SeePhys: Does Seeing Help Thinking? -- Benchmarking Vision-Based Physics Reasoning
Paper • 2505.19099 • Published • 7
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0 | A soap film ( $n=4 / 3$ ) of thickness $d$ is illuminated at normal incidence by light of wavelength 500 nm . Calculate the approximate intensities of interference maxima relative to the incident intensity as $d$ is varied, when viewed in reflected light. | 0.08 | The reflectivity at each surface of the soap film is given by
$$
R = \left[\frac{(n - n_{0})}{(n + n_{0})}\right]^{2} = \left[\frac{\left(\frac{4}{3} - 1\right)}{\left(\frac{4}{3} + 1\right)}\right]^{2} \approx 0.02.
$$
For a film of low reflectivity $R$, the intensity of the reflected light at either surface can be a... | 1 | 7 | OPT | English | optical_path | optional | IMAGE 1:
A light ray with intensity $I_0$ is incident on a plane-parallel slab, and two reflected rays, each labeled $RI_0$, are shown emerging from the upper and lower surfaces; all rays are indicated with solid arrows denoting their directions of propagation. | |
1 | As shown in Fig. 3.26, the switch has been in position A for a long time. At $t=0$ it is suddenly moved to position B. Immediately after contact with B: What is the time rate of change of the current through $R$ ? | $-10^{4} \mathrm{~A}/\mathrm{s}$ | As $-L \frac{d i_{L}}{d t}=i_{L} R$, $\left.\frac{d i_{L}}{d t}\right|_{t=0}=-i_{L}(0) \frac{R}{L}=1 \times \frac{10^{4}}{1}=-10^{4} \mathrm{~A}/\mathrm{s}$. | 1 | 7 | EM | English | circuit_diagram | essential | IMAGE 1:
Circuit diagram containing a $1\,\Omega$ resistor, a $1\,\text{V}$ battery, a switch S, a $10^4\,\Omega$ resistor labeled $R$, and an inductor labeled $L = 1\,\mathrm{H}$ in series; the switch S can connect point A or B, and a curved arrow by the inductor indicates conventional current direction in the loop co... | |
2 | At low temperatures, a mixture of ${ }^{3} \text{He}$ and ${ }^{4} \text{He}$ atoms form a liquid which separates into two phases: a concentrated phase (nearly pure ${ }^{3} \text{He}$), and a dilute phase (roughly $6.5 \text{\textperthousand} { }^{3} \text{He}$ for $T \text{\textless}= 0.1 \text{ K}$). The lighter ${ ... | $\frac{\alpha_{c} T^{2}}{2 T_{Fc}}, \frac{\alpha_{d} T^{2}}{2 T_{Fd}}$ | \[\n\text { (c) } \begin{aligned}\nQ_{\mathrm{c}} & =\int_{0}^{T} c_{v} d T\n\end{aligned}=\frac{\alpha_{\mathrm{c}} T^{2}}{2 T_{\mathrm{Fc}}}, ~ \begin{aligned}\nQ_{\mathrm{d}} & =\int_{0}^{T} c_{v} d T\n\end{aligned}=\frac{\alpha_{\mathrm{d}} T^{2}}{2 T_{\mathrm{Fd}}} .\n\] | 7 | TSM | English | thermodynamics | essential | IMAGE 1:
Rectangular container with two horizontal layers: the upper layer is labeled as "concentrated phase of $^3$He;" the lower, dotted layer is labeled as "dilute phase of $^3$He (in superfluid of $^4$He);" layers are separated flatly, showing spatial phase separation of helium-3 in a superfluid helium-4 background... | ||
3 | The current-voltage characteristic of the output terminals A, B (Fig. 3.3) is the same as that of a battery of emf $\varepsilon_{0}$ and internal resistance $r$. Find the short-circuit current provided by the battery. | $0.625 \mathrm{~A}$ | Then the short-circuit current provided by the battery is $I=\frac{\varepsilon_{0}}{r}=\frac{3 \mathrm{~V}}{4.8 \Omega}=0.625 \mathrm{~A}$. | 3 | 7 | EM | English | circuit_diagram | essential | IMAGE 1:
A 15 V voltage source is connected in series to two resistors of $24\,\Omega$ and $6\,\Omega$; points $A$ and $B$ are tapped across the $24\,\Omega$ and $6\,\Omega$ resistors, respectively, as shown in the circuit diagram. All components and labels are indicated. | |
4 | A block of mass $M$ is rigidly connected to a massless circular track of radius $a$ on a frictionless horizontal table as shown in Fig. 2.7. A particle of mass $m$ is confined to move without friction on the circular track which is vertical. Set up the Lagrangian, using $\theta$ as one coordinate. | $\frac{1}{2} M \dot{x}^{2}+\frac{1}{2} m\left[\dot{x}^{2}+a^{2} \dot{\theta}^{2}+2a\dot{x}\dot{\theta}\cos\theta\right]+mga\cos\theta$ | Using a fixed coordinate frame $x,y$ and choosing the $x$ coordinate of the center of the circular track and the angle $\theta$ as generalized coordinates, the coordinates of the mass $m$ are $(x+a\sin\theta, -a\cos\theta)$ while the mass $M$ (rigidly connected to the track) has velocity $(\dot{x},0)$. Thus, the kineti... | 7 | CM | English | static_force_analysis | essential | IMAGE 1:
A pulley of radius $a$ supports a mass $m$ at its rim and is connected via a string to a block of mass $M$ resting on a horizontal surface; the angle $\theta$ measures the position of $m$ on the pulley. The $x$- and $y$-axes are indicated, with $x$ horizontally and $y$ vertically. | ||
5 | A block of mass $M$ is rigidly connected to a massless circular track of radius $a$ on a frictionless horizontal table as shown in Fig. 2.7. A particle of mass $m$ is confined to move without friction on the circular track which is vertical. In the limit of small angles, solve that equations of motion for $\theta$ as a... | $\theta=A\sin(\omega t)+B\cos(\omega t)$, with $\omega=\sqrt{\frac{(M+m)g}{Ma}}$ | For small oscillations, $\theta$ and $\dot{\theta}$ are small so that higher order terms can be neglected. This simplifies the equations to\newline\quad $(M+m)\ddot{x}+ma\ddot{\theta}=0$,\newline\quad $a\ddot{\theta}+\ddot{x}+g\theta=0$. Eliminating $\ddot{x}$ leads to\newline\quad $\ddot{\theta}+\frac{(M+m)g}{Ma}\thet... | 7 | CM | English | static_force_analysis | optional | IMAGE 1:
A uniform disk of radius $a$ and mass $m$ is connected by a rigid, horizontal rod to a block of mass $M$; the system rests on a horizontal surface. The disk and block are joined such that their centers are aligned horizontally. The angle $\theta$ denotes the orientation of the radius with respect to the vertic... | ||
6 | A useful optical grating configuration is to diffract light back along itself. If there are $N$ grating lines per unit length, what wavelength(s) is (are) diffracted back at the incident angle $\theta$, where $\theta$ is the angle between the normal to the grating and the incident direction? | $\lambda = 2d \sin\left( \frac{\theta}{m} \right)$ | This kind of grating is known as blazed grating. As Fig. 2.59 shows, the angle of incidence, $\theta$, is equal to $\alpha$, which is the angle between the groove plane and the grating plane. Each line has across it an optical path difference $\Delta=2 d \sin \theta$, where $d=1 / N$. Wavelengths for which the grating ... | 7 | OPT | English | optical_path | essential | IMAGE 1:
Schematic of incident light (shown as parallel arrows) striking a blazed diffraction grating; the angle between the incident rays and the normal to the grating surface is labeled $\theta = \alpha$, with $\alpha$ also denoting the blaze (facet) angle of the grating. The normal to the grating is indicated by a d... | ||
7 | Static charges are distributed along the $x$-axis in the interval $-a \le x' \le a$. The charge density is $\rho(x')$ for $|x'| \le a$ and $0$ for $|x'| > a$. Derive a multipole expansion for the potential valid for $x>a$. | $\Phi(x)=\frac{1}{4\pi\varepsilon_{0}}\left[\frac{1}{x}\int_{-a}^{a}\rho(x')\,dx'+\frac{1}{x^2}\int_{-a}^{a}x'\rho(x')\,dx'+\frac{1}{x^3}\int_{-a}^{a}x'^2\rho(x')\,dx'+\ldots\right]$ | For $x>a$, $-a<x'<a$, we have $\frac{1}{|x-x'|}=\frac{1}{x}+\frac{x'}{x^2}+\frac{x'^2}{x^3}+\ldots$. Hence the multipole expansion of $\Phi(x)$ is $\Phi(x)=\frac{1}{4\pi\varepsilon_{0}}\left[\int_{-a}^{a}\frac{\rho(x')}{x}\,dx'+\int_{-a}^{a}\frac{x'\rho(x')}{x^2}\,dx'+\int_{-a}^{a}\frac{x'^2\rho(x')}{x^3}\,dx'+\ldots\r... | 7 | EM | English | charge_distribution | essential | IMAGE 1:
Three horizontal lines labeled (I), (II), and (III), each representing arrangements of point charges and a marked position $x$ on the $x$-axis:
- (I): A single point charge $q$ at $x' = 0$, with $x$ marked to its right.
- (II): Two point charges, $-q$ at $x' = -\frac{a}{2}$ and $+q$ at $x' = \frac{a}{2}$, ... | ||
8 | A horizontal ray of light passes through a prism of index 1.50 and apex angle $4^{\\circ}$ and then strikes a vertical mirror, as shown in the figure. Through what angle must the mirror be rotated if after reflection the ray is to be horisontal? | $1^{\circ}$ | As the apex angle is very small ($\alpha=4^\circ$), the angle of deviation $\delta$ can be obtained approximately:
$$
\delta = (n-1)\alpha = (1.5-1) \times 4^\circ = 2^\circ
$$
From Fig. 1.2 we see that if the reflected ray is to be horizontal, the mirror must be rotated clockwise through an angle $\gamma$ given by
$$... | 1 | 7 | OPT | English | optical_path | optional | IMAGE 1:
A light ray is incident from the left onto a prism with apex angle labeled $\angle A$, then exits the prism and strikes a vertical mirror; the reflected ray from the mirror travels downward and right, with all light ray directions shown by arrows. | |
9 | A simple pendulum is attached to a support which is driven horizontally with time as shown in Fig. 2.25. Set up the Lagrangian for the system in terms of the generalized coordinates $\theta$ and $y$, where $\theta$ is the angular displacement from equilibrium and $y(t)$ is the horizontal position of the pendulum suppor... | $L=\frac{m}{2}(\dot{y}_{s}^{2}+l^{2}\dot{\theta}^{2}+2l\dot{y}_{s}\dot{\theta}\cos\theta)+mgl\cos\theta$ | The mass $m$ has coordinates $\left(y_{s}+l\sin\theta, -l\cos\theta\right)$ and velocity $\left(\dot{y}_{s}+l\dot{\theta}\cos\theta, l\dot{\theta}\sin\theta\right)$. Hence the Lagrangian is $L=T-V=\frac{m}{2}(\dot{y}_{s}^{2}+l^{2}\dot{\theta}^{2}+2l\dot{y}_{s}\dot{\theta}\cos\theta)+mgl\cos\theta$. | 7 | CM | English | simple_harmonic_motion | essential | IMAGE 1:
A point mass $m$ is suspended by a massless rod of length $l$ from a block constrained to move horizontally along the $y$-axis, with its position denoted $y_s$. The angular displacement of the rod from the vertical $z$-axis is labeled $\theta$. The coordinate system has origin $O$, with $y$ and $z$ axes indica... | ||
10 | As in Fig. $2.39$, what is the direction of the current in the resistor $r$ (from $A$ to $B$ or from $B$ to $A$) when the following operations are performed? In each case give a brief explanation of your reasoning? The switch $S$ is closed. | from $B$ to $A$ | In all the three cases the magnetic field produced by coil 1 at coil 2 is increased. Lenz's law requires the magnetic field produced by the induced current in coil 2 to be such that as to prevent the increase of the magnetic field crossing coil 2. Applying the right-hand rule we see that the direction of the current in... | 7 | CM | English | electromagnetic_field | essential | IMAGE 1:
Two solenoids labeled $1$ and $2$ are shown side by side. Solenoid $1$ is connected in series with a resistor $R$, a voltage source $V$, and a switch $S$; a current path is indicated by an arrow. Solenoid $2$ has no external connections and its ends are labeled $A$ and $B$. Both solenoids are positioned parall... | ||
11 | The figure below shows an apparatus for the determination of $C_{p} / C_{v}$ for a gas, according to the method of Clement and Desormes. A bottle $G$, of reasonable capacity (say a few litres), is fitted with a tap $H$, and a manometer $M$. The difference in pressure between the inside and the outside can thus be deter... | $\frac{7}{5}$ | Oxygen consists of diatomic molecules. When $t=20^{\text{C}}$, only the translational and rotational motions of the molecules contribute to the specific heat. Therefore $$ C_{v}=\frac{5 R}{2}, \quad C_{p}=\frac{7 R}{2}, \quad \gamma=\frac{7}{5} . $$ | 7 | TSM | English | thermodynamics | essential | IMAGE 1:
A vessel $G$ is sealed with a stopper $H$ connected to a manometer $M$ via tubing; the manometer contains a fluid column of height $h$, indicating a pressure difference relative to atmospheric pressure. | ||
12 | A beam of neutrons (mass $m$ ) traveling with nonrelativistic speed $v$ impinges on the system shown in Fig. 1.7, with beam-splitting mirrors at corners $B$ and $D$, mirrors at $A$ and $C$, and a neutron detector at $E$. The corners all make right angles, and neither the mirrors nor the beam-splitters affect the neutro... | $I_{0} \cos ^{2}\left(\frac{m g H L}{2 \hbar v}\right)$ | Assume that when the system is in a horizontal plane the two split beams of neutrons have the same intensity when they reach $D$, and so the wave functions will each have amplitude $
\sqrt{I_{0}} / 2$. Now consider the system in a vertical plane. As BA and CD are equivalent dynamically, they need not be considered. The... | 7 | AMONP | English | atomic_physics | essential | IMAGE 1:
Diagram of a neutron beam apparatus: incident neutrons enter at point B and travel horizontally through a labeled "region of magnetic field," then follow two possible paths: B→A→D→E (top path) and B→C→D→E (bottom path). Distances are labeled $L$ (horizontal from A to D and from C to D) and $s$ (vertical from B... | ||
13 | A long solenoid of radius $b$ and length $l$ is wound so that the axial magnetic field is $$\mathbf{B}=\begin{cases} B_{0}\,\mathbf{e}_{z}, & r<b,\\ 0, & r>b, \end{cases}$$ A particle of charge $q$ is emitted with velocity $v$ perpendicular to a central rod of radius $a$ (see Fig. 2.59). The electric force on the parti... | $\mathbf{J}_{EM}=\frac{qB_{0}(b^{2}-a^{2})}{2}\,\mathbf{e}_{z}$ and $\mathbf{J}_{S}=\frac{qB_{0}}{2}(b-a)^{2}\,\mathbf{e}_{z}$ | After the particle has gone far away from the solenoid, the momentum density of the electromagnetic field at a point within the solenoid is $$\mathbf{g}=\frac{\mathbf{E}\times\mathbf{H}}{c^{2}}=\varepsilon_{0}\,\mathbf{E}\times\mathbf{B}$$ and the angular momentum density is $$\mathbf{j}=\mathbf{r}\times\mathbf{g}=\var... | 7 | EM | English | electromagnetic_field | optional | IMAGE 1:
A cross-sectional diagram showing a cylindrical conductor carrying current $I$ out of the page (cross-hatched inner region) of radius $a$, surrounded by a coaxial cylindrical shell of radius $b$ at distance $b$ from the center; a point $P$ is marked a distance $r$ from the axis. The direction of the current $I... | ||
14 | Three point particles, two of mass $m$ and one of mass $M$, are constrained to lie on a horizontal circle of radius $r$. They are mutually connected by springs, each of constant $K$, that follow the arc of the circle and that are of equal length when the system is at rest as shown in Fig. 2.48. Assuming motion that str... | $\omega_{1}=0,\quad \omega_{2}=\sqrt{\frac{3K}{m}},\quad \omega_{3}=\sqrt{\frac{(2m+M)K}{mM}}$ | Equation (1) shows that $\omega_{1}=0$, corresponding to the mode in which the system rotates as a whole. Equation (2) gives $\omega_{2}=\sqrt{\frac{3K}{m}}$. The equation of motion for $\zeta$ is $\ddot{\zeta}+\Big(\frac{(2m+M)K}{mM}\Big)\zeta=0$, yielding $\omega_{3}=\sqrt{\frac{(2m+M)K}{mM}}$. | 7 | CM | English | spring_force | optional | IMAGE 1:
A circular ring consisting of identical point masses $m$ connected by massless springs is shown, with an additional mass $M$ attached at the bottom of the ring. Three specific masses are labeled: two $m$ masses at the top and one $M$ mass at the bottom. | ||
15 | Four identical coherent monochromatic wave sources A, B, C, D, as shown in Fig. 4.2 produce waves of the same wavelength $\lambda$. Two receivers $R_{1}$ and $R_{2}$ are at great (but equal) distances from $B$. Which receiver, if any, picks up the greater signal if source $B$ is turned off? | Both receivers pick up signals of the same intensity | If source B is turned off, then $E_{10} \approx -E_{0} e^{i K r}$, $E_{20} \approx E_{0} e^{i K r}$. Thus $I_{1}=I_{2} \sim E_{0}^{2}$, that is, the two receivers pick up signals of the same intensity. | 7 | CM | English | charge_distribution | essential | IMAGE 1:
Diagram shows points $A$, $B$, $C$, and $D$ arranged with $B$ at the intersection; line segment $AC$ is horizontal with $B$ midway, $AB = BC = \frac{\lambda}{2}$; line $BD$ is vertical above $B$ of length $\frac{\lambda}{2}$; points $R_1$ and $R_2$ are located to the left of $A$ and below $B$, respectively. | ||
16 | Consider a solid cylinder of mass $m$ and radius $r$ sliding without rolling down the smooth inclined face of a wedge of mass $M$ that is free to move on a horizontal plane without friction (Fig. 1.174). Now suppose that the cylinder is free to roll down the wedge without slipping. How far does the wedge move in this c... | $\frac{m h}{M+m}\cot\theta$ | If the cylinder is allowed to roll, conservation of the horizontal component of the total linear momentum of the system still holds. It follows that the result obtained in (a) is also valid here. | 7 | CM | English | circular_motion | essential | IMAGE 1:
A uniform block of mass $M$ forms an inclined plane at angle $\theta$ with the horizontal $x$-axis and height $h$ above the ground; a homogeneous sphere of radius $r$ and mass $m$ is positioned on the incline at a distance $\xi$ up the slope. The weight vector $mg$ is shown acting vertically downward from the ... | ||
17 | The one-dimensional quantum mechanical potential energy of a particle of mass $m$ is given by $$ V(x)=\begin{cases} V_{0}\delta(x), & -a<x<\infty \\ \infty, & x<-a \end{cases} $$ as shown in Fig. 1.19. At time $t=0$, the wave function of the particle is completely confined to the region $-a<x<0$. [Define the quantities... | $\psi_{k}(x)=\begin{cases} c_{k}\sin k(x+a), & (-a<x<0)\\ c_{k}\sin k(x+a)+A_{k}\sin kx, & (x\ge0)\\ 0, & (x<-a) \end{cases},\quad A_{k}=\frac{c_{k}\alpha}{k}\sin ka,\quad c_{k}=\left\{\frac{\pi}{2}\Bigl[1+\frac{\alpha\sin2ka}{k}+\Bigl(\frac{\alpha\sin ka}{k}\Bigr)^2\Bigr]\right\}^{-\frac{1}{2}}$ | (c) In both regions I and II, the wave equation is $$ \psi^{\prime\prime}(x)+k^2\psi(x)=0. $$ Its real solutions are sinusoidal functions. The solutions satisfying the boundary conditions are $$ \psi_{k}(x)=\begin{cases} c_{k}\sin k(x+a), & -a<x<0,\\ c_{k}\sin k(x+a)+A_{k}\sin kx, & x\ge0,\\ 0, & x<-a, \end{cases} $$ w... | 7 | QMIT | English | coordinate_system | essential | IMAGE 1:
One-dimensional potential energy diagram $V(x)$ versus position $x$ showing a finite-width potential barrier centered at $x = 0$, with two regions labeled: region I for $x < 0$ and region II for $x > 0$; a vertical wall is located at $x = -a$. | ||
18 | A long‐range rocket is fired from the surface of the earth (radius $R$) with velocity $\mathbf{v}=(v_r, v_\theta)$ (Fig. 1.29). Neglecting air friction and the rotation of the earth (but using the exact gravitational field), obtain an equation to determine the maximum height $H$ achieved by the trajectory. | $H \approx \frac{v_{r}^{2}R}{2\left(\frac{GM}{R}-v_{\theta}^{2}\right)}$ | Both the angular momentum and mechanical energy of the rocket are conserved under the action of gravity, a central force. Considering the initial state and the final state when the rocket achieves maximum height, we have $$ mR v_{\theta} = m(R+H)v_{\theta}' $$ and $$ \frac{1}{2}m\left(v_{\theta}^{2}+v_{r}^{2}\right)-\f... | 7 | CM | English | projectile_motion | essential | IMAGE 1:
A circle of radius $R$ with center marked is shown; an arc on the circumference is depicted with height $H$ (drawn perpendicularly from the circle’s surface), where $H$ is shown as a vector directed outward and away from the circle. | ||
19 | 1093 A block of mass $m$ slides without friction on an inclined plane of mass $M$ which in turn is free to slide without friction on a horizontal table (Fig. 1.65). Write sufficient equations to find the motion of the block and the inclined plane. You do not need to solve these equations. (Wisconsin) | $M \ddot{X}=N \sin \alpha$, $m(\ddot{x}+\ddot{X}\cos \alpha)=-m g \sin \alpha$, $-m \ddot{X}\sin \alpha=N-m g\cos \alpha$ | As shown in Fig. 1.65, let $x, y$ be a coordinate frame attached to the inclined plane, whose horizontal coordinate in the laboratory frame is denoted by $X$. The forces on the block and the inclined plane are as shown in the diagram. We have for the inclined plane $M \ddot{X}=N \sin \alpha$, for the motion of the bloc... | 7 | CM | English | circular_motion | optional | IMAGE 1:
A particle of mass $2\,\mathrm{kg}$ moves in a horizontal circular path of radius $6\,\mathrm{m}$, with an arrow indicating the direction of motion as counterclockwise. | ||
20 | For a particle of charge $q$, what are the energies and energy eigenfunctions if a very long solenoid containing a magnetic flux passes the doubled ring of (b) as shown in the Fig. 4.9? Assume the system does not radiate electromagnetically. | $E_{n}=\frac{\hbar^{2}}{8mR^{2}}\left(n-\frac{q\phi}{\pi\hbar}\right)^{2} \quad \text{and} \quad \Psi_{n}(\theta)=\frac{1}{\sqrt{4\pi}}\exp\left(i\frac{n}{2}\theta\right)$ | Similarly for the ring of (b), we have $E_{n}=\frac{\hbar^{2}}{2I}\left(\frac{n}{2}-\frac{q\phi}{2\pi\hbar}\right)^2=\frac{\hbar^{2}}{8I}\left(n-\frac{q\phi}{\pi\hbar}\right)^2$ and $\Psi_{n}(\theta)=\frac{1}{\sqrt{4\pi}}\exp\Big(i\frac{n}{2}\theta\Big)$, where $n=0,\pm1,\pm2,\ldots$. | 7 | QMIT | English | quantum_mechanics | essential | IMAGE 1:
A circular loop of radius $R$ lies in the $xy$-plane, centered at the origin, with the $x$, $y$, and $z$ axes indicated; the angle $\theta$ is measured from the $z$-axis to a point on the loop in the $xz$-plane. | ||
21 | As in Fig. 4.17, two coaxial cylindrical conductors with $r_{1}$ and $r_{2}$ form a waveguide. The region between the conductors is vacuum for $z<0$ and is filled with a dielectric medium with dielectric constant $\varepsilon \neq 1$ for $z>0$. Describe the TEM mode for $z<0$ and $z>0$. | For $z>0$: $\mathbf{E}^{\prime}(x,t)=\frac{C}{r}e^{i(k^{\prime} x-\omega t)}\mathbf{e}_{r}$ and $\mathbf{B}^{\prime}(x,t)=\frac{C\sqrt{\varepsilon}}{rc}e^{i(k^{\prime} x-\omega t)}\mathbf{e}_{\theta}$; for $z<0$: $\mathbf{E}(x,t)=\frac{A}{r}e^{i(kz-\omega t)}\mathbf{e}_{r}$ and $\mathbf{B}(x,t)=\frac{A}{rc}e^{i(kz-\ome... | Interpret $\varepsilon$ as the relative dielectric constant (with permittivity $=\varepsilon\varepsilon_{0}$) and use SI units. For sinusoidal waves, $\frac{\partial}{\partial t}\rightarrow -i\omega$, making the wave equation $\left(\nabla^{2}+\varepsilon\frac{\omega^{2}}{c^{2}}\right)\{\mathbf{E}^{\prime}/\mathbf{B}^{... | 7 | EM | English | capacitance_resistance | essential | IMAGE 1:
A long cylindrical shell of inner radius $r_1$ and outer radius $r_2$ is oriented with its axis along the $z$-direction; the region $r_1 < r < r_2$ has permittivity $\epsilon = 1$, while the region outside ($r>r_2$) has permittivity $\epsilon > 1$. The $z$-axis and a circular Gaussian surface centered at $z = ... | ||
22 | Figure 1.10 shows two capacitors in series, the rigid center section of length $b$ being movable vertically. The area of each plate is $A$. If the voltage difference between the outside plates is kept constant at $V_{0}$, what is the change in the energy stored in the capacitors if the center section is removed? | $W-W'=\frac{A\varepsilon_{0}V_{0}^{2}}{2(a-b)}\frac{b}{a}$ | The total energy stored in the capacitor is $W=\frac{1}{2}CV_{0}^{2}=\frac{A\varepsilon_{0}V_{0}^{2}}{2(a-b)}$. The energy stored if the center section is removed is $W'=\frac{A\varepsilon_{0}V_{0}^{2}}{2a}$ and we have $W-W'=\frac{A\varepsilon_{0}V_{0}^{2}}{2(a-b)}\frac{b}{a}$. | 7 | EM | English | circuit_diagram | essential | IMAGE 1:
Schematic of a parallel plate capacitor with plate separation $a$ and dielectric slab of thickness $b$ inserted between the plates, connected in series with a voltage source labeled $V_0$; physical separations $a$ and $b$ are indicated by vertical dimension lines. | ||
23 | Suppose you have been supplied with a number of sheets of two types of optically active material. Sheets of type $P$ are perfect polarizers: they transmit (normally incident) light polarized parallel to some axis $\boldsymbol{n}$ and absorb light polarized perpendicular to $\boldsymbol{n}$. Sheets of type $Q$ are quart... | 0 | The circularly polarized light is reflected back through the $\lambda/4$ wave plate; the result is a plane polarized light, polarized at right angle to the transmission axis of the polarizer (A side). Therefore, no light is transmitted backwards through the combination.
| 7 | OPT | English | optical_path | essential | IMAGE 1:
A linearly polarized electromagnetic wave, with electric field initially along the $\hat{x}$ direction, passes through a polarizing filter; the transmitted wave emerges aligned with the $\hat{y}$ direction. The filter is represented as a rectangular barrier perpendicular to the wave propagation axis, with the ... | ||
24 | The potential curves for the ground electronic state (A) and an excited electronic state (B) of a diatomic molecule are shown in Fig. 8.2. Each electronic state has a series of vibrational levels which are labelled by the quantum number $\nu$. Some molecules were initially at the lowest vibrational level of the electro... | $\nu \approx 5$ | Electrons move much faster than nuclei in vibration. When an electron transits to another state, the distance between the vibrating nuclei remains practically unchanged. Hence the probability of an electron to transit to the various levels is determined by the electrons initial distribution probability. As the molecule... | 7 | AMONP | English | quantum_mechanics | essential | IMAGE 1:
Potential energy diagram showing two molecular potential energy curves labeled $A$ (lower) and $B$ (upper) as functions of internuclear distance $r$, with the energy axis labeled $E$; curve $A$ contains several discrete horizontal energy levels labeled 1 to 7, curve $B$ has an energy separation marked $\Delta_... | ||
25 | Assume all surfaces to be frictionless and the inertia of pulley and cord negligible (Fig. $1.6$). Find the horizontal force necessary to prevent any relative motion of $m_{1}$, $m_{2}$ and $M$. | $F=\frac{m_{2}\left(M+m_{1}+m_{2}\right)g}{m_{1}}$ | The forces $f_{1}, F$ and $m g$ are shown in Fig. $1.7$. The accelerations of $m_{1}, m_{2}$ and $M$ are the same when there is no relative motion among them. The equations of motion along the $x$-axis are $$\begin{array}{c} \left(M+m_{1}+m_{2}\right) \ddot{x}=F \\ m_{1} \ddot{x}=f_{1} \end{array}$$ As there is no rela... | 7 | CM | English | projectile_motion | essential | IMAGE 1:
A projectile is depicted launching from point $O$ with initial velocity $\vec{v}_1$ at an angle onto a stepped platform arrangement. The trajectory lands at horizontal distance $l$ and vertical height $l$ below, corresponding to velocity $\vec{v}_2$ shown at the impact point. Both $\vec{v}_1$ and $\vec{v}_2$ a... | ||
26 | Consider the circuit shown in Fig. 3.87. If one varies the frequency but not the amplitude of $V$, what is the maximum current that can flow? The minimum current? At what frequency will the minimum current be observed. | $\left(I_{0}\right)_{\max}=\frac{V_{0}}{R},\;\left(I_{0}\right)_{\min}=0,\; \omega=\frac{1}{\sqrt{L_{1}C_{1}}}$ | The complex current is $I=\frac{V}{Z}=\frac{V}{R+j\left(\omega L-\frac{1}{\omega C}+\frac{\omega L_{1}}{1-\omega^{2}L_{1}C_{1}}\right)}$. So its amplitude is $I_{0}=\frac{V_{0}}{\left[R^{2}+\left(\omega L-\frac{1}{\omega \bar{C}}+\frac{\omega L_{1}}{1-\omega^{2}L_{1}C_{1}}\right)^{2}\right]^{1/2}}$, where $V_{0}$ is th... | 7 | EM | English | circuit_diagram | essential | IMAGE 1:
Schematic of an electrical circuit showing, in series from left to right, an inductor $L$, resistor $R$, and capacitor $C$, followed by a parallel branch containing a capacitor $C_1$ and an inductor $L_1$. All components are represented by standard symbols and are connected between two terminals. | ||
27 | A $\pi$ meson with a momentum of $5 m_{\pi} c$ makes an elastic collision with a proton ($m_{p}=7 m_{\pi}$) which is initially at rest (Fig. 3.20). Find the momentum of the incident pion in the c.m. system. | $3.18 m_{\pi} c$ | Solution: (c) The total energy in the c.m. frame is $$E^{\prime}=\sqrt{p_{\pi}^{\prime 2}+m_{\pi}^{2} c^{4}}+\sqrt{p_{p}^{\prime 2}+m_{p}^{2} c^{4}},$$ $$=\sqrt{p_{\pi}^{\prime 2}+m_{\pi}^{2} c^{4}}+\sqrt{p_{\pi}^{\prime 2}+49 m_{\pi}^{2} c^{4}},$$ since $|\mathbf{p}_{p}^{\prime}|=|\mathbf{p}_{\pi}^{\prime}|$ in the c.... | 3 | 7 | AMONP | English | static_force_analysis | essential | IMAGE 1:
Feynman diagram showing a $\rho$ meson as an internal vertex, with an incoming $\pi$ meson and two outgoing lines: one $\pi$ meson and one $\rho$ meson; particle trajectories are indicated by directed lines labeled by their particle symbols ($\pi$, $\rho$), with all momenta implicitly following the directions ... | |
28 | A mass $M$ is constrained to slide without friction on the track $AB$ as shown in Fig. 2.24. A mass $m$ is connected to $M$ by a massless inextensible string. (Make small angle approximation.) Write a Lagrangian for this system. | $L = \frac{1}{2} M \dot{x}^{2} + \frac{1}{2} m \left( \dot{x}^{2} + b^{2}\dot{\theta}^{2} + 2b\dot{x}\dot{\theta}\cos\theta \right) + m g b\cos\theta$ | Use coordinates as shown in Fig. 2.24. $M$ and $m$ have coordinates $(x,0)$ and $(x+b\sin \theta, -b\cos \theta)$ respectively. The Lagrangian is then $L=T-V=\frac{1}{2} M \dot{x}^{2}+\frac{1}{2} m\left(\dot{x}^{2}+b^{2} \dot{\theta}^{2}+2b\dot{x}\dot{\theta}\cos \theta\right)+m g b\cos \theta$. | 7 | CM | English | simple_harmonic_motion | essential | IMAGE 1:
Block of mass $M$ is placed on a horizontal frictionless surface along the $x$-axis (from $A$ to $B$), connected at its side to a pendulum consisting of a mass $m$ suspended by a rigid, massless rod of length $b$; the pendulum makes an angle $\theta$ from the vertical. The $x$ and $y$ axes are indicated, and a... | ||
29 | An ideal gas is contained in a large jar of volume $V_{0}$. Fitted to the jar is a glass tube of cross-sectional area $A$ in which a metal ball of mass $M$ fits snugly. The equilibrium pressure in the jar is slightly higher than atmospheric pressure $p_{0}$ because of the weight of the ball. If the ball is displaced sl... | $f = \frac{1}{2 \pi} \sqrt{\frac{\gamma A^{2}\left(p_{0}+\frac{m g}{A}\right)}{V m}}$ | Assume the pressure in the jar is $p$. As the process is adiabatic, we have\n\[np V^{\gamma}=\text { const }\]\ngiving\n\[\frac{d p}{p}+\gamma \frac{\partial V}{V}=0 .\]\n\nThis can be written as $F=A d p=-k x$, where $F$ is the force on the ball, $x=d V / A$ and $k=\gamma A^{2} p / V$. Noting that $p=p_{0}+m g / A$, w... | 7 | TSM | English | thermodynamics | optional | IMAGE 1:
A rigid-walled vessel is shown with a vertical tube that is sealed at the top by a frictionless movable piston. The piston is depicted as a horizontal circle fitting tightly within the tube, allowing vertical motion. No external forces or fields are illustrated. | ||
30 | Four identical coherent monochromatic wave sources A, B, C, D, as shown in Fig. 4.2 produce waves of the same wavelength $\lambda$. Two receivers $R_{1}$ and $R_{2}$ are at great (but equal) distances from $B$. Which receiver picks up the greater signal? | $R_{2}$ picks up greater signal | Let $r$ be the distance of $R_{1}$ and $R_{2}$ from $B$. We are given $r \gg \lambda$. Suppose the amplitude of the electric vector of the electromagnetic waves emitted by each source is $E_{0}$. The total amplitudes of the electric field at the receivers are $R_{1}: \; E_{10}= E_{0} \exp\left[i K\left(r-\frac{\lambda}... | 7 | OPT | English | charge_distribution | essential | IMAGE 1:
Points $A$, $B$, $C$, and $D$ are arranged such that $AB = BC = \frac{\lambda}{2}$ and $BD = \frac{\lambda}{2}$, forming perpendicular axes at $B$; points $R_1$ and $R_2$ are marked to the left of $A$ and below $B$, respectively, with all spatial separations labeled; arrows indicate the positive directions fro... | ||
31 | An object is placed 10 cm in front of a convering lens of focal length 10 cm . A diverging lens of focal length -15 cm is placed 5 cm behind the converging lens (Fig. 1.27). Find the character of the final image. | upright, virtual | The negative sign indicates that the image is to the left of $F^{\text{′}}$. Then the image is upright, virtual and magnified 1.5 times. | 7 | OPT | English | optical_path | essential | IMAGE 1:
A vertical object is placed on the left, 10 cm from a convex lens labeled with focal length $f_1 = 10\,\text{cm}$, followed by a concave lens labeled $f_2 = -15\,\text{cm}$ placed 5 cm to the right of the convex lens; all components are aligned along a common optical axis. | ||
32 | Find the electric quadrupole moment for two point charges of charge $e$ located at the ends of a line of length $2l$ that rotates with a constant angular velocity $\omega/2$ about an axis perpendicular to the line and through its center. | $Q_{11}=el^{2}[1+3\cos(\omega t')]$, $Q_{12}=Q_{21}=3el^{2}\sin(\omega t')$, $Q_{22}=el^{2}[1-3\cos(\omega t')]$. | The electric dipole moment is calculated as $\mathbf{P}=e\mathbf{r}_{1}'+e\mathbf{r}_{2}'=0$. The magnetic dipole moment is obtained by $\mathbf{m}=IS\mathbf{e}_{z}=\frac{2e}{T}(\pi l^{2})\mathbf{e}_{z}=\frac{1}{2}e\omega l^{2}\mathbf{e}_{z}$, which is constant. The position vectors of the two point charges are $\mathb... | 7 | EM | English | charge_distribution | essential | IMAGE 1:
Two point charges $e$ are located at the ends of straight wires of length $l$, symmetrically placed about the origin $O$ in the $xy$-plane. The wires form an angle $\frac{\omega t}{2}$ with the $x$-axis. Each charge $e$ has a velocity vector $\vec{v}$ tangential to the ends of the wires, as indicated by the ar... | ||
33 | A mirror is moving through vacuum with relativistic speed $v$ in the $x$ direction. A beam of light with frequency $\omega_{i}$ is normally incident (from $x=+\infty$) on the mirror, as shown in Fig. 3.9. What is the frequency of the reflected light expressed in terms of $\omega_{i}$, $c$ and $v$? | $\left(\frac{c+v}{c-v}\right)\omega_{i}$ | Let $\Sigma$, $\Sigma^{\prime}$ be the rest frames of the light source and observer, and of the mirror respectively. The transformation for angular frequency is given by $\omega^{\prime}=\gamma\left(\omega-\beta c k_{x}\right), \quad \omega=\gamma\left(\omega^{\prime}+\beta c k_{x}^{\prime}\right)$ where $\beta=\frac{v... | 7 | OPT | English | static_force_analysis | essential | IMAGE 1:
Two Cartesian coordinate systems, $\Sigma$ (with axes $x$, $y$) and $\Sigma'$ (with axes $x'$, $y'$), are shown, where $\Sigma'$ moves to the right relative to $\Sigma$ with velocity $v$ along the $x$-axis; a light ray is depicted as an arrow pointing in the positive $x$, $x'$ direction. | ||
34 | Two small spheres of mass $M$ are suspended between two rigid supports with three identical springs of spring constant $K$ (each of unstretched length $\frac{a}{2}$). Assuming small oscillations about the equilibrium configuration, find the frequencies for the two normal modes corresponding to vertical oscillations. | $\omega_{3}=\sqrt{\frac{K}{2M}}, \quad \omega_{4}=\sqrt{\frac{3K}{2M}}$ | For the vertical oscillations, after defining shifted coordinates $$y_{1}'=y_{1}+\frac{2Mg}{K}, \quad y_{2}'=y_{2}+\frac{2Mg}{K},$$ the two Lagrange equations become $$M\ddot{y}_{1}'+K y_{1}'-\frac{K}{2}y_{2}'=0,\quad M\ddot{y}_{2}'+K y_{2}'-\frac{K}{2}y_{1}'=0.$$ Trying a solution of the type $y_{i}'=B_i e^{i\omega t}... | 7 | CM | English | spring_force | essential | IMAGE 1:
Three identical point masses are attached at equal intervals of distance $a$ along a horizontal straight line, connected by four identical springs linked end-to-end, with the outer springs attached to fixed supports on both the left and right sides. | ||
35 | $1254$ A rope is attached at one end to a wall and is wrapped around a capstan through an angle $\theta$. If someone pulls on the other end with a force $F$ as shown in Fig. 1.231(a), find the tension in the rope at a point between the wall and the capstan in terms of $F$, $\theta$ and $\mu_{s}$, the coefficient of fri... | $T=F e^{-\mu_{s} \theta}$ | Consider an element of the rope as shown in Fig. 1.231(b). The forces acting on the element are the tensions $T$ and $T+\Delta T$ at its two ends, the reaction $N$ exerted by the capstan, and the friction $f$. As the element is in equilibrium we have $$\begin{array}{l} f+(T+\Delta T) \cos \left(\frac{\Delta \theta}{2}\... | 7 | CM | English | circular_motion | essential | IMAGE 1:
Figure (a) shows a rope wound around a fixed circular cylinder, with the rope experiencing an external force $F$ at an angle $\theta$ relative to the contact point; tension $T$ is queried at the attachment point to the wall. Figure (b) depicts the contact region over a differential angle $\Delta\theta$, labeli... | ||
36 | A car is traveling in the $x$-direction and maintains constant horizontal speed $v$. The car goes over a bump whose shape is described by $y_{0}=A[1-\cos (\pi x / l)]$ for $0 \leq x \leq 2 l ; y_{0}=0$ otherwise. Determine the motion of the center of mass of the car while passing over the bump. Represent the car as a m... | $y(t)=C_{1}\cos (\omega t)+B \cos \left(\frac{\pi v t}{l}\right)+A+l_{0}-\frac{m g}{k}$ with $\omega=\sqrt{\frac{k}{m}},\ C_{1}=-\frac{m\pi^{2}v^{2}A}{m\pi^{2}v^{2}-kl^{2}},\ B=\frac{kl^{2}A}{m\pi^{2}v^{2}-kl^{2}}$ | Solution: Let the location of the mass at time $t$ be $(x, y)$. Choose the origin so that $x(0)=0$. Then $x(t)=vt$. The equation of the motion of the mass in the $y$-direction is $$\begin{aligned} m\ddot{y} &= -k\left(y-y_{0}-l_{0}\right)-mg \\ &= -k\left(y-A-l_{0}+\frac{mg}{k}\right)-kA\cos\left(\frac{\pi vt}{l}\right... | 7 | CM | English | static_force_analysis | optional | IMAGE 1:
A block of mass $m$ is attached to a vertical spring at point $O$ on the $y$-axis and is launched horizontally along the $x$-axis with velocity $v$ from the spring’s equilibrium position. The block moves over a smooth, hump-shaped surface, reaching a height before descending, with its position indicated at $2l... | ||
37 | Consider the ground state and \(n=2\) states of hydrogen atom. There are four corrections to the indicated level structure that must be considered to explain the various observed splitting of the levels. These corrections are: (a) Lamb shift, (b) fine structure, (c) hyperfine structure, (d) relativistic effects. Which ... | fine structure, hyperfine structure | For the $n=2$ state $(l=0$ and $l=1)$, the fine-structure correction causes the most splitting in the $l=1$ level, to which the hyperfine structure correction also contributes (see Fig. 1.17). | 7 | AMONP | English | atomic_physics | optional | IMAGE 1:
Diagram shows two sets of horizontal energy levels labeled $n=1$ (single line, lower position) and $n=2$ (three parallel lines, upper position) within a rectangular boundary, indicating distinct quantum states. No arrows, forces, connections, or other entities are present; only energy level positions and their... | ||
38 | Two uniform cylinders are spinning independently about their axes, which are parallel. One has radius $R_{1}$ and mass $M_{1}$, the other $R_{2}$ and $M_{2}$. Initially they rotate in the same sense with angular speeds $\Omega_{1}$ and $\Omega_{2}$ respectively as shown in Fig. 1.128. They are then displaced until they... | $\omega_{1}=\frac{M_{1} R_{1} \Omega_{1}-M_{2} R_{2} \Omega_{2}}{R_{1}(M_{1}+M_{2})}$, $\omega_{2}=\frac{M_{2} R_{2} \Omega_{2}-M_{1} R_{1} \Omega_{1}}{R_{2}(M_{1}+M_{2})}$ | $\omega_{1} R_{1}=-\omega_{2} R_{2}$\n\nLet $J_{1}$ and $J_{2}$ be the time-integrated torque 2 exerts on 1 and 1 on 2, then\n\n$\frac{J_{1}}{R_{1}}=\frac{J_{2}}{R_{2}}$\n\n$J_{1}=I_{1}(\omega_{1}-\Omega_{1}), \quad J_{2}=I_{2}(\omega_{2}-\Omega_{2})$\n\nor\n\n$\frac{I_{1}(\omega_{1}-\Omega_{1})}{R_{1}}=\frac{I_{2}(\om... | 7 | CM | English | circular_motion | optional | IMAGE 1:
Two uniformly rotating disks, one of radius $R_1$ and angular velocity $\Omega_1$ (left), and the other of radius $R_2$ and angular velocity $\Omega_2$ (right), are shown with their rotation axes perpendicular to the page; both disks have arrows indicating counterclockwise angular velocity vectors. | ||
39 | Consider the Babinet Compensator shown in the figure (Fig.~2.76). The device is constructed from two pieces of uniaxial optical material with indices $n_{e}$ and $n_{o}$ for light polarized perpendicular and parallel to the optic axis, respectively. A narrow beam of light of vacuum wavelength $\lambda$ is linearly pola... | $\frac{4 \pi}{\lambda}\left(n_{\mathrm{o}}-n_{\mathrm{e}}\right) \frac{x d}{L}$ | . As the incident light is polarized at $45^\circ$ to X- and Z-axes, it is equivalent to two components, polarized in $X$ and $Z$ directions, of amplitudes
$$
E_x = E_z = E/\sqrt{2}
$$
The relative phase shifts of the two components of the beam resulting from traversing the left prism and the right prism are respectiv... | 7 | OPT | English | optical_path | essential | IMAGE 1:
Diagram shows a rectangular birefringent crystal of length $L$ with optic axis OA, aligned either parallel to the $X$-axis in the plane of the paper (↑ OA) or perpendicular to the paper along the $Z$-axis (• OA); a narrow beam of laser light enters from the left at a lateral distance $x$ from the lower edge an... | ||
40 | Figure 2.75 shows two long parallel wires carrying equal and opposite steady currents $I$ and separated by a distance $2a$. Find an expression for the magnetic field strength at a point in the median plane (i.e. $xz$ plane in Fig. 2.75) lying a distance $z$ from the plane containing the wires. | $\mathbf{B}=-\frac{\mu_{0}I}{\pi}\frac{a}{(z^2+a^2)}\mathbf{e}_z$ | Suppose the long wires carrying currents $+I$ and $-I$ cross the $y$ axis at $+a$ and $-a$ respectively. Consider an arbitrary point $P$ and without loss of generality we can take the $yz$ plane to contain $P$. Let the distances of $P$ from the $y$- and $z$-axes be $z$ and $y$ respectively, and its distances from the w... | 7 | EM | English | electromagnetic_field | essential | IMAGE 1:
Three long straight wires are arranged in the $xy$-plane, extending along the $y$, $z$, and $-x$ axes, respectively, each carrying a current $I$ directed away from the origin; the perpendicular distance between the non-parallel wires is labeled $2a$. | ||
41 | A thin ring of mass $M$ and radius $r$ lies flat on a frictionless table. It is constrained by two extended identical springs with relaxed length $l_{0}$ ($l_{0} \gg r$) and spring constant $k$ as shown in Fig. 1.55. What are the normal modes of small oscillations and their frequencies? | $\omega_{x}=\sqrt{\frac{2k}{M}},\ \omega_{y}=\sqrt{\frac{k}{M}}$ | As $l_{0} \gg r$, any rotation of the ring will cause a negligible change of length in the springs, and any elastic force so arising is also negligible. Newton's second law then gives $M\ddot{x}=-k\left[\sqrt{(2l_{0}+x)^2+y^2}-l_{0}\right]\frac{2l_{0}+x}{\sqrt{(2l_{0}+x)^2+y^2}}+k\left[\sqrt{(2l_{0}-x)^2+y^2}-l_{0}\rig... | 7 | CM | English | spring_force | essential | IMAGE 1:
A ring of radius $r$ is connected between two identical horizontal springs with spring constants $k$, each attached to fixed walls; the unstretched lengths from each wall to the ring are $2\ell_0$, and the ring has a distance $2r$ between the spring endpoints. No external forces or other vectors are shown. | ||
42 | An electric circuit consists of two resistors (resistances $R_{1}$ and $R_{2}$), a single condenser (capacitor $C$) and a variable voltage source $V$ joined together as shown in Fig. 3.83. When $V(t)$ is a very sharp pulse at $t=0$, we approximate $V(t)=A\delta(t)$. What is the time history of the potential drop across... | $V_{1}\propto \exp\left(-\frac{R_{1}+R_{2}}{CR_{1}R_{2}}t\right)$ for $t>0$ and $V_{1}=0$ for $t<0$ | When $V(t)=A\delta(t)$, use the relation $\delta(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\omega t}d\omega$. The voltage drop across $R_{1}$ is written as $V_{1}=\frac{A}{2\pi}\int_{-\infty}^{\infty}\frac{R_{1}(R_{2}-\frac{i}{\omega C})}{R_{1}R_{2}-\frac{i}{\omega C}(R_{1}+R_{2})}e^{i\omega t}d\omega=\frac{A}{2\pi}\... | 7 | EM | English | circuit_diagram | essential | IMAGE 1:
A circuit diagram showing an AC voltage source labeled $V$ connected in series with resistor $R_1$ and capacitor $C$; resistor $R_2$ is connected to the right of $C$, forming a second branch. Currents $I_1$ and $I_2$ are indicated by curved arrows, with $I_1$ circulating clockwise through $R_1$ and $C$, and $I... | ||
43 | A very long conducting pipe has a square cross section of its inside surface, with side $D$ as in Fig. 1.47. Far from either end of the pipe is suspended a point charge located at the center of the square cross section. Determine the electric potential at all points inside the pipe, perhaps in the form of an infinite s... | $\varphi(x, y, z)= \frac{2 Q}{\pi \varepsilon_{0} D} \sum_{m, m^{\prime}=0}^{\infty} \frac{\cos\frac{(2m+1)\pi x}{D}\cos\frac{(2m^{\prime}+1)\pi y}{D}}{\sqrt{(2m+1)^2+(2m^{\prime}+1)^2}}\; e^{-\frac{x \mid 21}{D}\sqrt{(2m+1)^2+(2m^{\prime}+1)^2}}$ | Poisson's equation for the potential and the boundary conditions can be written as follows:\n$$\n\left.\begin{array}{l}\n\nabla^{2} \varphi=-\frac{Q}{\varepsilon_{0}} \delta(x) \delta(y) \delta(z), \\\n\left.\varphi\right|_{x=\pm D/2}=0, \\\n\left.\varphi\right|_{y=\pm D/2}=0.\n\end{array}\right\}\n$$\nBy Fourier trans... | 7 | EM | English | capacitance_resistance | essential | IMAGE 1:
A point charge $Q$ is located midway between two parallel conducting rails separated by distance $D$; the top view shows the charge $Q$ positioned equidistantly between the rails, with coordinate axes $x$ and $y$ in the plane of the rails and the $z$-axis oriented along the rails’ length (shown by the arrow). | ||
44 | In Fig. 5.10 a point charge $e$ moves with constant velocity $v$ in the $z$ direction so that at time $t$ it is at the point $Q$ with coordinates $x=0$, $y=0, z=v t$. Find at the time $t$ and at the point $P$ with coordinates $x=b, y=0, z=0$ the vector potential $\mathbf{A}$. | $\mathbf{A}=\frac{e v}{4 \pi \varepsilon_{0} c^{2} \sqrt{\left(1-\beta^{2}\right) b^{2}+v^{2} t^{2}}} \mathbf{e}_{z}$ | Solution:\n(b) The vector potential $\mathbf{A}$ is\n$$\n\mathbf{A}=\frac{e v}{4 \pi \varepsilon_{0} c^{2} \sqrt{\left(1-\beta^{2}\right) b^{2}+v^{2} t^{2}}} \mathbf{e}_{z}.\n$$ | 7 | EM | English | charge_distribution | essential | IMAGE 1:
Three-dimensional coordinate axes ($x$, $y$, $z$) are shown with two points: $P$ at position $(0, b, 0)$ along the $x$-axis, and $Q$ at $(0, 0, vt)$ along the $z$-axis; a velocity vector $\vec{v}$ points in the positive $z$-direction from the origin. | ||
45 | In the circuit shown in Fig. 3.23, the resistance of $L$ is negligible and initially the switch is open and the current is zero. Find the quantity of heat dissipated in the resistance $R_{2}$ when the switch is closed and remains closed for a long time. (Notice the circuit diagram and the list of values for $V, R_{1}, ... | $45.5\;\mathrm{J}$ | Consider a resistance $R$ and an inductance $L$ in series with a battery of emf $\varepsilon$. We have $\varepsilon - L \frac{d I}{d t} = I R$ or $\frac{-R d I}{\varepsilon - I R} = -R \frac{d t}{L}$. Integrating we have $\ln [\varepsilon - I(t) R] = -\frac{t}{\tau} + K$ where $\tau = \frac{L}{R}$ and $K$ is a constant... | 3 | 7 | EM | English | circuit_diagram | essential | IMAGE 1:
Schematic diagram of an electric circuit with a voltage source $V = 100~\text{V}$ connected in series to a resistor $R_1 = 10~\Omega$, followed by a parallel branch comprising a resistor $R_2 = 100~\Omega$ and an inductor $L = 10~\text{H}$ in series; all components are labeled with their values. | |
46 | A perfectly uniform ball 20 cm in diameter and with a density of $5\,\mathrm{g}/\mathrm{cm}^{3}$ is rotating in free space at $1\,\mathrm{rev}/\mathrm{s}$. An intelligent flea of $10^{-3}\,\mathrm{g}$ resides in a small (massless) house fixed to the ball's surface at a rotational pole as shown in Fig. 1.197. The flea d... | $6 \times 10^{6}\,\mathrm{s}$ | After the flea moves to a position of latitude $45^{\circ}$, the angular velocity $\omega$ no longer coincides with a principal axis of the system. This causes the ball to precess. As the mass of the flea is much smaller than that of the ball, the center of mass of the system can be taken to be at the center of the bal... | 1 | 7 | CM | English | circular_motion | optional | IMAGE 1:
A circular ring is shown with its vertical axis depicted as a dashed line; an angular velocity vector labeled $\omega$ points counterclockwise around this axis, indicating rotational motion of the ring about the vertical axis. | |
47 | Consider a square loop of wire, of side length $l$, lying in the $xy$ plane as shown in Fig. 2.43. Suppose a particle of charge $q$ is moving with a constant velocity $v$, where $v \ll c$, in the $xz$-plane at a constant distance $z_{0}$ from the $xy$-plane. (Assume the particle is moving in the positive $x$ direction.... | $-\frac{\mu_{0}qv^{2}}{4\pi}\left\{\frac{1}{\sqrt{v^{2}t^{2}+(z-z_{0})^{2}}}-\frac{1}{\sqrt{(l-vt)^{2}+(z-z_{0})^{2}}}-\frac{1}{\sqrt{(l-vt)^{2}+l^{2}+(z-z_{0})^{2}}}+\frac{1}{\sqrt{v^{2}t^{2}+l^{2}+(z-z_{0})^{2}}}\right\}$ | At time $t$, the position of $q$ is $\left(vt,0,z_{0}\right)$. The radius vector $\mathbf{r}$ from $q$ to a field point $(x,y,z)$ is $\left(x-vt,y,z_{0}\right)$. As $v\ll c$, the electromagnetic field due to the uniformly moving charge is given by $\mathbf{E}(\mathbf{r},t)=\frac{q}{4\pi\varepsilon_{0}}\frac{\mathbf{r}}... | 7 | EM | English | electromagnetic_field | essential | IMAGE 1:
A rectangular loop of length $l$ lies in the $xy$-plane, with sides aligned along the $x$ and $y$ axes. A point labeled $z_0$ is marked on the $z$-axis perpendicular to the plane. A particle of charge $q$ moves with velocity $\vec{v}$ parallel to the $xz$-plane. The directions of $x$, $y$, and $z$ axes are ind... | ||
48 | Consider an intrinsic semiconductor whose electronic density of states function $N(E)$ is depicted in Fig. Where is the Fermi level with respect to the valence and conduction bands? | $\varepsilon_{\mathrm{F}}=\frac{\varepsilon_{\mathrm{C}}+\varepsilon_{\mathrm{V}}}{2}$ | The number of electrons in the conduction band, whose bottom is $\n\varepsilon_{\mathrm{C}}$, and the number of holes in the valence band, whose top is $\n\varepsilon_{\mathrm{V}}$, are respectively\n\[\n\begin{array}{l}\nn=\int_{\varepsilon_{\mathrm{C}}}^{\infty} N(E) f(E) d E \\np=\int_{-\infty}^{\varepsilon_{\mathrm... | 7 | AMONP | English | atomic_physics | essential | IMAGE 1:
Energy band diagram showing density of states $N(E)$ on the vertical axis and energy $E$ on the horizontal axis; two regions are labeled: the valence band (shaded, centered at $E = -6.5\,\text{eV}$), and the conduction band (at $E = -3\,\text{eV}$), separated by an energy gap of $1.5\,\text{eV}$; the maximal d... | ||
49 | Four identical masses are connected by four identical springs and constrained to move on a frictionless circle of radius $b$ as shown in Fig. 2.30. What are the frequencies of small oscillations? | $\sqrt{\frac{k}{m}}$ and $\sqrt{\frac{2 k}{m}}$ | The $T$ and $V$ matrices are $\mathbf{T}=\begin{pmatrix} m & 0 & 0 & 0 \\ 0 & m & 0 & 0 \\ 0 & 0 & m & 0 \\ 0 & 0 & 0 & m \end{pmatrix}$, $\mathbf{V}=\begin{pmatrix} k & -\frac{k}{2} & 0 & -\frac{k}{2} \\ -\frac{k}{2} & k & -\frac{k}{2} & 0 \\ 0 & -\frac{k}{2} & k & -\frac{k}{2} \\ -\frac{k}{2} & 0 & -\frac{k}{2} & k \... | 7 | CM | English | spring_force | essential | IMAGE 1:
A loop of wire (square-shaped with coiled segments) is positioned inside a circle, with each corner of the square loop touching the circumference of the circle. No labels, vectors, or additional symbols are shown. | ||
50 | The betatron accelerates particles through the emf induced by an increasing magnetic field within the particle's orbit. Let $\bar{B}_{1}$ be the average field within the particle orbit of radius $R$, and let $B_{2}$ be the field at the orbit (see Fig. 2.80). What must be the relationship between $\bar{B}_{1}$ and $B_{2... | $B_{2}=\frac{\bar{B}_{1}}{2}$ | Suppose the magnetic field is oriented in the $z$ direction, i.e., $\mathbf{B}_{2}=B_{2}\,\mathrm{e}_{x}$. From $\nabla \times \mathbf{E}=-\frac{\partial \mathrm{B}}{\partial t}$, where $\frac{\partial \mathrm{B}}{\partial t}>0$, we see that the electric field is along the $-\mathbf{e}_{\theta}$ direction and has axial... | 7 | EM | English | electromagnetic_field | essential | IMAGE 1:
A charged particle beam travels along the $z$-axis through a solenoidal region of uniform magnetic field $\vec{B}_1$ (direction along $+z$) of radius $R$, flanked by regions of magnetic field $\vec{B}_2$ with the same direction. The setup includes a cylindrical coordinate system $(x, y, z)$ with associated uni... | ||
51 | The conductors of a coaxial cable are connected to a battery and resistor as shown in Fig. 2.15. Starting from first principles find, in the region between $r_{1}$ and $r_{2}$, the Poynting vector. | $\mathbf{N}=\frac{V^{2}}{2 \pi r^{2} R \ln \frac{r_{2}}{r_{1}}}\mathbf{e}_{z}$ | $\mathbf{N}=\mathbf{E}\times\mathbf{H}=\mathbf{E}\times\frac{\mathbf{B}}{\mu_{0}}=\frac{V}{r \ln \frac{r_{2}}{r_{1}}}\mathbf{e}_{\mathrm{r}}\times\frac{V}{2\pi r R}\mathbf{e}_{\theta}=\frac{V^{2}}{2\pi r^{2} R \ln \frac{r_{2}}{r_{1}}}\mathbf{e}_{z}$. | 7 | EM | English | capacitance_resistance | essential | IMAGE 1:
A cylindrical conductor with inner radius $r_1$ and outer radius $r_2$ is connected in series with a resistor $R$ and a voltage source $V$; a cross-sectional view shows the inner and outer cylindrical surfaces labeled $r_1$ and $r_2$, with current-carrying wires attached at each end. | ||
52 | Two infinite parallel wires separated by a distance $d$ carry equal currents $I$ in opposite directions, with $I$ increasing at the rate $\frac{dI}{dt}$. A square loop of wire of length $d$ on a side lies in the plane of the wires at a distance $d$ from one of the parallel wires, as illustrated in Fig. 2.30. Find the e... | $\varepsilon=-\frac{\mu_{0}d}{2\pi}\ln\left(\frac{4}{3}\right)\frac{dI}{dt}$ | The magnetic field produced by an infinite straight wire carrying current $I$ at a point distance $r$ from the wire is given by $B=\frac{\mu_{0}I}{2\pi r}$. Thus the magnetic flux crossing the loop due to the wire farther away from the loop is $\phi_{1}=\int_{2d}^{3d}\frac{\mu_{0}Id}{2\pi r}\,dr=\frac{\mu_{0}Id}{2\pi}\... | 7 | EM | English | circuit_diagram | essential | IMAGE 1:
Two parallel, horizontal straight wires labeled 1 and 2 carry currents $I$ to the right; they are separated vertically by a distance $d$. A square loop of side length $d$ is placed below wire 2 such that its top edge is at a distance $d$ vertically beneath wire 2 and its left edge is at a distance $d$ horizont... | ||
53 | Two uniform discs in a vertical plane of masses $M_{1}$ and $M_{2}$ with radii $R_{1}$ and $R_{2}$ respectively have a thread wound about their circumferences, and are thus connected as shown in Fig. 1.158. The first disc has fixed frictionless horizontal axis of rotation through its center. Set up the equations to det... | $M_{2}\ \ddot{x}=M_{2}g-F$, $I_{1}\ \ddot{\theta}_{1}=FR_{1}$, $I_{2}\ \ddot{\theta}_{2}=FR_{2}$, $\ddot{x}=R_{1}\ \ddot{\theta}_{1}+R_{2}\ \ddot{\theta}_{2}$, with $I_{1}=\frac{m_{1}R_{1}^{2}}{2}$ and $I_{2}=\frac{m_{2}R_{2}^{2}}{2}$ | Let $F$ be the tension in the thread, $x_{1}$ the distance of the center of mass of disc 2 from that of disc 1, and $\dot{\theta}_{1},\dot{\theta}_{2}$ the angular velocities of the discs, as shown in Fig. 1.158. We have the equations of motion $$ M_{2}\ \ddot{x}=M_{2}g-F,\quad I_{1}\ \ddot{\theta}_{1}=FR_{1},\quad I_{... | 7 | CM | English | circular_motion | essential | IMAGE 1:
Two solid cylinders of masses $M_1$ and $M_2$ and radii $R_1$ and $R_2$ are connected by a massless string that passes over both, with a vertical section of length $x$ between them; the string applies equal tensions $F$ upwards and downwards. Each cylinder rotates, with angular velocities $\dot{\theta}_1$ (clo... | ||
54 | In Fig. 3.37 the capacitor is originally charged to a potential difference $V$. The transformer is ideal: no winding resistance, no losses. At $t=0$ the switch is closed. Assume that the inductive impedances of the windings are very large compared with $R_{p}$ and $R_{s}$. Calculate: The initial secondary current. | $i_{s}(0)=\frac{N_{p} N_{s} V}{N_{s}^{2} R_{p}+N_{p}^{2} R_{s}$ | Using the result from part (a), $i_{p}(0)=\frac{V}{R_{p}+\left(\frac{N_{p}}{N_{s}}\right)^2 R_{s}}$, the initial secondary current is computed from $i_{s}(0)=i_{p}(0)\frac{N_{p}}{N_{s}}$, which gives $i_{s}(0)=\frac{N_{p} N_{s} V}{N_{s}^{2} R_{p}+N_{p}^{2} R_{s}}$. | 7 | EM | English | circuit_diagram | essential | IMAGE 1:
A circuit diagram showing a transformer with primary coil $N_p$ and series resistor $R_p$ connected to a voltage source $V$ and capacitor $C$ via a closed switch; the secondary coil $N_s$ forms a separate loop with resistor $R_s$; primary and secondary coils are magnetically coupled, with no vector or current ... | ||
55 | A rectangular coil of dimensions $a$ and $b$ and resistance $R$ moves with constant velocity $v$ into a magnetic field $\mathbf{B}$ as shown in Fig. 2.36. Derive an expression for the vector force on the coil in terms of the given parameters. | $-\frac{v b^{2} B^{2}}{R}$ | As it starts to cut across the magnetic field lines, an emf is induced in the coil of magnitude $\varepsilon=-\int \mathbf{B} \times \mathbf{v} \cdot d \mathbf{l}=-B v b$ and produces a current of $I=\frac{\varepsilon}{R}=-\frac{B v b}{R}$. The minus sign indicates that the current flows counterclockwise. The force on ... | 7 | EM | English | electromagnetic_field | optional | IMAGE 1:
A rectangular loop with horizontal length $b$ and vertical height $a$ is positioned such that its lower edge is inside a uniform magnetic field $\vec{B}$ (indicated by crosses, pointing into the page), while the upper edge is outside the field; all dimensions are labeled. | ||
56 | A parallel-plate capacitor is made of circular plates as shown in Fig. 2.10. The voltage across the plates (supplied by long resistanceless lead wires) has the time dependence $V=V_{0} \cos \omega t$. Assume $d \ll a \ll c / \omega$, so that fringing of the electric field and retardation may be ignored. What current fl... | $I=-\frac{\pi a^{2}\varepsilon_{0}V_{0}\omega}{d}\sin\omega t,\quad j_{r}(r)=\frac{(a^{2}-r^{2})\varepsilon_{0}V_{0}\omega}{2dr}\sin\omega t\,e_{r}$ | Let $\sigma$ be the surface charge density of the upper plate which is the interface between regions I and II. We have $$\sigma=-\varepsilon_{0}E_{z}^{(\mathrm{I})}=\frac{\varepsilon_{0}V_{0}}{d}\cos\omega t.$$ Then the total charge on the plate is $$Q=\pi a^{2}\sigma=\frac{\pi a^{2}\varepsilon_{0}V_{0}}{d}\cos\omega t... | 7 | EM | English | capacitance_resistance | essential | IMAGE 1:
A conducting sphere of radius $a$ is positioned above two parallel conducting plates (labeled (II) and (III)), separated by distance $d$, with plate (II) below plate (III); a uniform electric field $\vec{E}$ is shown directed downward, perpendicular to the plates. The outward normal vector $\vec{n}$ is indicat... | ||
57 | A ray of light enters a spherical drop of water of index $n$ as shown (Fig. 1.21). Find the angle $\phi$ which produces minimum deflection. | $\cos ^{2} \phi = \frac{n^{2}-1}{3}$ | For minimum deflection, we require $\frac{d \delta}{d \phi} = -4 \frac{d \alpha}{d \phi} + 2 = 0$, or $\frac{d \alpha}{d \phi} = \frac{1}{2}$.
As
$$
\alpha = \sin^{-1}\left(\frac{1}{n} \sin \phi\right),
$$
we have $\frac{d \alpha}{d \phi} = \frac{1}{n} \frac{\cos \phi}{\cos \alpha}$ and the above gives
$$
1 - \frac{1}... | 7 | OPT | English | optical_path | essential | IMAGE 1:
A circular object is shown at the center, with solid radial lines extending from its edge to a point labeled $\delta$ outside the circle; three dashed lines representing rays or lines of sight pass through the circle to $\delta$, defining an angle $\alpha$ at the circle's center, while another angle $\varphi$ ... | ||
58 | Consider a particle of mass m moving in a time-dependent potential $V(x, t)$ in one dimension. Write down the Schrödinger equations appropriate for two reference systems $(x, t)$ and $(x', t)$ moving with respect to each other with velocity $v$ (i.e. $x=x'+vt$). | For the $(x,t)$ system: $\left[-\frac{\hbar^{2}}{2m}\frac{d^2}{dx^2}+V(x,t)\right]\psi(x,t)=i\hbar\frac{\partial}{\partial t}\psi(x,t)$; for the $(x',t)$ system: $\left[-\frac{\hbar^{2}}{2m}\frac{d^2}{dx'^2}+V'(x',t)\right]\psi(x',t)=i\hbar\frac{\partial}{\partial t}\psi(x',t)$ where $V'(x',t)=V(x'-vt,t)$ | Both $(x,t)$ and $(x',t)$ are inertial systems, and so the Schrödinger equations are: for the $(x,t)$ system, \n\[\n\left[-\frac{\hbar^{2}}{2m}\frac{d^2}{dx^2}+V(x,t)\right]\psi(x,t)=i\hbar\frac{\partial}{\partial t}\psi(x,t)\,,\n\]\nfor the $(x',t)$ system, \n\[\n\left[-\frac{\hbar^{2}}{2m}\frac{d^2}{dx'^2}+V'(x',t)\r... | 7 | QMIT | English | quantum_mechanics | essential | IMAGE 1:
A graph of potential energy $V(x)$ versus position $x$, depicting a parabolic (harmonic oscillator) potential well for time $t < 0$. The $x$-axis represents position, and the $V(x)$-axis represents potential energy; both axes are indicated by arrows pointing to the right ($x$) and upward ($V(x)$). | ||
59 | Consider a rectangular waveguide, infinitely long in the $x$-direction, with a width ($y$-direction) 2 cm and a height ($z$-direction) 1 cm. The walls are perfect conductor, as in Fig. 4.13. For the lowest mode that can propagate, find the phase velocity and the group velocity. | Phase velocity: $v=\frac{\omega}{\sqrt{\frac{\omega^{2}}{c^{2}}-\frac{\pi^{2}}{4}}}>c$, Group velocity: $v_{g}=\frac{c^{2}}{v}$. | For the lowest mode, the wave can be represented by $$E_{z}=Y(y)Z(z)e^{i\left(k' x-\omega t\right)}.$$ Helmholtz's equation separates into $$\frac{d^{2}Y}{dy^{2}}+k_{1}^{2}Y=0,\quad \frac{d^{2}Z}{dz^{2}}+k_{2}^{2}Z=0,$$ with $$k_{1}^{2}+k_{2}^{2}=k^{2}-k'^{2}.$$ The solutions are $$\begin{array}{l} Y=A_{1}\cos\left(k_{... | 7 | EM | English | capacitance_resistance | optional | IMAGE 1:
A rectangular plate is positioned in the $xy$-plane with one corner at the origin ($0$), extending $2\,\mathrm{cm}$ along the $y$-axis and $1\,\mathrm{cm}$ along the $z$-axis; the $x$, $y$, and $z$ axes are shown with their positive directions indicated by arrows, and the orientation of the plate is parallel t... | ||
60 | Let us apply a shearing force on a rectangular solid block as shown in Fig. 2.77. Find the relation between the displacement $u$ and the applied force within elastic limits. | $u=l\varphi=\frac{lF}{An}$ | Hooke's law for shearing $\frac{F}{A}=n\varphi$, where $F$ is the shearing force, $n$ the shear modulus of the material of the block, $\varphi$ the shear angle, and $A$ the cross sectional area of the block parallel to $F$, gives the resulting displacement as $u=l\varphi=\frac{lF}{An}$ as $\varphi$ is a small angle. | 7 | AMONP | English | static_force_analysis | essential | IMAGE 1:
A uniform rod of length $l$ is suspended between two points on a horizontal surface at equal angles $\varphi$ to the vertical $y$-axis; a horizontal force $F$ acts to the right at the upper right attachment, and velocity $u$ is directed to the right; coordinate axes $x$ (vertical) and $y$ (horizontal) are show... | ||
61 | A plane monochromatic wave (wavelength $\lambda$ ) is incident on a set of 5 slits spaced at a distance $d$ (Fig. 2.55). You may assume that the width of the individual slits is much less than $d$. For the resulting interference pattern, which is focused on a screen, compute either analytically or approximately the fol... | $\frac{\lambda}{5 d}$ | Angular width of central image is $\theta \approx \frac{\lambda}{5 d}$. | 7 | OPT | English | optical_path | essential | IMAGE 1:
A parallel beam of light of wavelength $\lambda$ is incident on a diffraction grating with slit separation $d$; the diffracted beams at angle $\theta$ relative to the optical axis are focused by a convex lens (shown in side view) of focal length $f$ onto a screen at its focal plane. The $y$-axis is vertical, a... | ||
62 | Consider the ground state and \(n=2\) states of hydrogen atom. Indicate in the diagram (Fig. 1.14) the complete spectroscopic notation for all four states. There are four corrections to the indicated level structure that must be considered to explain the various observed splitting of the levels. These corrections are: ... | hyperfine structure | For the $n=1$ state $(l=0), E_{m}, E_{D}, E_{L}$ can only cause the energy level to shift as a whole. As $E_{\text {so }}=0$ also, the fine-structure correction does not split the energy level. On the other hand, the hyperfine structure correction can cause a splitting as shown in Fig. 1.16. | 7 | AMONP | English | atomic_physics | optional | IMAGE 1:
Energy level diagram showing two sets of horizontal lines labeled $n = 1$ (single line) and $n = 2$ (three closely spaced lines), representing possible quantum states, with no transitions or vectors indicated. | ||
63 | Consider the following energy level structure (Fig. 2.30): The ground states form an isotriplet as do the excited states (all states have a spin-parity of $0^{+}$). The ground state of $_{21}^{42}\text{Sc}$ can $\beta$-decay to the ground state of $_{20}^{42}\text{Ca}$ with a kinetic end-point energy of 5.4 MeV (transi... | $\left|\left\langle \alpha, 1 \left\| P_{1} \right\| \alpha^{\prime}, 1 \right\rangle\right|^{2} = 1.48 \times 10^{-3} \mathrm{MeV}^{2},\ \left|\left\langle \alpha, 1 \left\| P_{1} \right\| \alpha^{\prime}, 1 \right\rangle\right| = 38 \mathrm{keV}$ | 3 | 7 | AMONP | English | atomic_physics | essential | IMAGE 1:
Energy level diagram showing the $0.0~\mathrm{MeV}~(0^+)$ and $1.8~\mathrm{MeV}~(0^+)$ states for ${}^{42}_{20}\mathrm{Ca}$, ${}^{42}_{21}\mathrm{Sc}$, and ${}^{42}_{22}\mathrm{Ti}$; arrows labeled I and II indicate transitions from ${}^{42}_{20}\mathrm{Ca}$ to ${}^{42}_{21}\mathrm{Sc}$ at $0.0~\mathrm{MeV}~(0... | ||
64 | Find the expression for the speed of the transverse elastic wave. | $v=\sqrt{\frac{n}{\rho}}$ | The equation shows that $u$, which is in the $y$-direction, propagates along the $x$-direction as a transverse wave with speed $v=\sqrt{\frac{n}{\rho}}$. | 7 | CM | English | static_force_analysis | essential | IMAGE 1:
A uniform rod of length $l$ is suspended by two identical inclined supports forming angles $\varphi$ with the horizontal surface ($y$-axis), aligned parallel to the $x$-axis. A horizontal force $\vec{F}$, labeled $F$, acts to the right at the rod’s right end, where velocity $\vec{u}$ is also shown to the right... | ||
65 | According to the Weinberg-Salam model, the Higgs boson $\phi$ couples to every elementary fermion $f$ ( $f$ may be a quark or lepton) in the form
$$
\frac{e m_{f}}{m_{W}} \phi \bar{f} f,
$$
where $m_{f}$ is the mass of the fermion $f, e$ is the charge of the electron, and $m_{W}$ is the mass of the $W$ boson. Some theo... | $1.5 x 10^{-4}$ | The process $e^{+} e^{-} \rightarrow \bar{f} f$ consists of the following interactions:
$$
e^{+} e^{-} \xrightarrow{\gamma, Z^{0}} \bar{f} f \quad \text { and } \quad e^{+} e^{-} \xrightarrow{H} \bar{f} f .
$$
When $\sqrt{S}=10 \mathrm{GeV}, Z^{0}$ exchange can be ignored. Consider $e^{+} e^{-} \xrightarrow{\gamma} \b... | 2 | 7 | AMONP | English | feynman_diagram | essential | IMAGE 1:
Feynman diagram depicting an incoming electron ($e^-$) and positron ($e^+$) vertex annihilation, creating an intermediate virtual boson $\phi(k)$ (dashed line) with four-momentum $k$, decaying into an outgoing fermion $f$ with spinor $u(p)$ and four-momentum $p$, and antifermion $\bar{f}$ with spinor $v(q)$ an... | |
66 | Consider a two-dimensional classical system with Hamiltonian\n\nH=\frac{1}{2 m}\left(P_{1}^{2}+P_{2}^{2}\right)+\frac{1}{2} \mu^{2}\left(x_{1}^{2}+x_{2}^{2}\right)-\frac{1}{4} \lambda\left(x_{1}^{2}+x_{2}^{2}\right)^{2} .\n\nA system of $N$ particles of mass $m$ each is in thermal equilibrium at temperature $T$ within ... | $\frac{N \mu^{3}}{\sqrt{2 \pi m \lambda k T}} \cdot e^{-\frac{\mu^{4}}{6 k T \lambda}}$ | Putting $x_{2}=0$ and $\frac{\partial H}{\partial x_{1}}=0$, we obtain $b=\mu / \sqrt{\lambda}$ corresponding to the peak of potential barrier. Assume $b \gg l$, where $l$ is the mean free path of the particles, so that even near the peak the particles are in thermal equilibrium. We need consider only the escape rate n... | 7 | TSM | English | coordinate_system | essential | IMAGE 1:
A graph of $v(x)$ versus $x_1$ showing a single-peaked curve that reaches its maximum value at $x_1 = b$ (indicated by a dashed vertical line); $v(x)$ is plotted on the vertical axis and $x_1$ on the horizontal axis. | ||
67 | One mole of the paramagnetic substance whose $T S$ diagram is shown below is to be used as the working substance in a Carnot refrigerator operating between a sample at 0.2 K and a reservoir at 1 K: How much work will be performed on the paramagnetic substance per cycle? | $6.6 \times 10^{7} \mathrm{ergs} / \mathrm{mol}$ | \[\begin{aligned} Q_{\mathrm{rel}} & =T_{\mathrm{high}} \Delta S_{D \rightarrow A}=1 \times(1.5-0.5) R \\ & =8.3 \times 10^{7} \mathrm{ergs} / \mathrm{mol} . \end{aligned}\] The work done is \[ W=Q_{\mathrm{rel}}-Q_{\mathrm{abs}}=6.6 \times 10^{7} \mathrm{ergs} / \mathrm{mol} \] | 2 | 7 | TSM | English | coordinate_system | essential | IMAGE 1:
Contour plot showing temperature $T$ (in K) on the vertical axis versus the entropy-to-gas constant ratio $S/R$ on the horizontal axis, with curves labeled by values of magnetic field $H$ (0, 2, 3, 4, 5, 7, 10, 15, and 20 kG); key points A, B, C, and D are marked; all curves and axes are explicitly labeled. | |
68 | A simple molecular beam apparatus is shown in Fig. 2.40. The oven contains $\mathrm{H}_{2}$ molecules at 300 K and at a pressure of 1 mm of mercury. The hole on the oven has a diameter of $100 \mu \mathrm{~m}$ which is much smaller than the molecular mean free path. After the collimating slits, the beam has a divergenc... | $\frac{3}{2} \sqrt{\frac{\pi k T}{2 m}}$ | The mean speed is\n\[\n\langle v\rangle=\frac{\int v \cdot v^{3} e^{-\frac{m}{2 k T} v^{2}} d v}{\int v^{3} e^{-\frac{m}{2 k T} v^{2}} d v}=\frac{3}{2} \sqrt{\frac{\pi k T}{2 m}} .\n\] | 7 | TSM | English | atomic_physics | optional | IMAGE 1:
Rectangular vacuum chamber labeled "vacuum" connected to a pump ("to pump") at the bottom, contains two collimating slits aligned vertically and an oven attached at the left; dashed lines indicate the path of a particle or atomic/molecular beam emerging from the oven, passing through the slits inside the chamb... | ||
69 | In Fig. 2.71, the cylindrical cavity is symmetric about its long axis. For the purposes of this problem, it can be approximated as a coaxial cable (which has inductance and capacitance) shorted at one end and connected to a parallel plate disk capacitor at the other. Find the direction and radial-dependence of the Poyn... | At point $A$, $\mathbf{N} \sim \frac{1}{r^{2}}\mathbf{e}_{z}$; at point $B$, $\mathbf{N} \sim r\mathbf{e}_{r}$ | At point $A$, $r_{1}<r<r_{2}$, $\mathbf{E}(r) \sim \frac{\mathbf{e}_{r}}{r}$, $\mathbf{B}(r) \sim \frac{e_{e}}{r}$, so $\mathbf{N} \sim \frac{1}{r^{2}} \mathbf{e}_{z}$. At point $B$, $0<r<r_{1}$, $\mathbf{E}(r) \sim -\mathbf{e}_{z}$, $\mathbf{B}(r) \sim r\mathbf{e}_{\theta}$, so $\mathrm{N} \sim r \mathbf{e}_{r}$. | 7 | EM | English | electromagnetic_field | essential | IMAGE 1:
Cross-sectional diagram of a cylindrical coaxial structure with two conducting cylindrical shells; the inner shell has radius $r_1$, the outer shell has inner radius $r_2$, separated by a gap $a$, both of height $h$. Points $A$ and $B$ are marked inside the gap. No additional vectors or fields are depicted. | ||
70 | Refer to Fig. 3.67. When this monostable circuit is triggered how long will $Q_{2}$ be off? | 7 \mu \mathrm{~s} | The monostable pulse width is\n$$\n\begin{aligned}\n\Delta t=R C \ln 2 & =100 \times 10^{3} \times 100 \times 10^{-12} \times 0.7 \\\n& =7 \times 10^{-6} \mathrm{~s}=7 \mu \mathrm{~s}\n\end{aligned}\n$$\nduring which $Q_{2}$ is off. | 1 | 7 | CM | English | circuit_diagram | essential | IMAGE 1:
Electronic circuit diagram showing two NPN transistors $Q_1$ and $Q_2$ arranged in a bistable multivibrator configuration: $Q_1$ and $Q_2$ have collector resistors of $5\,\text{K}\Omega$ each connected to $-20\ \text{V}$, a coupling capacitor $C$ between the base of $Q_2$ and the collector of $Q_1$, a $100\,\t... | |
71 | A dipole of fixed length $2 R$ has mass $m$ on each end, charge $+Q_{2}$ on one end and $-Q_{2}$ on the other. It is in orbit around a fixed point charge $+Q_{1}$. (The ends of the dipole are constrained to remain in the orbital plane.) Figure 1.57 shows the definitions of the coordinates $r, \theta, lpha$. Figure 1.58... | $T = 2 \pi \sqrt{\frac{4 \pi \varepsilon_{0}}{Q_{1} Q_{2}} \cdot m R r^{2}}$ | As $\dot{r} \approx \ddot{r} \approx 0$, $r$ is a constant. Also, with $\ddot{\theta}=0$ and $\alpha \ll 1$ (i.e., $\sin \alpha \approx \alpha$), Eq. (3) becomes\n$$\nmR\ddot{\alpha}+\frac{Q_{1}Q_{2}}{4\pi\varepsilon_{0}} \cdot \frac{\alpha}{r^{2}} = 0.\n$$\nThis shows that the motion in $\alpha$ is simple harmonic wit... | 7 | CM | English | capacitance_resistance | essential | IMAGE 1:
**Fig. 1.57:** Point charge $Q_1$ is located at the origin, with another charge configuration consisting of two point charges, $+Q_2$ and $-Q_2$, separated by distance $R$ along a direction forming an angle $\alpha$ with the horizontal. The observation point is at distance $r$ from $Q_1$ at an angle $\theta$ f... | ||
72 | A pinhole camera consists of a box in which an image is formed on the film plane which is a distance $P$ from a pinhole of diameter $d$. The object is at a distance $L$ from the pinhole, and light of wavelength $\nabla$ is used (Fig. 2.66). Approximately what diameter $d$ of the pinhole will give the best image resolut... | $\sqrt{\frac{\lambda * L * P}{L + P}}$ | By geometrical optics, a point on the object would cast a bright disk on the film of diameter $\Delta_1$ given by
$$
\frac{\Delta_1}{L+P} = \frac{d}{L}.
$$
Due to diffraction by the pinhole, the point would form a bright Airy disk on the film of diameter
$$
\Delta_2 \approx \frac{2\lambda P}{d}.
$$
The resultant diamet... | 7 | OPT | English | optical_path | essential | IMAGE 1:
Two rays, separated by a distance $d$ at a slit or obstacle, diverge at an angle and pass through a rectangular aperture; the distance from the source to the slit is labeled $L$, the distance from the slit to the aperture is labeled $P$, and the directions of ray propagation are indicated by arrows. | ||
73 | Figure 1.24 is an energy level diagram for the ground state and first four excited states of a helium atom. Given electrons of sufficient energy, which levels could be populated as the result of electrons colliding with ground state atoms? | $(1 s 2 s)^{1} S_{0}, (1 s 2 s)^{3} S_{1}$ | The $(1 s 2 s)^{1} S_{0}$ and $(1 s 2 s)^{3} S_{1}$ states are metastable. So, besides the ground state, these two levels could be populated by many electrons due to electrons colliding with ground state atoms. | 7 | AMONP | English | atomic_physics | essential | IMAGE 1:
Energy level diagram showing singlet states labeled as $A$ ($1^1S_0$), $B$ ($2^1S_0$), $C$ ($2^1P_1$), and triplet states labeled as $D$ ($2^3S_1$), $E$ ($2^3P_{2,1,0}$), with quantum numbers $n=1$ and $n=2$ indicated for respective groups; singlet and triplet states are shown in separate columns, and all stat... | ||
74 | What are the neutron separation energies for ${ }_{20}^{40}\text{Ca}$ and ${ }_{82}^{208}\text{Pb}$ ? | 13 MeV, 3 MeV | The outermost neutron of ${ }^{40}\text{Ca}$ is the twentieth one. Figure 2.16 gives for $A=40$ that the last neutron is in $1 d_{3 / 2}$ shell with separation energy of about 13 MeV .
${ }^{208}\text{Pb}$ has full shells, the last pair of neutrons being in $3 p_{1 / 2}$ shell. From Fig. 2.16 we note that for $A=208$, ... | 7 | AMONP | English | atomic_physics | essential | IMAGE 1:
Single-particle energy levels (labeled by quantum numbers, e.g., $1\mathrm{s}_{1/2}$, $1\mathrm{p}_{3/2}$, $1\mathrm{d}_{5/2}$, $2\mathrm{s}_{1/2}$, etc.) are plotted as functions of the mass number $A$ (horizontal axis, ranging from $A=0$ to $A \simeq 220$); energy $E$ is plotted in MeV on the vertical axis. ... | ||
75 | Consider the Babinet Compensator shown in the figure (Fig.~2.76). The device is constructed from two pieces of uniaxial optical material with indices $n_{e}$ and $n_{o}$ for light polarized perpendicular and parallel to the optic axis, respectively. A narrow beam of light of vacuum wavelength $\lambda$ is linearly pola... | $x=\frac{(2 N+1)}{8} \frac{\lambda L}{\left(n_{0}-n_{e}\right) d}$ | The emerging light would be circularly polarized if
$$
\Delta\varphi = \frac{(2N + 1)\pi}{2}, \text{ i.e., } x = \frac{(2N + 1)}{8}\frac{\lambda L}{(n_o - n_e)d}
$$
where $N = 0$ or integer. | 7 | OPT | English | optical_path | essential | IMAGE 1:
Diagram showing a narrow beam of laser light entering a rectangular crystal (vertical rectangle), with horizontal entry parallel to the $y$-axis at a distance $x$ from the left edge. Two possible optic axes, labeled $OA$, are depicted: one is upward (parallel to the $x$-axis and lying in the plane of the paper... | ||
76 | A very long conducting pipe has a square cross section of its inside surface, with side $D$ as in Fig. 1.47. Far from either end of the pipe is suspended a point charge located at the center of the square cross section. Give the asymptotic expression for this potential for points far from the point charge. | $\varphi=\frac{\sqrt{2} Q}{\varepsilon_{0} \pi D}\cos\frac{\pi x}{D}\cos\frac{\pi y}{D}e^{-\frac{\sqrt{3} x}{\partial}|z|}$ | For points far from the point charge we need only choose the terms with $m=m^{\prime}=0$ for the potential, i.e.,\n$$\n\varphi=\frac{\sqrt{2} Q}{\varepsilon_{0}\pi D}\cos\frac{\pi x}{D}\cos\frac{\pi y}{D}e^{-\frac{\sqrt{3} x}{\partial}|z|}.\n$$ | 7 | EM | English | capacitance_resistance | essential | IMAGE 1:
Two parallel conducting rods separated by distance $D$ are shown, with a point charge $Q$ positioned midway between them; the top view indicates the charge $Q$ traveling along the positive $z$-axis, while the cross-sectional view shows $Q$ at the center with coordinate axes $x$ and $y$ indicated, where $x$ is ... | ||
77 | The space between two long thin metal cylinders is filled with a material with dielectric constant $\varepsilon$. The cylinders have radii $a$ and $b$, as shown in Fig. 1.19. What is the electric field between the cylinders? | $\mathbf{E}=-\frac{V}{r \ln \left(\frac{a}{b}\right)} \mathbf{e}_{r}$ | Gauss' law then gives the electric field intensity in the capacitor: $\mathbf{E}=\frac{\lambda_{\mathrm{i}}}{2 \pi \varepsilon r} \mathbf{e}_{r}=-\frac{V}{r \ln \left(\frac{a}{b}\right)} \mathbf{e}_{r}$. | 7 | EM | English | capacitance_resistance | optional | IMAGE 1:
A long cylindrical shell is shown with inner radius labeled $a$ and outer radius labeled $b$. Points $e$ and $d$ are marked on the shaded annular cross-section between radii $a$ and $b$. The longitudinal axis of the cylinder is indicated but not labeled. No external forces, fields, or currents are depicted. | ||
78 | A particle of mass m is released at $t=0$ in the one-dimensional double square well shown in the figure in such a way that its wave function at $t=0$ is just one sinusoidal loop (half a sine wave with nodes just at the edges of the left half of the potential as shown). Find the average value of the energy at $t=0$ (in ... | $\frac{\hbar^{2}\pi^{2}}{2ma} - V_{0}$ | The normalized wave function at $t=0$ is $\psi(x,0)=\sqrt{\frac{2}{a}}\sin\frac{\pi x}{a}$. Thus $$\langle\hat{H}\rangle_{t=0} = -V_{0} -\frac{\hbar^{2}}{2m}\frac{2}{a}\int_{0}^{a}\sin\left(\frac{\pi x}{a}\right)\frac{d^{2}}{dx^{2}}\sin\left(\frac{\pi x}{a}\right)dx = \frac{\hbar^{2}\pi^{2}}{2ma}-V_{0}.$$ | 7 | QMIT | English | coordinate_system | essential | IMAGE 1:
Graph of a one-dimensional periodic potential $V(x)$ as a function of position $x$, showing rectangular barriers of height $V_0$ and width $a$ repeated at regular intervals; the wavefunction $\psi(x,0)$ is sketched below the potential, and the vertical axes are labeled $V(x)$ and $\psi(x,0)$. | ||
79 | Inelastic neutrino scattering in the quark model. Consider the scattering of neutrinos on free, massless quarks. We will simplify things and discuss only strangeness‐conserving reactions, i.e. transitions only between the $u$ and $d$ quarks. Assume that inelastic $u$ (or $\bar{u}$)-nucleon cross sections are given by t... | 3 | For the reactions $
u d \rightarrow \mu^{-} u$ and $\bar{
u} \bar{d} \rightarrow \mu^{+} \bar{u}$ (Fig. 3.21) we have, similarly,
$$
\begin{array}{l}
\frac{d \sigma}{d \Omega}\left(
u d \rightarrow \mu^{-} u\right)_{\mathrm{cm}}=\frac{G_{F}^{2} S}{4 \pi^{2}} \cos ^{2} \theta_{c}
\frac{d \sigma}{d \Omega}\left(\bar{
u}... | 1 | 7 | AMONP | English | feynman_diagram | essential | IMAGE 1:
Feynman diagram showing an incoming neutrino $\nu$ with four-momentum $k$ and a down quark $d$ with four-momentum $p$ interacting via exchange of a virtual $W$ boson (dashed line labeled $W$); outgoing particles are a muon $\mu^-$ with four-momentum $k'$ and an up quark $u$ with four-momentum $p'$. All particl... | |
80 | Three identical objects, each of mass $m$, are connected by springs of spring constant $K$, as shown in Fig. 1.95. The motion is confined to one dimension. At $t=0$, the masses are at rest at their equilibrium positions. Mass $A$ is then subjected to an external time‐dependent driving force $F(t)=f \cos(\omega t)$, $t>... | $x_{C}=2a+\frac{f}{3m\omega^{2}}[1-\cos(\omega t)]+\frac{f}{2m(\omega_{2}^{2}-\omega^{2})}[\cos(\omega t)-\cos(\omega_{2}t)] + \frac{f}{6m(\omega_{3}^{2}-\omega^{2})}[\cos(\omega t)-\cos(\omega_{3}t)]$, where $\omega_{2}=\sqrt{\frac{K}{m}}$ and $\omega_{3}=\sqrt{\frac{3K}{m}}$ | Let $x_{A}, x_{B}, x_{C}$ be the coordinates of the three masses and $a$ the relaxed length of each spring. The equations of motion are $f\cos(\omega t)+K(x_{B}-x_{A}-a)=m\ddot{x}_{A}$, $K(x_{C}-x_{B}-a)-K(x_{B}-x_{A}-a)=m\ddot{x}_{B}$, and $-K(x_{C}-x_{B}-a)=m\ddot{x}_{C}$. This set can be written as $f\cos(\omega t)=... | 7 | CM | English | spring_force | essential | IMAGE 1:
Three blocks labeled $A$, $B$, and $C$ are aligned along the $x$-axis on a horizontal surface; block $A$ is connected to block $B$ and block $B$ to block $C$ via springs. A horizontal force $\vec{F}$ is applied to the leftmost block $A$ in the positive $x$-direction from the origin $O$. | ||
81 | A simple molecular beam apparatus is shown in Fig. 2.40. The oven contains $\mathrm{H}_{2}$ molecules at 300 K and at a pressure of 1 mm of mercury. The hole on the oven has a diameter of $100 \mu \mathrm{~m}$ which is much smaller than the molecular mean free path. After the collimating slits, the beam has a divergenc... | $n v \left(\frac{m}{2 \pi k T}\right)^{3/2} e^{-m v^2/(2 k T)} v^2 dv d\Omega$ | The Maxwell distribution is given by\n\[\nn\left(\frac{m}{2 \pi k T}\right)^{3 / 2} e^{-\frac{m}{2 k T} v^{2}} d v\n\]\n\nThe speed distribution of molecules in the beam is given by\n\[\nn v\left(\frac{m}{2 \pi k T}\right)^{3 / 2} e^{-\frac{m}{2 k T} v^{2}} v^{2} d v d_{\Omega} .\n\] | 7 | TSM | English | atomic_physics | optional | IMAGE 1:
Schematic showing an oven (left) emitting a particle or molecular beam (dashed line) directed through two collimating slits into a vacuum chamber; a port labeled "to pump" at the bottom provides vacuum maintenance. | ||
82 | Figure 2.19 gives the low-lying states of ${}^{18}\text{O}$ with their spin-parity assignments and energies (in MeV) relative to the $0^{+}$ ground state. What $J^{p}$ values are expected for the low-lying states of ${}^{19}\text{O}$? | $\frac{5}{2}^+$, $\frac{1}{2}^+$, $\frac{3}{2}^+$, $\frac{7^{+}}{2}$, $\frac{9^{+}}{2}$ | To calculate the lowest levels of ${}^{19}\text{O}$ using the simple shell model and ignoring interconfiguration interactions, we consider the last unpaired neutron. According to Fig. 2.16, it can go to $1 d_{5 / 2}, 2 s_{1 / 2}$, or $1 d_{3 / 2}$. So the ground state is $\left(\frac{5}{2}\right)^{+}$, the first excite... | 7 | AMONP | English | atomic_physics | essential | IMAGE 1:
Energy level diagram for an $^{18}\mathrm{O}$ nucleus, showing discrete states with energies $E$ (in MeV) labelled as $0$, $1.98$, and $3.55$, each associated respectively with spin-parity quantum numbers $J^P = 0^+, 2^+$, and $4^+$. | ||
83 | Consider a $2-\text{cm}$ thick plastic scintillator directly coupled to the surface of a photomultiplier with a gain of $10^{6}$. A $10-\text{GeV}$ particle beam is incident on the scintillator as shown in Fig. 4.8(a). Suppose one could detect a signal on the anode of as little as $10^{-12}$ coulomb. If the beam partic... | $5.0 \times 10^{-4} \mathrm{rad}$ | Figure $4.8(\text{b})$ shows a neutron scatters by a small angle $\theta$ in the laboratory frame. Its momentum is changed by an amount $p \theta$ normal to the direction of motion. This is the momentum of the recoiling nucleus. Then the kinetic energy acquired by it is
$$
\frac{p^{2} \theta^{2}}{2 m}
$$
where $m$ is t... | 2 | 7 | AMONP | English | atomic_physics | essential | IMAGE 1:
Figure 4.8:
(a) Schematic of a scintillator coupled to a photomultiplier tube, with a particle beam indicated by a right-pointing arrow passing horizontally through the scintillator.
(b) Diagram showing a particle of type $n$ with incident momentum $\vec{p}$ (arrow) scattering at a point into momentum $\ve... | |
84 | Suppose you have been supplied with a number of sheets of two types of optically active material. Sheets of type $P$ are perfect polarizers: they transmit (normally incident) light polarized parallel to some axis $\boldsymbol{n}$ and absorb light polarized perpendicular to $\boldsymbol{n}$. Sheets of type $Q$ are quart... | $\frac{I_{0}}{4}$ | Place two P-type sheets together such that the first sheet has its $\mathbf{n}$ axis at $45^\circ$ to the $\mathbf{x}$ axis and the second has its $\mathbf{n}$ axis at $90^\circ$ to the $\mathbf{x}$ axis. The combined sheet will convert an incident light polarized parallel to $\mathbf{x}$ into an outgoing light polariz... | 2 | 7 | OPT | English | optical_path | essential | IMAGE 1:
A linearly polarized electromagnetic wave is incident on a polarizing filter (central rectangular plate); the incident wave’s electric field oscillates in the $\hat{x}$ direction, as indicated by vector $\hat{x}$, and after passing through the polarizer, the transmitted wave’s electric field oscillates in the ... | |
85 | In an experiment a beam of silver atoms emerges from an oven, which contains silver vapor at $T=1200 \, \mathrm{K}$. The beam is collimated by being passed through a small circular aperture. If the screen is at $L=1$ meter from the aperture, estimate numerically the smallest $D$ that can be obtained by varying $a$. (Yo... | $8.0 \times 10^{-6} \mathrm{~m}$ | Using the uncertainty principle, we obtain the angle of deflection of the outgoing atoms\n\[\\n\theta \approx \frac{\lambda}{a} \approx \frac{h}{p a}=\frac{h}{a \sqrt{3 m k T}} .\\n\]\n\nThus, $D=a+2 \theta L=a+\frac{2 h L}{a \sqrt{3 m k T}} \geq 2 \sqrt{\frac{2 h L}{\sqrt{3 m k T}}}$.\[\\nD_{\min }=2 \frac{(2 h L)^{1 ... | 2 | 7 | TSM | English | thermodynamics | essential | IMAGE 1:
A rectangular oven emits radiation confined within two dashed lines diverging at an angle $\theta$, striking a vertical screen at distance $L$ from the oven, producing a spread of width $D$ on the screen. The diagram labels the oven, angle $\theta$, distance $L$, and width $D$. | |
86 | The pions that are produced when protons strike the target at Fermilab are not all moving parallel to the initial proton beam. A focusing device, called a "horn", (actually two of them are used as a pair) is used to deflect the pions so as to cause them to move more closely towards the proton beam direction. This devic... | $\theta \approx 0.0013\,\mathrm{rad}$ | For a $100\,\mathrm{GeV}/c$ meson, noting that $p=\frac{m_{0}v}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=100\,\mathrm{GeV}/c$ so that $v\approx c$, the deflecting force is $F=eBv=1.6\times10^{-19}\times0.41\times3\times10^{8}=2.0\times10^{-11}\,\mathrm{N}$. The deflected transverse distance is $x=\frac{1}{2}\frac{F}{m}\left(\frac... | 2 | 7 | AMONP | English | circuit_diagram | essential | IMAGE 1:
Figure (a) shows a long cylindrical solenoid with axis labeled $G$–$C$, radius $5\,\text{cm}$, and point $P$ located $40\,\text{cm}$ away from the solenoid along the direction indicated by an arrow labeled $\vec{\pi}$.
Figure (b) depicts an RLC circuit consisting of a voltage source $V_0$ in series with a ca... | |
87 | In Fig. 3.70 the circuit is a "typical" TTL totom pole output circuit. You should assume that all the solid state devices are silicon unless you specifically state otherwise. Give the voltages requested within 0.1 volt. Case 2: $V_{\mathrm{A}}=0.2$ volts, give $V_{\mathrm{B}}, V_{\mathrm{C}}, V_{\mathrm{D}}, V_{\mathrm... | $V_{\mathrm{B}}=0\,\mathrm{V},\; V_{\mathrm{C}}=5\,\mathrm{V},\; V_{\mathrm{D}}=4.3\,\mathrm{V},\; V_{\mathrm{E}}=3.6\,\mathrm{V}$ | Case 2: As $V_{\mathrm{A}}=0.2\,\mathrm{V}$, $T_{1}$ is in a cutoff state, so $V_{\mathrm{B}}=0$ and $T_{3}$ is also in a cutoff state. For $T_{2}$, $\beta=\frac{I_{c}}{I_{b}}=\frac{1400}{100}=14$ so that $T_{2}$ is saturated. Thus $V_{\mathrm{C}}=5\,\mathrm{V},\; V_{\mathrm{D}}=5-0.7=4.3\,\mathrm{V},\; V_{\mathrm{E}}=... | 2 | 7 | AMONP | English | circuit_diagram | essential | IMAGE 1:
Schematic of an electronic circuit featuring three transistors ($T_1$, $T_2$, $T_3$), resistors ($1~\text{k}\Omega$, $1.4~\text{k}\Omega$, $5~\text{k}\Omega$, $100~\Omega$), and a load resistor $R_\text{LOAD} = 1~\text{k}\Omega$, with nodes labeled $A$, $F$, $C$, $B$, $G$, $D$, $E$; a $+5~\text{V}$ power suppl... | |
88 | A thin spherical shell of radius $R$ has a fixed charge $+q$ distributed uniformly over its surface. A small circular section (radius $r \ll R$) of charge is removed from the surface. The cut section is replaced and the sphere is set in motion rotating with constant angular velocity $\omega=\omega_{0}$ about the $z$-ax... | $\frac{\mu_{0} k r_{P}^{2} q}{6 R}$ | Consider a ring of width $R\,d\theta$ as shown in Fig. 2.25. The surface current density on the ring is $\frac{\theta}{4 \pi R^{2}} \cdot \omega R \sin\theta$. The contributions of a pair of symmetrical points on the ring to the magnetic field at the center of the sphere will sum up to a resultant in the $z$-direction.... | 7 | EM | English | circuit_diagram | essential | IMAGE 1:
**Caption:**
Two diagrams of a uniformly rotating solid sphere of radius $R$ about the $z$-axis with angular velocity $\omega$; the sphere's center is labeled $O$ and an observation point is labeled $P$. In Fig. 2.24 (left), the sphere and $P$ are shown in the $z$ plane, with the $+z$ and $-z$ axes and the di... | ||
89 | A circular aperture of radius $a$ is uniformly illuminated by plane waves of wavelength $\lambda$. Let the $\boldsymbol{z}$-axis coincide with the aperture axis, with $z=0$ at the aperture, and with the incident flux travelling from negative values of $z$ toward $z=0$ (Fig.~2.30). Find the values of $z$ at which the in... | $z = a^2 / (\lambda N) \text{for} N = 2, 4, 6, ...$ | Fresnel's half-period zone construction shows that if the number of half-period zones is even for a point on the axis, the intensity of illumination at this point will be zero. As
$$
N = \frac{a^{2}}{\lambda z}
$$
or equivalently
$$
z = \frac{a^{2}}{\lambda N}
$$
the intensity at the point $z$ will be zero for $N = 2, ... | 7 | OPT | English | optical_path | essential | IMAGE 1:
A plane perpendicular to the $z$-axis contains a circular aperture of radius $a$ centered at point $O$; the $z$-axis is indicated with an arrow pointing to the right through the center of the aperture. | ||
90 | The Stark effect. The energy levels of the $n=2$ states of atomic hydrogen are illustrated in Fig. 5.14. The $S_{1/2}$ and $P_{1/2}$ levels are degenerate at an energy $\varepsilon_{0}$ and the $P_{3/2}$ level is degenerate at an energy $\varepsilon_{0}+A$. A uniform static electric field $E$ applied to the atom shifts... | \(\varepsilon_{0}+\Delta+\frac{|b|^{2}}{A}\) | Suppose the matrix elements of the perturbation Hamiltonian \(H'=-e\,\mathbf{E}\cdot\mathbf{r}\) are\n\begin{tabular}{cccc}\n & \(P_{3/2}\) & \(P_{1/2}\) & \(S_{1/2}\) \\\n\(P_{3/2}\) & 0 & 0 & \(b\) \\\n\(P_{1/2}\) & 0 & 0 & \(a\) \\\n\(S_{1/2}\) & \(b^{*}\) & \(a^{*}\) & 0\n\end{tabular},\nsince \(\langle l'|H'|l\ran... | 7 | AMONP | English | quantum_mechanics | essential | IMAGE 1:
Energy level diagram showing three horizontal lines labeled $S_{1/2}$ (at energy $\varepsilon_0$), $P_{1/2}$ (at energy $\varepsilon_0$), and $P_{3/2}$ (at energy $\varepsilon_0 \cdot \Delta$), with a vertical energy axis marked at $\varepsilon_0$ and $\varepsilon_0 \cdot \Delta$. | ||
91 | A flyball governor consists of two masses $m$ connected to arms of length $l$ and a mass $M$ as shown in Fig. 2.68. The assembly is constrained to rotate around a shaft on which the mass $M$ can slide up and down without friction. Neglect the mass of the arms, air friction, and assume that the diameter of the mass $M$ ... | $f=\frac{1}{2\pi}\sqrt{\frac{(m+M)g(1+3\cos^{2}\theta_{0})}{(m+2M\sin^{2}\theta_{0})l\cos\theta_{0}}}$ | When the shaft is free to rotate, using the Lagrangian with $\omega=\dot{\varphi}$ leads to the conserved quantity $\dot{\varphi}\sin^{2}\theta=c$. The subsequent Lagrange equations yield, at equilibrium ($\ddot{\theta}=0$, $\dot{\theta}=0$, $\theta=\theta_{0}$), the condition $$mc^{2}\frac{\cos\theta_{0}}{\sin^{3}\the... | 7 | CM | English | static_force_analysis | essential | IMAGE 1:
A rigid vertical shaft (mass $M$) oriented along the $z$-axis rotates with constant angular velocity $\omega_0$; two identical point masses $m$ are attached via four rods of length $l$, symmetrically on either side of the shaft, forming a diamond shape within the $xy$-plane. The angle $\theta$ is marked at eac... | ||
92 | A ray of light enters a spherical drop of water of index $n$ as shown (Fig. 1.21). Find an expression for the angle of deflection $\delta$. | $\pi - 4 \alpha + 2 \phi$ | As $\alpha=(\phi-\alpha)+x$, or $x=2 \alpha-\phi$, we have $\delta=\pi-2 x=\pi-4 \alpha+2 \phi$. | 7 | OPT | English | optical_path | essential | IMAGE 1:
A spherical object is positioned with its center as the origin; dashed lines indicate light rays passing through or tangent to the sphere, intersecting at the external point labeled $\delta$. Angles $\phi$ and $\alpha$ are marked: $\phi$ at the ray-sphere interface, and $\alpha$ at the angle subtending the sph... | ||
93 | Consider a $2-\text{cm}$ thick plastic scintillator directly coupled to the surface of a photomultiplier with a gain of $10^{6}$. A $10-\text{GeV}$ particle beam is incident on the scintillator as shown in Fig. 4.8(a). Same as Part (b), but it scatters elastically from a carbon nucleus. | $1.73 \times 10^{-3} \mathrm{rad}$ | If the recoiling particle is a carbon nucleus, then
$$
\theta_{\text{min}}^{2}=\frac{2 m_{c}}{p_{n}^{2}} \times 12.5 \times 10^{3}=\frac{2 \times 12 \times 10^{9}}{\left(10^{10}\right)^{2}} \times 12.5 \times 10^{3}=3.0 \times 10^{-6} \text{rad}^{2}
$$
i.e.,
$$
\theta_{\text{min}}=1.73 \times 10^{-3} \text{rad}
$$ | 3 | 7 | AMONP | English | atomic_physics | essential | IMAGE 1:
(a) A particle beam passes horizontally through a scintillator connected to a photomultiplier tube; (b) a diagram showing vectors labeled $\vec{p}$ (initial momentum), $\vec{p}'$ (final momentum) at an angle $\theta$ with respect to the initial direction, and $n$ (unit vector) indicating their respective direc... | |
94 | For the combination of one prism and 2 lenses shown (Fig. 1.42), find the location and size of the final image when the object, length 1 cm, is located as shown in the figure. | $10 \mathrm{cm}$ to the left of the second lens; $0.5 \mathrm{cm}$ | For the right-angle prism, $n=1.5$, the critical angle $\alpha=\sin ^{-1}\left(\frac{1}{n}\right)=$ $42^{\circ}$, which is smaller than the angle of incidence, $45^{\circ}$, at the hypotenuse of the prism. Therefore total internal reflection occurs, which forms a virtual image. The prism, equivalent to a glass plate of... | 2 | 7 | OPT | English | optical_path | essential | IMAGE 1:
A parallel light beam enters a 45°-45° glass prism of refractive index $n = 1.5$, with incident height $6\,\mathrm{cm}$, passes $10\,\mathrm{cm}$ to a convex lens ($f_1 = 20\,\mathrm{cm}$), then $5\,\mathrm{cm}$ further to a concave lens ($f_2 = -10\,\mathrm{cm}$); all elements are aligned along a common axis,... | |
95 | Looking through a small hole is a well-known method to improve sight. If your eyes are near-sighted and can focus an object 20 cm away without using any glasses, estimate the required diameter of the hole through which you would have good sight for objects far away. | $0.12 \mathrm{mm}$ | Inside the human eye the distance between the crystalline lens and the retina is 20 mm . Using the lens formula $\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$, we get $f=19 \mathrm{~cm}$ for the image distance 2.0 cm and the object 20 cm away. As Fig. 1.51 shows, a lens focuses an object infinitely far away at the focal point F... | 2 | 7 | OPT | English | optical_path | essential | IMAGE 1:
Three identical convex lenses are arranged coaxially with their principal axes aligned horizontally; each consecutive pair of lenses is separated by a distance $f$ (the focal length labeled above). An upright arrow object is placed to the left of the first lens on the axis. The diagram shows no vectors or ray ... | |
96 | In Fig. 2.71, the cylindrical cavity is symmetric about its long axis. For the purposes of this problem, it can be approximated as a coaxial cable (which has inductance and capacitance) shorted at one end and connected to a parallel plate disk capacitor at the other. Derive an expression for the lowest resonant frequen... | $\omega_{0}=\frac{2 d c^{2}}{h\left(2 d h+r_{1}^{2}\ln \frac{r_{2}}{r_{1}}\right)}$ | To find the inductance and capacitance per unit length of the coaxial cable, we suppose that the inside and outside conductors respectively carry currents $I$ and $-I$ and uniform linear charges $\lambda$ and $-\lambda$. Use cylindrical coordinates $(r, \theta, z)$ with the $z$-axis along the axis of the cable. Let the... | 7 | CM | English | electromagnetic_field | essential | IMAGE 1:
Cross-sectional diagram of a coaxial cylindrical system showing two concentric cylindrical shells; inner shell (inner radius $r_1$), outer shell (inner radius $r_2$), with vertical length $h$ and gap $a$ between the shells. Points $A$ and $B$ are marked inside the gap. No vectors are shown in the diagram. | ||
97 | For the circuit shown in Fig. 3.35, the coupling coefficient of mutual inductance for the two coils $L_{1}$ and $L_{2}$ is unity. What is the average power supplied by the oscillator as a function of frequency? | $P=\frac{RV_{0}^{2}/2}{R^{2}+\left(\frac{\omega L}{1-\omega^{2}LC}\right)^{2}}$ | $$\\np(t)=V(t)i_{1}(t)=\frac{V_{0}^{2}}{Z}\cos(\omega t-\varphi)\cos\omega t\n$$\nAveraging over one cycle we have\n$$\n\begin{aligned}\nP &= \bar{p}=\frac{V_{0}^{2}}{Z}\overline{\cos(\omega t-\varphi)\cos\omega t}=\frac{V_{0}^{2}}{2Z}\cos\varphi \\\n&= \frac{R}{2Z^{2}}V_{0}^{2}=\frac{RV_{0}^{2}/2}{R^{2}+\left(\frac{\o... | 7 | EM | English | circuit_diagram | essential | IMAGE 1:
Schematic of two magnetically coupled circuits: the left circuit is an AC driven ($V = V_0 \cos \omega t$) series $R$-$L_1$ loop with current $i_1$, and the right circuit is an $L_2$-$C$ loop with current $i_2$; $L_1$ and $L_2$ represent inductors, and both currents $i_1$ and $i_2$ are indicated with arrows sh... | ||
98 | A sound field is created by an arrangement of identical line sources grouped into two identical arrays of $N$ sources each as shown (Fig. 2.62) below.\nAll of the radiators lie in a plane perpendicular to the page, and produce waves of wavelength $\theta, N, c$ and $d$, the distance between the centers of the arrays. | $I = \frac{I_{m}}{N^{2}}\left(\frac{\sin \frac{N \delta}{2}}{\sin \frac{\delta}{2}}\right)^{2} \cos ^{2}\left(\frac{\pi d \sin \theta}{\lambda}\right)$ | The intensity distribution produced by each array is represented by
$$
I \sim \left(\frac{\sin \frac{N \delta}{2}}{\sin \frac{\delta}{2}}\right)^{2},
$$
where $\delta = 2\pi c \sin \theta / \lambda$. The expression
$$
I = I_{1} + I_{2} + 2\sqrt{I_{1} I_{2}} \cos \varphi
$$
for the resultant intensity of two coherent so... | 7 | OPT | English | optical_path | essential | IMAGE 1:
Two identical linear radiators are positioned vertically, separated by a distance $d$, with each radiator having a length $c$. A point at distance $r$ from the origin forms an angle $\theta$ with the horizontal. Dashed arrows indicate the direction from the radiators to the observation point, and geometric var... |
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SeePhys: Does Seeing Help Thinking? -- Benchmarking Vision-Based Physics Reasoning
Can AI truly see the Physics? Test your model with the newly released SeePhys Benchmark! Covering 2,000 vision-text multimodal physics problems spanning from middle school to doctoral qualification exams, the SeePhys benchmark systematically evaluates LLMs/MLLMs on tasks integrating complex scientific diagrams with theoretical derivations. Experiments reveal that even SOTA models like Gemini-2.5-Pro and o4-mini achieve accuracy rates below 55%, with over 30% error rates on simple middle-school-level problems, highlighting significant challenges in multimodal reasoning.
The benchmark is now open for evaluation at the ICML 2025 AI for MATH Workshop. Academic and industrial teams are invited to test their models!
🔗 Key Links:
📜Paper: http://arxiv.org/abs/2505.19099
⚛️Project Page: https://seephys.github.io/
🏆Challenge Submission: https://www.codabench.org/competitions/7925/
➡️Competition Guidelines: https://sites.google.com/view/ai4mathworkshopicml2025/challenge
The answer will be announced on July 1st, 2025 (Anywhere on Earth, AoE), which is after the submission deadline for the ICML 2025 Challenges on Automated Math Reasoning and Extensions.
If you find SeePhys useful for your research and applications, please kindly cite using this BibTeX:
@article{xiang2025seephys,
title={SeePhys: Does Seeing Help Thinking? -- Benchmarking Vision-Based Physics Reasoning},
author={Kun Xiang, Heng Li, Terry Jingchen Zhang, Yinya Huang, Zirong Liu, Peixin Qu, Jixi He, Jiaqi Chen, Yu-Jie Yuan, Jianhua Han, Hang Xu, Hanhui Li, Mrinmaya Sachan, Xiaodan Liang},
journal={arXiv preprint arXiv:2505.19099},
year={2025}
}
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