shayekh/physics_gre huggingface dataset:
The positronium "atom" consists of an electron and a positron bound together by their mutual Coulomb attraction and moving about their center of mass, which is located halfway between them. Thus the positronium "atom" is somewhat analogous to a hydrogen atom. The ground-state binding energy of hydrogen is 13.6 electron volts. What is the ground-state binding energy of positronium?

A. $\left(\frac{1}{2}\right)^{2} \times 13.6 \mathrm{eV}$

B. $\frac{1}{2} \times 13.6 \mathrm{eV}$

C. $13.6 \mathrm{eV}$

D. $2 \times 13.6 \mathrm{eV}$

E. $(2)^{2} \times 13.6 \mathrm{eV}$


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The screen of a pinhole camera is at a distance $D$ from the pinhole, which has a diameter $d$. The light has an effective wavelength $\lambda .(\lambda<<D)$ For which of the following values of $d$ will the image be sharpest?
A. $\sqrt{\lambda D}$
B. $\lambda$
C. $\frac{\lambda}{10}$
D. $\frac{\lambda^{2}}{D}$
E. $\frac{D^{2}}{\lambda}$



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

A.



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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. $0.024 \mathrm{~m}$

B. $0.048 \mathrm{~m}$

C. $0.121 \mathrm{~m}$

D. $0.242 \mathrm{~m}$

E. $0.484 \mathrm{~m}$


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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 $\sqrt{m}$

C. linear in $R$

D. proportional to $R^{3 / 2}$

E. proportional to $R^{2}$


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In a nonrelativistic, one-dimensional collision, a particle of mass $2 m$ 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. $\frac{1}{4}$

C. $\frac{1}{3}$

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

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

SCRATCHWORK


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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. $\frac{1}{2} k T$

B. $k T$

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

D. $3 k T$

E. $6 k T$


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An ideal monatomic gas expands quasi-statically to twice its volume. If the process is isothermal, the work done by the gas is $W_{i}$. If the process is adiabatic, the work done by the gas is $W_{a}$. Which of the following is true?

A. $W_{i}=W_{a}$

B. $0=W_{i}<W_{a}$

C. $0<W_{i}<W_{a}$

D. $0=W_{a}<W_{i}$

E. $0<W_{a}<W_{i}$



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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?

A.



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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. $2 Q$

B. $Q$

C. 0

D. $-Q$

E. $-2 Q$


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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? $\left(k=1 / 4 \pi \epsilon_{0}\right)$

A. 0

B. $k q / r^{2}$

C. $5 k q / r^{2}$

D. $\left(k q / r^{2}\right) \cos (2 \pi / 5)$

E. $\left(5 k q / r^{2}\right) \cos (2 \pi / 5)$


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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. $0.09 \mathrm{~J}$

B. $0.18 \mathrm{~J}$

C. $0.27 \mathrm{~J}$

D. $0.41 \mathrm{~J}$

E. $0.81 \mathrm{~J}$



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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. $5.0 \mathrm{~cm}$ to the right of the second lens

B. $13.3 \mathrm{~cm}$ to the right of the second lens

C. infinitely far to the right of the second lens

D. $13.3 \mathrm{~cm}$ to the left of the second lens

E. $100 \mathrm{~cm}$ to the left of the second lens



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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. $\mathrm{V}$

SCRATCHWORK


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Two stars are separated by an angle of $3 \times 10^{-5}$ radians. What is the diameter of the smallest telescope that can resolve the two stars using visible light ( $\lambda \cong 600$ nanometers) ? (Ignore any effects due to Earth's atmosphere.)

A. $1 \mathrm{~mm}$

B. $2.5 \mathrm{~cm}$

C. $10 \mathrm{~cm}$

D. $2.5 \mathrm{~m}$

E. $10 \mathrm{~m}$


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An 8-centimeter-diameter by 8 -centimeter-long $\mathrm{NaI}(\mathrm{Tl})$ 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. $2 \times 10^{-4}$

C. $4 \times 10^{-4}$

D. $8 \pi \times 10^{-4}$

E. $16 \pi \times 10^{-4}$

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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?

A.



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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. $80 \mathrm{~s}$

B. $160 \mathrm{~s}$

C. $2,000 \mathrm{~s}$

D. $5,000 \mathrm{~s}$

E. $6,400 \mathrm{~s}$


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The ground state electron configuration for phosphorus, which has 15 electrons, is

A. $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{1} 3 p^{4}$

B. $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{3}$

C. $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 d^{3}$

D. $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{1} 3 d^{4}$

E. $1 s^{2} 2 s^{2} 2 p^{6} 3 p^{2} 3 d^{3}$


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The energy required to remove both electrons from the helium atom in its ground state is $79.0 \mathrm{eV}$. How much energy is required to ionize helium (i.e., to remove one electron)?

A. $24.6 \mathrm{eV}$

B. $39.5 \mathrm{eV}$

C. $51.8 \mathrm{eV}$

D. $54.4 \mathrm{eV}$

E. $65.4 \mathrm{eV}$

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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


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In the production of $X$ rays, the term "bremsstrahlung" refers to which of the following?

A. The cut-off wavelength, $\lambda_{\min }$, of the $X$-ray tube

B. The discrete $\mathrm{X}$-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 $\mathrm{X}$-ray tube

E. The smooth, continuous X-ray spectra produced by rapidly decelerating electrons in the target metal of the X-ray tube


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In the hydrogen spectrum, the ratio of the wavelengths for Lyman- $\alpha$ radiation ( $n=2$ to $n=1)$ to Balmer- $\alpha$ radiation $(n=3$ to $n=2)$ is

A. $5 / 48$

B. $5 / 27$

C. $1 / 3$

D. 3

E. $27 / 5$

SCRATCHWORK


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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

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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. $0^{\circ}$

B. $30^{\circ}$

C. $45^{\circ}$

D. $60^{\circ}$

E. $90^{\circ}$

SCRATCHWORK


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A stone is thrown at an angle of $45^{\circ}$ 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 $v_{x}$ versus $t$ and $v_{y}$ versus $t$, respectively?

\begin{tabular}{|c|c|c|}
\hline & $v_{x}$ vs. $t$ & $v_{y} \mathrm{vs}$ 

A. & I & IV 

B. & II & 

C. & II & III 

D. & II & V 

E. & IV & V \\
\hline
\end{tabular}



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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. $(7 / 2) m r^{2}$

B. $(13 / 2) m r^{2}$

C. $(29 / 2) m r^{2}$

D. $(49 / 2) m r^{2}$

E. $(55 / 2) m r^{2}$



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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. $\sqrt{\frac{1}{3} g L}$

B. $\sqrt{g L}$

C. $\sqrt{3 g L}$

D. $\sqrt{12 g L}$

E. $12 \sqrt{g L}$

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The eigenvalues of a Hermitian operator are always

A. real

B. imaginary

C. degenerate

D. linear

E. positive

$$
\begin{gathered}
\left|\psi_{1}\right\rangle=5|1\rangle-3|2\rangle+2|3\rangle \\
\left|\psi_{2}\right\rangle=|1\rangle-5|2\rangle+x|3\rangle
\end{gathered}
$$


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The states $|1\rangle,|2\rangle$, and $|3\rangle$ are orthonormal. For what value of $x$ are the states $\left|\psi_{1}\right\rangle$ and $\left|\psi_{2}\right\rangle$ given above orthogonal?

A. 10

B. 5

C. 0

D. -5

E. -10


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The state $\psi=\frac{1}{\sqrt{6}} \psi_{-1}+\frac{1}{\sqrt{2}} \psi_{1}+\frac{1}{\sqrt{3}} \psi_{2}$ is a linear combination of three orthonormal eigenstates of the operator $\hat{O}$ corresponding to eigenvalues $-1,1$, and 2 . What is the expectation value of $\hat{O}$ for this state?

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

B. $\sqrt{\frac{7}{6}}$

C. 1

D. $\frac{4}{3}$

E. $\frac{(\sqrt{3}+2 \sqrt{2}-1)}{\sqrt{6}}$

SCRATCHWORK


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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. $A \mathrm{e}^{-b r}$

II. $A \sin (b r)$

III. $A / r$

A. I only

B. II only

C. I and II only

D. I and III only

E. I, II, and III


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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. $6.0 \mathrm{eV}$

B. $6.8 \mathrm{eV}$

C. $12.2 \mathrm{eV}$

D. $13.6 \mathrm{eV}$

E. $24.2 \mathrm{eV}$


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C: 0
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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. $m c / 2$

B. $m c / \sqrt{2}$

C. $m c$

D. $\sqrt{3} m c$

E. $2 m c$

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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. $0.43 \times 10^{8} \mathrm{~m} / \mathrm{s}$

B. $2.84 \times 10^{8} \mathrm{~m} / \mathrm{s}$

C. $2.90 \times 10^{8} \mathrm{~m} / \mathrm{s}$

D. $2.98 \times 10^{8} \mathrm{~m} / \mathrm{s}$

E. $3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}$


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In an inertial reference frame $S$, two events occur on the $x$-axis separated in time by $\Delta t$ and in space by $\Delta x$. In another inertial reference frame $S^{\prime}$, 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 $\Delta x$ and $\Delta t$

B. Only if $|\Delta x / \Delta t|<c$

C. Only if $|\Delta x / \Delta t|>c$

D. Only if $|\Delta x / \Delta t|=c$

E. Under no condition


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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 .

SCRATCHWORK


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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 $-\int P d V$.

D. The mechanical work done by the gas is $\int P d V$.

E. The temperature of the gas remains constant.



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A constant amount of an ideal gas undergoes the cyclic process $A B C A$ in the $P V$ diagram shown above. The path $B C$ is isothermal. The work done by the gas during one complete cycle, beginning and ending at $A$, is most nearly

A. $600 \mathrm{~kJ}$

B. $300 \mathrm{~kJ}$

C. 0

D. $-300 \mathrm{~kJ}$

E. $-600 \mathrm{~kJ}$



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An AC circuit consists of the elements shown above, with $R=10,000 \mathrm{ohms}, L=25$ millihenries, and $C$ an adjustable capacitance. The $\mathrm{AC}$ 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. $4 \mathrm{nF}$

B. $40 \mathrm{nF}$

C. $4 \mu \mathrm{F}$

D. $40 \mu \mathrm{F}$

E. $400 \mu \mathrm{F}$


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Which two of the following circuits are high-pass filters? [FIGURE] 

A. I and II 

B. I and III 

C. I and IV 

D. II and III 

E. II and IV

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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?

A. Voltage



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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. $\nabla \cdot \mathbf{E}=\rho / \epsilon_{\mathrm{o}}$

II. $\nabla \cdot \mathbf{B}=0$

III. $\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

IV. $\nabla \times \mathbf{B}=\mu_{\mathrm{o}} \mathbf{J}+\mu_{\mathrm{o}} \epsilon_{\mathrm{o}} \frac{\partial \mathbf{E}}{\partial t}$

A. I only

B. II only

C. III only

D. I and IV

E. II and III


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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 $v$. 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 $B$

SCRATCHWORK


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The components of the orbital angular momentum operator $\mathbf{L}=\left(L_{x}, L_{y}, L_{z}\right)$ satisfy the following commutation relations.

$\left[L_{x}, L_{y}\right]=i \hbar L_{z}$,

$\left[L_{y}, L_{z}\right]=i \hbar L_{x}$,

$\left[L_{z}, L_{x}\right]=i \hbar L_{y}$.

What is the value of the commutator $\left[L_{x} L_{y}, L_{z}\right]$ ?

A. $2 i \hbar L_{x} L_{y}$

B. $i \hbar\left(L_{x}^{2}+L_{y}^{2}\right)$

C. $-i \hbar\left(L_{x}^{2}+L_{y}^{2}\right)$

D. $i \hbar\left(L_{x}^{2}-L_{y}^{2}\right)$

E. $-i \hbar\left(L_{x}^{2}-L_{y}^{2}\right)$


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The energy eigenstates for a particle of mass $m$ in a box of length $L$ have wave functions

$\phi_{n}(x)=\sqrt{2 / L} \sin (n \pi x / L)$ and energies $E_{n}=n^{2} \pi^{2} \hbar^{2} / 2 m L^{2}$, where $n=1,2,3, \ldots$

At time $t=0$, the particle is in a state described as follows.

$$
\Psi(t=0)=\frac{1}{\sqrt{14}}\left[\phi_{1}+2 \phi_{2}+3 \phi_{3}\right]
$$

Which of the following is a possible result of a measurement of energy for the state $\Psi$ ?

A. $2 E_{1}$

B. $5 E_{1}$

C. $7 E_{1}$

D. $9 E_{1}$

E. $14 E_{1}$

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Let $|n\rangle$ represent the normalized $n^{\text {th }}$ energy eigenstate of the one-dimensional harmonic oscillator, $H|n\rangle=\hbar \omega\left(n+\frac{1}{2}\right)|n\rangle$. If $|\psi\rangle$ is a normalized ensemble state that can be expanded as a linear combination

$|\psi\rangle=\frac{1}{\sqrt{14}}|1\rangle-\frac{2}{\sqrt{14}}|2\rangle+\frac{3}{\sqrt{14}}|3\rangle$ of the eigenstates, what is the expectation value of the energy operator in this ensemble state?

A. $\frac{102}{14} \hbar \omega$

B. $\frac{43}{14} \hbar \omega$

C. $\frac{23}{14} \hbar \omega$

D. $\frac{17}{\sqrt{14}} \hbar \omega$

E. $\frac{7}{\sqrt{14}} \hbar \omega$


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A free particle with initial kinetic energy $E$ and de Broglie wavelength $\lambda$ enters a region in which it has potential energy $V$. What is the particle's new de Broglie wavelength?

A. $\lambda(1+E / V)$

B. $\lambda(1-V / E)$

C. $\lambda(1-E / V)^{-1}$

D. $\lambda(1+V / E)^{1 / 2}$

E. $\lambda(1-V / E)^{-1 / 2}$

SCRATCHWORK


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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. $2 n R \ln 2$

B. $n R \ln 2$

C. $\frac{1}{2} n R \ln 2$

D. $-n R \ln 2$

E. $-2 n R \ln 2$


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A gaseous mixture of $\mathrm{O}_{2}$ (molecular mass $32 \mathrm{u}$ ) and $\mathrm{N}_{2}$ (molecular mass $28 \mathrm{u}$ ) is maintained at constant temperature. What is the ratio $\frac{v_{r m s}\left(\mathrm{~N}_{2}\right)}{v_{r m s}\left(\mathrm{O}_{2}\right)}$ of the root-mean-square speeds of the molecules?

A. $\frac{7}{8}$

B. $\sqrt{\frac{7}{8}}$

C. $\sqrt{\frac{8}{7}}$

D. $\left(\frac{8}{7}\right)^{2}$

E. $\ln \left(\frac{8}{7}\right)$


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In a Maxwell-Boltzmann system with two states of energies $\epsilon$ and $2 \epsilon$, respectively, and a degeneracy of 2 for each state, the partition function is

A. $\mathrm{e}^{-\epsilon / k T}$

B. $2 \mathrm{e}^{-2 \epsilon / k T}$

C. $2 \mathrm{e}^{-3 \epsilon / k T}$

D. $\mathrm{e}^{-\epsilon / k T}+\mathrm{e}^{-2 \epsilon / k T}$

E. $2\left[\mathrm{e}^{-\epsilon / k T}+\mathrm{e}^{-2 \epsilon / k T}\right]$

Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
At $20^{\circ} \mathrm{C}$, 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 $20^{\circ} \mathrm{C}$ ?

A. $414 \mathrm{~Hz}$

B. $427 \mathrm{~Hz}$

C. $433 \mathrm{~Hz}$

D. $440 \mathrm{~Hz}$

E. $453 \mathrm{~Hz}$


Answer scores (1 = correct):
A: 0
B: 1
C: 0
D: 0
E: 0
Unpolarized light of intensity $I_{0}$ is incident on a series of three polarizing filters. The axis of the second filter is oriented at $45^{\circ}$ to that of the first filter, while the axis of the third filter is oriented at $90^{\circ}$ to that of the first filter. What is the intensity of the light transmitted through the third filter?

A. 0

B. $I_{0} / 8$

C. $I_{0} / 4$

D. $I_{0} / 2$

E. $I_{0} / \sqrt{2}$



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A: 0
B: 1
C: 0
D: 0
E: 0
The conventional unit cell of a body-centered cubic Bravais lattice is shown in the figure above. The conventional cell has volume $a^{3}$. What is the volume of the primitive unit cell?

A. $a^{3} / 8$

B. $a^{3} / 4$

C. $a^{3} / 2$

D. $a^{3}$

E. $2 a^{3}$

SCRATCHWORK


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C: 1
D: 0
E: 0
Which of the following best represents the temperature dependence of the resistivity of an undoped semiconductor?

A.



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B: 1
C: 0
D: 0
E: 0
The figure above shows a plot of the timedependent force $F_{x}(t)$ acting on a particle in motion along the $x$-axis. What is the total impulse delivered to the particle?

A. 0

B. $1 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$

C. $2 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$

D. $3 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$

E. $4 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$


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A: 0
B: 0
C: 1
D: 0
E: 0