shayekh/physics_gre huggingface dataset:
$X$ rays of wavelength $\lambda=0.250 \mathrm{~nm}$ 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 $\theta=14.5^{\circ}$. Which of the following gives the value of the interplanar spacing $d ?\left(\sin 14.5^{\circ} \approx 1 / 4\right)$

A. $0.125 \mathrm{~nm}$

B. $0.250 \mathrm{~nm}$

C. $0.500 \mathrm{~nm}$

D. $0.625 \mathrm{~nm}$

E. $0.750 \mathrm{~nm}$


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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., $T_{1}=3 T_{2}$. The ratio $m_{1} / m_{2}$ of the masses of the two planets is

A. 1

B. $\sqrt{3}$

C. 2

D. 3

E. 9


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If the Sun were suddenly replaced by a black hole of the same mass, it would have a Schwarzschild radius of $3,000 \mathrm{~m}$. 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.

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A distant galaxy is observed to have its hydrogen- $\beta$ line shifted to a wavelength of $580 \mathrm{~nm}$, away from the laboratory value of $434 \mathrm{~nm}$. Which of the following gives the approximate velocity of recession of the distant galaxy? (Note: $\frac{580}{434} \approx \frac{4}{3}$ )

A. $0.28 c$

B. $0.53 c$

C. $0.56 c$

D. $0.75 c$

E. $0.86 c$


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A small plane can fly at a speed of $200 \mathrm{~km} / \mathrm{h}$ in still air. A $30 \mathrm{~km} / \mathrm{h}$ wind is blowing from west to east. How much time is required for the plane to fly $500 \mathrm{~km}$ due north?

A. $\frac{50}{23} \mathrm{~h}$

B. $\frac{50}{\sqrt{409}} \mathrm{~h}$

C. $\frac{50}{20} \mathrm{~h}$

D. $\frac{50}{\sqrt{391}} \mathrm{~h}$

E. $\frac{50}{17} \mathrm{~h}$



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Each of the figures above shows blocks of mass $2 m$ and $m$ acted on by an external horizontal force $\mathbf{F}$. For each figure, which of the following statements about the magnitude of the force that one block exerts on the other $\left(F_{12}\right)$ is correct? (Assume that the surface on which the blocks move is frictionless.)

\section*{Figure 1}

A. $F_{12}=\frac{F}{3}$

B. $F_{12}=\frac{F}{3}$

C. $F_{12}=\frac{2 F}{3}$

D. $F_{12}=\frac{2 F}{3}$

E. $F_{12}=F$

\section*{Figure 2}

$$
F_{12}=\frac{F}{3}
$$

$F_{12}=\frac{2 F}{3}$

$F_{12}=\frac{F}{3}$

$F_{12}=\frac{2 F}{3}$

$F_{12}=F$



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In the figure above, block $A$ has mass

$m_{A}=25 \mathrm{~kg}$ and block $B$ has mass $m_{B}=10 \mathrm{~kg}$.

Both blocks move with constant acceleration $a=2 \mathrm{~m} / \mathrm{s}^{2}$ to the right, and the coefficient of static friction between the two blocks is $\mu_{s}=0.8$. The static frictional force acting between the blocks is

A. $20 \mathrm{~N}$

B. $50 \mathrm{~N}$

C. $78 \mathrm{~N}$

D. $196 \mathrm{~N}$

E. $274 \mathrm{~N}$


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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

A. $T=2 \pi \sqrt{\frac{l}{g}}$

B. $T=2 \pi \sqrt{\frac{l}{g-a}}$

C. $T=2 \pi \sqrt{\frac{l}{g+a}}$

D. $T=2 \pi \sqrt{\frac{l}{g} \frac{a}{(g+a)}}$

E. $T=2 \pi \sqrt{\frac{l}{g} \frac{(g+a)}{a}}$



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Three long, straight wires in the $x z$-plane, each carrying current $I$, cross at the origin of coordinates, as shown in the figure above. Let $\hat{\mathbf{x}}, \hat{\mathbf{y}}$, and $\hat{\mathbf{z}}$ denote the unit vectors in the $x$-, $y$-, and $z$-directions, respectively. The magnetic field $\mathbf{B}$ as a function of $x$, with $y=0$ and $z=0$, is

A. $\mathbf{B}=\frac{3 \mu_{0} I}{2 \pi x} \hat{\mathbf{x}}$

B. $\mathbf{B}=\frac{3 \mu_{0} I}{2 \pi x} \hat{\mathbf{y}}$

C. $\mathbf{B}=\frac{\mu_{0} I}{2 \pi x}(1+2 \sqrt{2}) \hat{\mathbf{y}}$

D. $\mathbf{B}=\frac{\mu_{0} I}{2 \pi x} \hat{\mathbf{x}}$

E. $\mathbf{B}=\frac{\mu_{0} I}{2 \pi x} \hat{\mathbf{y}}$



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A particle with mass $m$ and charge $q$, moving with a velocity $\mathbf{v}$, enters a region of uniform magnetic field $\mathbf{B}$, as shown in the figure above. The particle strikes the wall at a distance $d$ from the entrance slit. If the particle's velocity stays the same but its charge-to-mass ratio is doubled, at what distance from the entrance slit will the particle strike the wall?

A. $2 d$

B. $\sqrt{2} d$

C. $d$

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

E. $\frac{1}{2} d$



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Consider the closed cylindrical Gaussian surface above. Suppose that the net charge enclosed within this surface is $+1 \times 10^{-9} \mathrm{C}$ and the electric flux out through the portion of the surface marked $A$ is $-100 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$. The flux through the rest of the surface is most nearly given by which of the following?

A. $-100 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$

B. $\quad 0 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$

C. $10 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$

D. $100 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$

E. $200 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$

$$
{ }^{13} \mathrm{~N} \rightarrow{ }^{13} \mathrm{C}+e^{+}+v_{e}
$$


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The nuclear decay above is an example of a process induced by the

A. Mössbauer effect

B. Casimir effect

C. photoelectric effect

D. weak interaction

E. strong interaction


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Consider a single electron atom with orbital angular momentum $L=\sqrt{2} \hbar$. Which of the following gives the possible values of a measurement of $L_{z}$, the $z$-component of $L$ ?

A. 0

B. $0, \hbar$

C. $0, \hbar, 2 \hbar$

D. $-\hbar, 0, \hbar$

E. $-2 \hbar,-\hbar, 0, \hbar, 2 \hbar$


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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

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A muon can be considered to be a heavy electron with a mass $m_{\mu}=207 m_{e}$. 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 $n$ ?

A. $E_{n}=\frac{-E_{0}}{n^{2}}\left(\frac{m_{\mu}}{m_{e}}\right)$

B. $E_{n}=\frac{-E_{0}}{n^{2}}\left(\frac{m_{e}}{m_{\mu}}\right)$

C. $E_{n}=\frac{-E_{0}}{n^{2}}\left(\frac{\left(m_{p}+m_{e}\right)}{\left(m_{p}+m_{\mu}\right)}\right)$

D. $E_{n}=\frac{-E_{0}}{n^{2}}\left(\frac{m_{\mu}\left(m_{p}+m_{e}\right)}{m_{e}\left(m_{p}+m_{\mu}\right)}\right)$

E. $E_{n}=\frac{-E_{0}}{n^{2}}\left(\frac{m_{e}\left(m_{p}+m_{\mu}\right)}{m_{\mu}\left(m_{p}+m_{e}\right)}\right)$


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A large, parallel-plate capacitor consists of two square plates that measure $0.5 \mathrm{~m}$ on each side. A charging current of $9 \mathrm{~A}$ is applied to the capacitor. Which of the following gives the approximate rate of change of the electric field between the plates?

A. $2 \frac{\mathrm{V}}{\mathrm{m} \cdot \mathrm{s}}$

B. $40 \frac{\mathrm{V}}{\mathrm{m} \cdot \mathrm{s}}$

C. $1 \times 10^{12} \frac{\mathrm{V}}{\mathrm{m} \cdot \mathrm{s}}$

D. $4 \times 10^{12} \frac{\mathrm{V}}{\mathrm{m} \cdot \mathrm{s}}$

E. $2 \times 10^{13} \frac{\mathrm{V}}{\mathrm{m} \cdot \mathrm{s}}$



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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. $\frac{1}{3} \frac{V}{R}$

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

C. $\frac{V}{R}$

D. $\frac{3}{2} \frac{V}{R}$

E. $3 \frac{V}{R}$



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In the AC circuit above, $V_{i}$ is the amplitude of the input voltage and $V_{o}$ is the amplitude of the output voltage. If the angular frequency $\omega$ of the input voltage is varied, which of the following gives the ratio $V_{o} / V_{i}=G$ as a function of $\omega$ ?

A.



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A wire loop that encloses an area of $10 \mathrm{~cm}^{2}$ has a resistance of $5 \Omega$. The loop is placed in a magnetic field of $0.5 \mathrm{~T}$ 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. $10^{-4} \mathrm{C}$

B. $10^{-3} \mathrm{C}$

C. $10^{-2} \mathrm{C}$

D. $10^{-1} \mathrm{C}$

E. $1 \mathrm{C}$



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Two nonrelativistic electrons move in circles under the influence of a uniform magnetic field $\mathbf{B}$, as shown in the figure above. The ratio $r_{1} / r_{2}$ of the orbital radii is equal to $1 / 3$. Which of the following is equal to the ratio $v_{1} / v_{2}$ of the speeds?

A. $1 / 9$

B. $1 / 3$

C. 1

D. 3

E. 9

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Which of the following statements about bosons and/or fermions is true?

A. Bosons have symmetric wave functions and obey the Pauli 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 Pauli exclusion principle.


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The discovery of the $J / \psi$ 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



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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 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 mirror and upright

D. a distance $\frac{R}{3}$ to the left of the mirror and inverted

E. a distance $\frac{R}{3}$ to the right of the mirror and upright


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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 $\lambda=540 \mathrm{~nm}$. What is the next shorter wavelength at which the film will appear bright on reflection?

A. $135 \mathrm{~nm}$

B. $180 \mathrm{~nm}$

C. $270 \mathrm{~nm}$

D. $320 \mathrm{~nm}$

E. $405 \mathrm{~nm}$



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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. $\theta>\sin ^{-1}\left(\sqrt{n^{2}-1}\right)$

B. $\theta<\sin ^{-1}\left(\sqrt{n^{2}-1}\right)$

C. $\theta>\sin ^{-1}\left(\sqrt{n^{2}+1}\right)$

D. $\theta<\sin ^{-1}\left(\sqrt{n^{2}+1}\right)$

E. $\sin ^{-1}\left(\sqrt{n^{2}-1}\right)<\theta<\sin ^{-1}\left(\sqrt{n^{2}+1}\right)$

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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 $\frac{1}{m}$.

B. It is proportional to $\sqrt[4]{m}$.

C. It is proportional to $\sqrt{m}$.

D. It is proportional to $m$.

E. It is proportional to $m^{2}$.


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A particle can occupy two possible states with energies $E_{1}$ and $E_{2}$, where $E_{2}>E_{1}$. At temperature $T$, the probability of finding the particle in state 2 is given by which of the following?

A. $\frac{\mathrm{e}^{-E_{1} / k T}}{\mathrm{e}^{-E_{1} / k T}+\mathrm{e}^{-E_{2} / k T}}$

B. $\frac{\mathrm{e}^{-E_{2} / k T}}{\mathrm{e}^{-E_{1} / k T}+\mathrm{e}^{-E_{2} / k T}}$

C. $\frac{\mathrm{e}^{-\left(E_{1}+E_{2}\right) / k T}}{\mathrm{e}^{-E_{1} / k T}+\mathrm{e}^{-E_{2} / k T}}$

D. $\frac{\mathrm{e}^{-E_{1} / k T}+\mathrm{e}^{-E_{2} / k T}}{\mathrm{e}^{-E_{2} / k T}}$

E. $\frac{\mathrm{e}^{-E_{1} / k T}+\mathrm{e}^{-E_{2} / k T}}{\mathrm{e}^{-E_{1} / k T}}$

$$
\left(p+\frac{a}{V^{2}}\right)(V-b)=R T
$$


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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 $T_{0}$ from volume $V_{1}$ to volume $V_{2}$, which of the following gives the work done by the gas?

A. 0

B. $R T_{0} \ln \left(\frac{V_{2}}{V_{1}}\right)$

C. $R T_{0} \ln \left(\frac{V_{2}-b}{V_{1}-b}\right)$

D. $R T_{0} \ln \left(\frac{V_{2}-b}{V_{1}-b}\right)+a\left(\frac{1}{V_{2}}-\frac{1}{V_{1}}\right)$

E. $R T_{0}\left(\frac{1}{\left(V_{2}-b\right)^{2}}-\frac{1}{\left(V_{1}-b\right)^{2}}\right)+a\left(\frac{1}{V_{2}^{3}}-\frac{1}{V_{1}^{3}}\right)$


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A $1 \mathrm{~kg}$ block attached to a spring vibrates with a frequency of $1 \mathrm{~Hz}$ on a frictionless horizontal table. Two springs identical to the original spring are attached in parallel to an $8 \mathrm{~kg}$ block placed on the same table. Which of the following gives the frequency of vibration of the $8 \mathrm{~kg}$ block?

A. $\frac{1}{4} \mathrm{~Hz}$

B. $\frac{1}{2 \sqrt{2}} \mathrm{~Hz}$

C. $\frac{1}{2} \mathrm{~Hz}$

D. $1 \mathrm{~Hz}$

E. $2 \mathrm{~Hz}$



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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. $h=\frac{v^{2}}{2 g}$

B. $h=\frac{3 v^{2}}{4 g}$

C. $h=\frac{v^{2}}{g}$

D. $h=\frac{3 v^{2}}{2 g}$

E. $h=\frac{2 v^{2}}{g}$


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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?

A. $L=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2}-\frac{1}{2} k(r \cos \theta-s)^{2}$

B. $L=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2}-\frac{1}{2} k(r \sin \theta-s)^{2}$

C. $L=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2}+\frac{1}{2} k(r-s)^{2}$

D. $L=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2}-\frac{1}{2} k(r-s)^{2}$

E. $L=-\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2}+\frac{1}{2} k(r-s)^{2}$



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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

$$
H=\frac{p_{\theta}^{2}}{2 m L^{2}}+\frac{p_{\phi}^{2}}{2 m L^{2} \sin ^{2} \theta}-m g L \cos \theta
$$

where $\theta$ and $\phi$ are defined as shown in the figure. On the basis of Hamilton's equations of motion, the generalized coordinate or momentum that is a constant in time is

A. $\theta$

B. $\phi$

C. $\dot{\theta}$

D. $p_{\theta}$

E. $p_{\phi}$



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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 $\lambda=\frac{2 M}{L^{2}} x$, where $x$ is the distance from the origin. Which of the following gives the $x$-coordinate of the rod's center of mass?

A. $\frac{1}{12} L$

B. $\frac{1}{4} L$

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

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

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


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A particle is in an infinite square well potential with walls at $x=0$ and $x=L$. If the particle is in the state $\psi(x)=A \sin \left(\frac{3 \pi x}{L}\right)$, where $A$ is a constant, what is the probability that the particle is between $x=\frac{1}{3} L$ and $x=\frac{2}{3} L$ ?

A. 0

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

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

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

E. 1

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Which of the following are the eigenvalues of the Hermitian matrix $\left(\begin{array}{cc}2 & i \\ -i & 2\end{array}\right)$ ?

A. 1,0

B. 1,3

C. 2,2

D. $i,-i$

E. $1+i, 1-i$

$$
\sigma_{x}=\left(\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right), \quad \sigma_{y}=\left(\begin{array}{cc}
0 & -i \\
i & 0
\end{array}\right), \quad \sigma_{z}=\left(\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right), \quad I=\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right)
$$


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Consider the Pauli spin matrices $\sigma_{x}, \sigma_{y}$, and $\sigma_{z}$ and the identity matrix $I$ given above. The commutator $\left[\sigma_{x}, \sigma_{y}\right] \equiv \sigma_{x} \sigma_{y}-\sigma_{y} \sigma_{x}$ is equal to which of the following?

A. $I$

B. $2 i \sigma_{x}$

C. $2 i \sigma_{y}$

D. $2 i \sigma_{z}$

E. 0


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A spin- $\frac{1}{2}$ particle is in a state described by the spinor

$$
\chi=A\left(\begin{array}{c}
1+i \\
2
\end{array}\right)
$$

where $A$ is a normalization constant. The probability of finding the particle with spin projection $S_{z}=-\frac{1}{2} \hbar$ is

A. $\frac{1}{6}$

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

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

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

E. 1



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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 function for $x \leq 0$ is given by

$$
\psi=A \mathrm{e}^{i k_{1} x}+B \mathrm{e}^{-i k_{1} x}, \text { where } k_{1}=\sqrt{\frac{2 m E}{\hbar^{2}}}
$$

and the wave function for $x>0$ is given by

$$
\psi=C \mathrm{e}^{i k_{2} x}, \text { where } k_{2}=\sqrt{\frac{2 m\left(E-V_{0}\right)}{\hbar^{2}}}
$$

Which of the following gives the reflection coefficient for the system?

A. $R=0$

B. $R=1$

C. $R=\frac{k_{2}}{k_{1}}$

D. $R=\left(\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\right)^{2}$

E. $R=\frac{4 k_{1} k_{2}}{\left(k_{1}+k_{2}\right)^{2}}$



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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 $(r>b)$ ?

\section*{Region I}

\section*{Region II}

A. $\frac{Q}{4 \pi \varepsilon_{0}} \frac{1}{r}$

B. $\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{1}{a}\right)$ 0

C. $\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{1}{b}\right)$ $-\frac{Q}{4 \pi \varepsilon_{0}} \frac{1}{r}$

D. $\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{1}{a}\right)$

$\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{1}{b}-\frac{1}{a}\right)$

E. $\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{1}{b}\right)$

$\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{1}{a}-\frac{1}{b}\right)$


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In static electromagnetism, let $\mathbf{E}, \mathbf{B}, \mathbf{J}$, and $\rho$ be the electric field, magnetic field, current density, and charge density, respectively. Which of the following conditions allows the electric field to be written in the form $\mathbf{E}=-\nabla \phi$, where $\phi$ is the electrostatic potential?

A. $\nabla \cdot \mathbf{J}=0$

B. $\nabla \cdot \mathbf{E}=\rho / \epsilon_{0}$

C. $\nabla \times \mathbf{E}=\mathbf{0}$

D. $\nabla \times \mathbf{B}=\mu_{0} \mathbf{J}$

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


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A long, straight, hollow cylindrical wire with an inner radius $R$ and an outer radius $2 R$ carries a uniform current density. Which of the following graphs best represents the magnitude of the magnetic field as a function of the distance from the center of the wire?

A.



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A parallel-plate capacitor has plate separation $d$. The space between the plates is empty. A battery supplying voltage $V_{0}$ is connected across the capacitor, resulting in electromagnetic energy $U_{0}$ stored in the capacitor. A dielectric, of dielectric constant $\kappa$, 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 $V_{0}$ and $U_{0}$ ?
$\underline{E}$
$\underline{U}$
A. $\frac{V_{0}}{d}$
$U_{0}$
B. $\frac{V_{0}}{d}$
$\kappa U_{0}$
C. $\frac{V_{0}}{d}$
$\kappa^{2} U_{0}$
D. $\frac{V_{0}}{\kappa d}$
$U_{0}$
E. $\frac{V_{0}}{\kappa d}$
$\kappa U_{0}$


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An observer $O$ at rest midway between two sources of light at $x=0$ and $x=10 \mathrm{~m}$ observes the two sources to flash simultaneously. According to a second observer $O^{\prime}$, moving at a constant speed parallel to the $x$-axis, one source of light flashes $13 \mathrm{~ns}$ before the other. Which of the following gives the speed of $O^{\prime}$ relative to $O$ ?

A. $0.13 c$

B. $0.15 c$

C. $0.36 c$

D. $0.53 c$

E. $0.62 c$


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Let $\hat{\mathbf{J}}$ be a quantum mechanical angular momentum operator. The commutator $\left[\hat{J}_{x} \hat{J}_{y}, \hat{J}_{x}\right]$ is equivalent to which of the following?

A. 0

B. $i \hbar \hat{J}_{z}$

C. $i \hbar \hat{J}_{z} \hat{J}_{x}$

D. $-i \hbar \hat{J}_{x} \hat{J}_{z}$

E. $i \hbar \hat{J}_{x} \hat{J}_{y}$


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Which of the following ions CANNOT be used as a dopant in germanium to make an $n$-type semiconductor?

A. As

B. $\mathrm{P}$

C. $\mathrm{Sb}$

D. $\mathrm{B}$

E. $\mathrm{N}$

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In the Compton effect, a photon with energy $E$ scatters through a $90^{\circ}$ angle from a stationary electron of mass $m$. The energy of the scattered photon is

A. $E$

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

C. $\frac{E^{2}}{m c^{2}}$

D. $\frac{E^{2}}{E+m c^{2}}$

E. $\frac{E \cdot m c^{2}}{E+m c^{2}}$


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Which of the following is the principal decay mode of the positive muon $\mu^{+}$?

A. $\mu^{+} \rightarrow e^{+}+v_{e}$

B. $\mu^{+} \rightarrow p+v_{\mu}$

C. $\mu^{+} \rightarrow n+e^{+}+v_{e}$

D. $\mu^{+} \rightarrow e^{+}+v_{e}+\bar{v}_{\mu}$

E. $\mu^{+} \rightarrow \pi^{+}+\bar{v}_{e}+v_{\mu}$



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