text { for } \frac{-a}{2} \leqq x \leqq \frac{a}{2}, \text { and } \\
V^{\prime}=0 & \text { for } x<\frac{-a}{2}, \frac{a}{2}<x .
\end{array}
$$

If $\psi_{0}, \psi_{1}, \psi_{2}, \psi_{3}, \ldots$ are the energy eigenfunctions for a particle in the infinitely deep square well potential, with $\psi_{0}$ being the ground state, which of the following statements is correct about the eigenfunction $\psi_{0}^{\prime}$ of a particle in the perturbed potential $V+V^{\prime}$ ?

A. $\psi_{0}^{\prime}=a_{00} \psi_{0}, a_{00} \neq 0$

B. $\psi_{0}^{\prime}=\sum_{n=0}^{\infty} a_{0 n} \psi_{n}$ with $a_{0 n}=0$ for all odd values of $n$

C. $\psi_{0}^{\prime}=\sum_{n=0}^{\infty} a_{0 n} \psi_{n}$ with $a_{0 n}=0$ for all even values of $n$

D. $\psi_{0}^{\prime}=\sum_{n=0}^{\infty} a_{0 n} \psi_{n}$ with $a_{0 n} \neq 0$ for all values of $n$

E. None of the above



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Two uniform cylindrical disks of identical mass $M$, radius $R$, and moment of inertia $\frac{1}{2} M R^{2}$, as shown above, collide on a frictionless, horizontal surface. Disk 1 , having an initial counterclockwise angular velocity $\omega_{0}$ and a center-of-mass velocity $v_{0}=\frac{1}{2} \omega_{0} R$ to the right, makes a grazing collision with disk II initially at rest. If after the collision the two disks stick together, the magnitude of the total angular momentum about the point $P$ is

A. zero

B. $\frac{1}{2} M R^{2} \omega_{0}$

C. $\frac{1}{2} M R v_{0}$

D. $M R v_{0}$

E. dependent on the time of the collision



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A: 1
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The long thin cylindrical glass rod shown above has length $\ell$ and is insulated from its surroundings. The rod has an excess charge $Q$ uniformly distributed along its length. Assume the electric potential to be zero at infinite distances from the rod. If $k$ is the constant in Coulomb's law, the electric potential at a point $P$ along the axis of the rod and a distance $\ell$ from one end is $\frac{k Q}{\ell}$
multiplied by

A. $\frac{4}{9}$

B. $\frac{1}{2}$

C. $\frac{2}{3}$

D. $\ln 2$

E. 1

Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
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