index string | id string | context string | question string | marking string | answer string | answer_type string | unit string | points string | modality string | field string | source string | image_question string | information string | image images list |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | APhO_2025_1_A_1 | [Precession of the Earth's axis]
[Introduction]
It has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus co... | Find the values of exponents: (1) $\beta$, (2) $\gamma$, and (3) $\delta$. | [["Award 0.2 pt if the answer correctly expresses the dimension of $G$ as $[G] = L^3 M^{-1} T^{-2}$, where $L$ is the base dimensions length, $M$ is mass, and $T$ is time. Otherwise, award 0 pt.", "Award 0.1 pt if the answer correctly sets up the exponent equation $0 = 2 - \\beta$. Otherwise, award 0 pt.", "Award 0.1 p... | ["\\boxed{$\\beta = 2$}", "\\boxed{$\\gamma = -1$}", "\\boxed{$\\delta = 4$}"] | ["Numerical Value", "Numerical Value", "Numerical Value"] | [null, null, null] | [0.3, 0.3, 0.2] | text+illustration figure | Mechanics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAKKBuEDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
1 | APhO_2025_1_A_2 | [Precession of the Earth's axis]
[Introduction]
It has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus co... | Calculate the numerical value of $h_{\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. | [["Award 0.1 pt if the answer correctly calculates $\\omega = \\frac{2\\pi}{24 h} = 7.27 \\times 10^{-5} s^{-1}$, where $\\omega$ is the angular speed of the Earth's rotation. Otherwise, award 0 pt.", "Award 0.1 pt if the answer correctly gives $h_{\\max} = 21.9 \\mathrm{km}$ from the relation $h_{\\max} \\propto \\fra... | ["\\boxed{21.9}"] | ["Numerical Value"] | ["km"] | [0.2] | text+illustration figure | Mechanics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAKKBuEDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
2 | APhO_2025_1_B_1 | [Precession of the Earth's axis]
[Introduction]
It has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus co... | Find (1) the direction and (2) magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \ll d_{SE}$. | [["Award 0.2 pt if the answer correctly expresses the magnitude of $U(z) = -G \\frac{M_S}{\\sqrt{z^2 + d_{SE}^2}}$, where $M_S$ is the Sun's mass, $z$ is the axis coordinate, $d_{SE}$ is the Earth-Sun distance, $G$ is the gravitational constant. Otherwise, award 0 pt.", "Award 0.1 pt if the answer includes the correct ... | ["\\boxed{The direction of the gravitational field is toward the center of the Sun ring.}", "\\boxed{$g_z(z) \\approx -\\frac{G M_S}{d_{SE}^3} z$}"] | ["Open-Ended", "Expression"] | [null, null] | [0.2, 0.8] | text+illustration figure | Mechanics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAKKBuEDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
3 | APhO_2025_1_B_2 | [Precession of the Earth's axis]
[Introduction]
It has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus co... | Find (1) the direction and (2) magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \ll d_{SE}$. | [["Award 0.5 pt if the answer presents the idea of using Gauss's law to calculate the gravitational field in the plane of the Sun ring. Otherwise, award 0 pt.", "Award 0.4 pt if the answer correctly identifies a cylindrical Gaussian surface with axis along $z$ near the center of the Sun ring. Otherwise, award 0 pt.", "... | ["\\boxed{The direction of the gravitational field is outward along the radial direction.}", "\\boxed{$g_r(r) = \\frac{G M_S}{2 d_{SE}^3} r$}"] | ["Open-Ended", "Expression"] | [null, null] | [0.2, 2.0] | text+illustration figure | Mechanics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAKKBuEDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
4 | APhO_2025_1_C_1 | [Precession of the Earth's axis]
[Introduction]
It has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus co... | Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$. | [["Award 0.2 pt if the answer includes the idea of transforming the ellipsoid of revolution into a perfect sphere of radius $R_e$ by stretching uniformly along the polar diameter with factor $R_e/R_p$. Otherwise, award 0 pt.", "Award 0.3 pt if the answer correctly computes the volume of one of the excess regions as $V ... | ["\\boxed{$m = \\frac{h_{\\max}}{2 R_p} M_E$}"] | ["Expression"] | [null] | [0.8] | text+variable figure | Mechanics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAKKBuEDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
5 | APhO_2025_1_C_2 | [Precession of the Earth's axis]
[Introduction]
It has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus co... | Given this idea, find the torque $\tau$ exerted by the Sun ring on the Earth. Express your answer in terms of $M_{E}, M_{S}, d_{SE}, R$ (the average radius), $h_{\max}$ and the angle $\alpha$. You can use that $h_{\max} \ll R$. | [["Award 0.1 pt if the answer mentions that the net torque acting on a perfect sphere of radius $R_e$ is zero due to symmetry. Otherwise, award 0 pt.", "Award 0.2 pt if the answer states the idea that the torque on the ellipsoid-shaped Earth is given by $\\vec{\\tau} = - \\vec{\\tau}'$. Otherwise, award 0 pt.", "Award ... | ["\\boxed{$|\\tau| = \\frac{3}{5} \\cdot \\frac{G M_E M_S}{d_{SE}^3} \\cdot R h_{\\max} \\sin \\alpha \\cos \\alpha$}"] | ["Expression"] | [null] | [1.8] | text+variable figure | Mechanics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAKKBuEDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
6 | APhO_2025_1_D_1 | [Precession of the Earth's axis]
[Introduction]
It has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus co... | Give an expression for the period $T_{1}$ of precession of the Earth's axis. Express your answer in terms of $M_{S}, d_{SE}$, the angular speed $\omega$ of the Earth's rotation, $h_{\text{max}}, R$ and $\alpha$. | [["Award 0.2 pt if the answer applies Newton's second law for rotational motion, $\\vec{\\tau} = \\dfrac{d \\vec{L}}{dt}$, where $\\tau$ is torque and $\\vec{L}$ is angular momentum. Otherwise, award 0 pt.", "Award 0.2 pt if the answer expresses angular momentum as $|\\vec{L}| = I \\omega$, where $I$ is the moment of i... | ["\\boxed{$T_1 = \\frac{4 \\pi}{3} \\cdot \\frac{d_{SE}^3 R \\omega}{G M_S h_{\\max} \\cos \\alpha}$}"] | ["Expression"] | [null] | [1.8] | text+variable figure | Mechanics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAKKBuEDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
7 | APhO_2025_1_D_2 | [Precession of the Earth's axis]
[Introduction]
It has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus co... | Calculate the precession period $T_{1}$ in years. | [["Award 0.2 pt if the answer gives the correct numerical result for the precession period as $T_1 \\approx 80600$ years, obtained by correctly substituting the given data into a dimensionally correct formula. Partial points: award 0 pt if the substitution is incorrect or if the formula used has a dimensional error."]] | ["\\boxed{80600}"] | ["Numerical Value"] | ["years"] | [0.2] | text+variable figure | Mechanics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAKKBuEDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
8 | APhO_2025_1_E_1 | [Precession of the Earth's axis]
[Introduction]
It has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus co... | By what factor $T_{2} / T_{1}$ does the period of precession of the Earth's axis change if we also take into account the torque exerted by the Moon? Give your answer in terms of $d_{ME}, d_{SE}, M_{S}$ and $M_{M}$. | [["Award 0.3 pt if the answer explicitly states that the torques exerted by the Sun and the Moon add up. Otherwise, award 0 pt.", "Award 0.4 pt if the answer correctly calculates the torque exerted by the Moon or states that it is proportional to $M_M/d_{ME}^3$, where $M_M$ is the Moon's mass and $d_{ME}$ is the Earth-... | ["\\boxed{$T_2 / T_1 = \\frac{M_S / d_{SE}^3}{M_M / d_{ME}^3 + M_S / d_{SE}^3}$}"] | ["Expression"] | [null] | [1.0] | text+variable figure | Mechanics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAKKBuEDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
9 | APhO_2025_1_E_2 | [Precession of the Earth's axis]
[Introduction]
It has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus co... | By substituting the data, calculate the period of precession $T_{2}$ in years. | [["Award 0.2 pt if the answer gives the correct numerical result for the precession period $T_2 \\approx 25400$ years, obtained by substituting values into a dimensionally correct formula. Partial points: award 0 pt if the result does not come from explicit substitution (e.g. if it is taken from the introduction), if t... | ["\\boxed{25400}"] | ["Numerical Value"] | ["years"] | [0.2] | text+variable figure | Mechanics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAKKBuEDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
10 | APhO_2025_2_A_1 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | It is possible to write the ring's magnetic moment $\vec{\mu}$ in terms of its angular momentum $\vec{L}$ as $\vec{\mu} = \gamma \vec{L}$. Find the constant $\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. | [["Award 0.1 pt if the answer gives the correct result for the angular momentum of the planar loop, $\\vec{L} = M R^2 \\vec{\\omega}$, where $M$ is the mass, $R$ the radius, and $\\vec{\\omega}$ the angular velocity (Award 0.1 pt if only the magnitude is given in the answer). Otherwise, award 0 pt.", "Award 0.1 pt if t... | ["\\boxed{$\\gamma = \\frac{Q}{2M}$}"] | ["Expression"] | [null] | [0.3] | text-only | Electromagnetism | APhO_2025 | None. | ||
11 | APhO_2025_2_A_2 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | Find the angular frequency $\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. | [["Award 0.1 pt if the answer writes the torque equation for a magnetic dipole correctly as $\\vec{\\tau} = \\vec{\\mu} \\times \\vec{B} = \\dfrac{d \\vec{L}}{dt}$, where $\\mu$ is the magnetic dipole moment, $\\vec{B}$ the magnetic field, and $\\vec{L}$ the angular momentum. Otherwise, award 0 pt.", "Award 0.1 pt if t... | ["\\boxed{$\\omega_L = -\\gamma B$}"] | ["Expression"] | [null] | [0.4] | text+illustration figure | Electromagnetism | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAHuAjQDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
12 | APhO_2025_2_A_3 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | The magnetic interaction energy between the two rings can be written as $U = J_{0} \vec{L}_{1} \cdot \vec{L}_{2}$, where $J_{0}$ is a constant and $\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\gamma$, $d$ and fundamental constants. | [["Award 0.1 pt if the answer writes the interaction energy correctly as $U = - \\vec{\\mu}_2 \\cdot \\vec{B}_{1 on 2} = - \\frac{\\mu_0}{4 \\pi d^3} \\mu_1 \\mu_2 \\cos(\\pi - \\theta)$. Otherwise, award 0 pt.", "Award 0.1 pt if the answer gives the correct magnetic field magnitude, $|\\vec{B}_{1 on 2}| = \\frac{\\mu_... | ["\\boxed{$J_0 = \\frac{\\mu_0 \\gamma^2}{4 \\pi d^3}$}"] | ["Expression"] | [null] | [0.5] | text+illustration figure | Electromagnetism | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAHuAjQDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
13 | APhO_2025_2_B_1 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | The energy terms containing $\vec{S}_{i}$ in the sum above can be viewed as the interaction energy between an effective magnetic field $\vec{B}_{i, \text{eff}}$ and the magnetic moment of $\vec{S}_{i}$. Find $\vec{B}_{i, \text{eff}}$ and express your answer in terms of $J$, the gyromagnetic ratio $\gamma$, and other sp... | [["Award 0.2 pt if the answer explicitly shows the understanding that spins $i-1$ and $i+1$ are the contributors to the interaction energy of spin $i$. Otherwise, award 0 pt.", "Award 0.1 pt if the answer gives the correct effective magnetic field as $\\vec{B}{i, \\text{eff}} = \\frac{J}{\\gamma}( \\vec{S}{i-1} + \\vec... | ["\\boxed{$\\vec{B}_{i,\\text{eff}} = \\frac{J}{\\gamma} (\\vec{S}_{i-1} + \\vec{S}_{i+1})$}"] | ["Expression"] | [null] | [0.3] | text+variable figure | Modern Physics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAHuAjQDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
14 | APhO_2025_2_B_2 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | Using the concept of effective magnetic field, express the rate of change of the $i$-th spin vector, $d \vec{S}_{i} / d t$, in terms of $J$, $\vec{S}_{i}$, and other spins $\vec{S}_{j}$ (specify the indices $j$ in relation to $i$). | [["Award 0.1 pt if the answer writes the rate of change of spin using the effective magnetic field, i.e. $\\frac{d \\vec{S}_i}{dt} = \\vec{\\mu}_i \\times \\vec{B}_{i,\\text{eff}}$. Otherwise, award 0 pt.", "Award 0.2 pt if the answer gives the correct explicit equation $\\frac{d \\vec{S}_i}{dt} = J \\vec{S}_i \\times ... | ["\\boxed{$\\frac{d \\vec{S}_i}{dt} = J \\vec{S}_i \\times (\\vec{S}_{i-1} + \\vec{S}_{i+1})$}"] | ["Expression"] | [null] | [0.3] | text+variable figure | Modern Physics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAHuAjQDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
15 | APhO_2025_2_B_3 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | Find the relationship between $\omega$ and $k$ (known as the dispersion relation, $\omega(k)$) for the spin waves in terms of $J$, $S$ and $a$. Hint: express the position of the $i$-th spin as $x = a \cdot i$. | [["Award 0.25 pt if the answer writes the traveling wave as a function of $kx \\pm \\omega t$ (either sign and either trigonometric functions $\\cos$ or $\\sin$ or complex exponentials are acceptable). Otherwise, award 0 pt.", "Award 0.25 pt if the answer explicitly shows that the amplitudes of $S_x$ and $S_y$ are equa... | ["\\boxed{$\\omega(k) = 2JS[1 - \\cos(ka)]$}"] | ["Expression"] | [null] | [2.0] | text+variable figure | Modern Physics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAHuAjQDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
16 | APhO_2025_2_B_4 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | For small $k$ ($k \ll 1 / a$), find the effective mass $m_{\text{eff}}$ of the spin wave. Express your answer in terms of $J, S, a$ and fundamental constants. | [["Award 0.2 pt if the answer gives the correct Taylor expansion for small $k$, namely $\\omega(k) \\approx 2JS \\left[1 - 1 + \\frac{1}{2}(ka)^2 \\right] = JSa^2 k^2$, where $J$ is the exchange coupling constant, $S$ is the spin magnitude, and $a$ is the lattice spacing. Otherwise, award 0 pt.", "Award 0.1 pt if the a... | ["\\boxed{$m_{\\text{eff}} = \\frac{\\hbar}{2 J S a^2}$}"] | ["Expression"] | [null] | [0.6] | text+variable figure | Modern Physics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAHuAjQDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
17 | APhO_2025_2_B_5 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | Suppose that initially, all the spins in the chain are pointing along the $z$ direction. A neutron with low energy travels on the $x-y$ plane making an incident angle $\theta_{\text{in}}$ with the chain and scatters with an angle $\theta_{\text{out}}$ as shown in Figure B.3. Assuming the neutron excites a single low wa... | [["Award 0.4 pt if the answer applies conservation of momentum along the $y$-axis, i.e., $p_{\\text{in}} \\cos \\theta_{\\text{in}} = p_{\\text{out}} \\cos \\theta_{\\text{out}}$, where $p_{\\text{in}}$ and $p_{\\text{out}}$ are the incident and outgoing neutron momenta, and $\\theta_{\\text{in}}$, $\\theta_{\\text{out... | ["\\boxed{$m_{\\text{eff}} = \\frac{\\sin(\\theta_{\\text{in}} - \\theta_{\\text{out}})}{\\sin(\\theta_{\\text{in}} + \\theta_{\\text{out}})} m_n$}"] | ["Expression"] | [null] | [1.3] | text+variable figure | Modern Physics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAHuAjQDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
18 | APhO_2025_2_C_1 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | Assume first that $\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\uparrow}$ to being anti-aligned to the magnetic field $p_{\downarrow}$? Express $p_{\uparrow} / p_{\downarrow}$ in terms of $h, T$ and fundamental constants. | [["Award 0.2 pt if the answer uses the Boltzmann factor $p_i \\propto \\exp(-\\varepsilon_i / k_B T)$, where $\\varepsilon_i$ is the energy of state $i$, $k_B$ is the Boltzmann constant, and $T$ is the temperature. Otherwise, award 0 pt.", "Award 0.1 pt if the answer gives the correct Boltzmann factor for the spin-up s... | ["\\boxed{$p_{\\uparrow} / p_{\\downarrow} = e^{2h/k_B T}$}"] | ["Expression"] | [null] | [0.5] | text+illustration figure | Thermodynamics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAIyA+IDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
19 | APhO_2025_2_C_2 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | Find the average polarization of the system $\bar{s} \equiv \frac{1}{N} \sum_{i} s_{i}$ for $N \gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\bar{s}$ as a function of $h$ for the cases $h_{o} \gg k_{B} T$, $h_{o} \approx k_{B} T$ and... | [["Award 0.2 pt if the answer deduces the expression for the average polarization as $\\bar{s} = p_{\\uparrow} - p_{\\downarrow}$, where $p_{\\uparrow}$ and $p_{\\downarrow}$ are the probabilities of spin-up and spin-down states, respectively. Otherwise, award 0 pt.", "Award 0.1 pt if the answer uses the normalization ... | ["\\boxed{$\\bar{s} = \\tanh(\\frac{h}{k_B T})$}"] | ["Expression"] | [null] | [1.0] | text+illustration figure | Thermodynamics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAIyA+IDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
20 | APhO_2025_2_C_3 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | What is the energy $E_{g}$ of the ground state (the lowest energy state)? Express your answer in terms of $\tilde{J}$ and $N$. | [["Award 0.1 pt if the answer recognizes that the energy of the system is minimized when all spins align in the same direction. Otherwise, award 0 pt.", "Award 0.1 pt if the answer gives the correct ground state energy as $E_g = -\\tilde{J}(N-1) \\approx -\\tilde{J}N$, where $\\tilde{J}$ is the effective exchange coupl... | ["\\boxed{$E_g \\simeq -\\tilde{J} N$}"] | ["Expression"] | [null] | [0.2] | text+illustration figure | Modern Physics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAIyA+IDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
21 | APhO_2025_2_C_4 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | Approximate the energy of the system as a sum over all spins $E = -\tilde{J}_{\text{eff}} \sum_i s_i$ and express $\tilde{J}_{\text{eff}}$ in terms of $\tilde{J}$ and $\bar{s}$. | [["Award 0.1 pt if the answer realizes that $s_{i+1}$ can be replaced with the average polarization $\\bar{s}$. Otherwise, award 0 pt.", "Award 0.1 pt if the answer gives the correct final result $E = -\\tilde{J}_{\\text{eff}} \\sum_i s_i = -\\tilde{J} \\sum_i s_i \\bar{s}$, where $\\tilde{J}_{\\text{eff}} = \\tilde{J}... | ["\\boxed{$\\tilde{J}_{\\text{eff}} = \\tilde{J} \\bar{s}$}"] | ["Expression"] | [null] | [0.2] | text+illustration figure | Modern Physics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAIyA+IDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
22 | APhO_2025_2_C_5 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | (1) Using your result from C.2, find an equation that the average polarization $\bar{s}$ must satisfy.
(2) The number of solutions to this equation depends on $T$. Find the critical temperature $T_{c}$ at which the number of solutions changes. Express your answer in terms of $\tilde{J}$ and fundamental constants. | [["Award 0.1 pt if the answer correctly states the self-consistent equation for the polarization as $\\bar{s} = \\tanh\\left( \\frac{\\tilde{J}_{\\text{eff}}}{k_B T} \\right)$. Otherwise, award 0 pt.", "Award 0.2 pt if the answer replaces $\\tilde{J}_{\\text{eff}} = \\tilde{J} \\bar{s}$ into the result from C.2, leadin... | ["\\boxed{$\\bar{s} = \\tanh\\left(\\frac{\\tilde{J} \\bar{s}}{k_B T}\\right)$}", "\\boxed{$T_c = \\frac{\\tilde{J}}{k_B}$}"] | ["Equation", "Expression"] | [null, null] | [0.3, 0.9] | text+illustration figure | Thermodynamics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAIyA+IDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
23 | APhO_2025_2_C_6 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | Find all possible values of $\bar{s}$ when $T < T_{c}$ and $T_{c} - T \ll T_{c}$. Express your answers in terms of $T$ and $T_{c}$. Sketch all possible values of $\bar{s}$ for the temperature $T$ in the range $0 \leq T \leq 2 T_{c}$. | [["Award 0.1 pt if the answer uses the proper approximation $\\tanh(x) \\approx x - \\frac{1}{3}x^3$ for small $x$, leading to $\\bar{s} = \\frac{T_c}{T} \\bar{s} - \\frac{1}{3}\\left( \\frac{T_c}{T} \\bar{s} \\right)^3$. Otherwise, award 0 pt.", "Award 0.1 pt if the answer derives a correct non-trivial solution for $\... | ["\\boxed{$\\bar{s} = 0$}", "\\boxed{$\\bar{s} = \\sqrt{3 \\cdot \\frac{T_c - T}{T_c}}$}", "\\boxed{$\\bar{s} = -\\sqrt{3 \\cdot \\frac{T_c - T}{T_c}}$}"] | ["Numerical Value", "Expression", "Expression"] | [null, null, null] | [0.2, 0.4, 0.4] | text+illustration figure | Thermodynamics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAIyA+IDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
24 | APhO_2025_2_C_7 | [Waves and Phase Transitions in Spin Systems]
[Introduction]
In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and q... | Write the option letter (A or B) in your answer.
(1) What magnetic phase of matter does $T > T_{c}$ correspond to? (A) Paramagnetic (B) Ferromagnetic.
(2) What magnetic phase of matter does $T < T_{c}$ correspond to? (A) Paramagnetic (B) Ferromagnetic. | [["Award 0.1 pt if the answer correctly classifies the phase for $T > T_c$ as paramagnetic (option A). Otherwise, award 0 pt.", "Award 0.1 pt if the answer correctly classifies the phase for $T < T_c$ as ferromagnetic (option B). Otherwise, award 0 pt."]] | ["\\boxed{A}", "\\boxed{B}"] | ["Multiple Choice", "Multiple Choice"] | [null, null] | [0.1, 0.1] | text+illustration figure | Thermodynamics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAIyA+IDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
25 | APhO_2025_3_A_1 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Express the average net solar power received by the Earth and atmosphere system $P_{0}$ in terms of $F_{s}$, $a$ and $R_{E}$, the radius of the Earth. | [["Award 0.1 pt if the answer identifies the correct effective cross-sectional area as $A = \\pi R_E^2$, where $R_E$ is the Earth's radius. Otherwise, award 0 pt.", "Award 0.1 pt if the answer gives the correct final absorbed power expression $P_0 = (1 - a) \\pi R_E^2 F_s$, where $a$ is the albedo and $F_s$ is the sola... | ["\\boxed{$P_0 = (1 - a) \\pi R_E^2 F_s$}"] | ["Expression"] | [null] | [0.2] | text-only | Thermodynamics | APhO_2025 | None. | ||
26 | APhO_2025_3_A_2 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Estimate the temperature of the Earth's surface $T_{g 0}$ assuming that it is at a steady state. Ignore the atmosphere. Express your answer in $K$. | [["Award 0.1 pt if the answer sets up the energy balance condition $P_{bd} = P_0$, where $P_{bd}$ is the blackbody radiation power and $P_0$ is the absorbed solar power. Otherwise, award 0 pt.", "Award 0.1 pt if the answer writes the correct explicit blackbody radiation formula $P_{bd} = \\sigma A T^4$ with $A = 4 \\pi... | ["\\boxed{255}"] | ["Numerical Value"] | ["K"] | [0.3] | text-only | Thermodynamics | APhO_2025 | None. | ||
27 | APhO_2025_3_A_3 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Assuming the system is in a steady state, calculate $T_{g}$, the temperature of the ground. Use $t_{\mathrm{sw}} = 0.9$ and $t_{\mathrm{lw}} = 0.2$. Express your answer in $K$. | [["Award 0.1 pt if the answer includes a correct statement of radiation balance in the region outside the atmosphere, e.g., $t_{lw} P_E + P_{atmo} = P_0$, where $t_{lw}$ is the longwave transmission coefficient, $P_E$ is the Earth's emitted power, $P_{atmo}$ is the atmospheric radiation, and $P_0$ is the absorbed solar... | ["\\boxed{286}"] | ["Numerical Value"] | ["K"] | [0.7] | text+illustration figure | Thermodynamics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAHMAbsDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
28 | APhO_2025_3_B_1 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\omega_{d}$? | [["Award 0.1 pt if the answer writes the correct equation of motion for particle A: $\\frac{d^2 x_A}{dt^2} = +\\frac{k}{m_A}(\\ell - \\ell_0)$, where $x_A$ is the position of particle A, $m_A$ is its mass, $k$ is the spring constant, $\\ell$ is the instantaneous spring length, and $\\ell_0$ is the natural spring length... | ["\\boxed{$\\omega_d = \\sqrt{k \\frac{m_A + m_B}{m_A m_B}}$}"] | ["Expression"] | [null] | [0.5] | text-only | Mechanics | APhO_2025 | None. | ||
29 | APhO_2025_3_B_2 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_{p}$ that can excite the vibration in B.1? Neglect recoil effects. | [["Award 0.2 pt if the answer gives the correct photon energy as $E = \\hbar \\omega_d$ (where $\\hbar$ is the reduced Planck constant and $\\omega_d$ is the angular frequency). Partial points: only award 0.1 pt if $h$ is used instead of $\\hbar$; no other numerical factors receive credit."]] | ["\\boxed{$E_p = \\hbar \\omega_d$}"] | ["Expression"] | [null] | [0.2] | text-only | Modern Physics | APhO_2025 | None. | ||
30 | APhO_2025_3_B_3 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | What is the shift in the spectral line $f - f_{0}$ if the molecule is moving with velocity $v$ towards the emitter such that $|v| \ll c$, where $c$ is the speed of light. | [["Award 0.1 pt if the answer writes down an expression for the Doppler effect, even if it is incorrect. Otherwise, award 0 pt.", "Award 0.1 pt if the answer gives the correct frequency shift as $f - f_0 = \\frac{v}{c} f_0$, where $f$ is the observed frequency, $f_0$ is the source frequency, $v$ is the velocity of the ... | ["\\boxed{$f - f_0 = \\frac{v}{c} f_0$}"] | ["Expression"] | [null] | [0.2] | text-only | Optics | APhO_2025 | None. | ||
31 | APhO_2025_3_B_4 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Find the normalization constant $C$, assuming that the velocity $v$ could range from $-\infty$ to $\infty$. | [["Award 0.1 pt if the answer writes down the normalization condition $\\int_{-\\infty}^{\\infty} p(v) dv = 1$, even if it is incorrectly applied from 0 to $\\infty$. Otherwise, award 0 pt.", "Award 0.1 pt if the answer gives the correct result for the normalization constant as $C = \\sqrt{ \\frac{m}{2 \\pi k_B T} }$, ... | ["\\boxed{$C = \\sqrt{\\frac{m}{2 \\pi k_B T}}$}"] | ["Expression"] | [null] | [0.2] | text-only | Thermodynamics | APhO_2025 | None. | ||
32 | APhO_2025_3_B_5 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Find the probability distribution function $p_{2}(f)$ to find a molecule with a spectral line $f_{0}$ shifted to $f$ due to thermal motion, up to a normalization factor, in terms of $f, f_{0}, T, m$ and fundamental constants. Use $C$ to represent the normalization factor. | [["Award 0.1 pt if the answer correctly replaces the velocity $v$ in the distribution using the Doppler effect relation $v = \\frac{f - f_0}{f_0} c$, where $f$ is the observed frequency, $f_0$ is the source frequency, and $c$ is the speed of light. Otherwise, award 0 pt.", "Award 0.2 pt if the answer writes the correct... | ["\\boxed{$p_2(f) = C \\exp\\left[-\\frac{mc^2}{2k_B T} \\left(\\frac{f - f_0}{f_0}\\right)^2\\right]$}"] | ["Expression"] | [null] | [0.3] | text-only | Thermodynamics | APhO_2025 | None. | ||
33 | APhO_2025_3_B_6 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Sketch $p_{2}(f)$ as a function of $f - f_{0}$, and determine the shift $f^{\star} - f_{0}$ at which $p_{2}(f^{\star})$ is a fraction $1 / e$ of its peak value, where $e$ is the natural number. | [["Award 0.1 pt if the answer states that the probability distribution $p(f)$ has a single peak at zero frequency shift ($f - f_0 = 0$). Otherwise, award 0 pt.", "Award 0.1 pt if the answer correctly identifies that the distribution is symmetric about $f - f_0 = 0$. Otherwise, award 0 pt.", "Award 0.1 pt if the answer ... | ["\\boxed{$f^{\\star} - f_0 = f_0 \\sqrt{\\frac{2k_B T}{mc^2}}$}"] | ["Expression"] | [null] | [0.4] | text-only | Thermodynamics | APhO_2025 | None. | ||
34 | APhO_2025_3_C_1 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Assuming that the small air mass is at hydrostatic equilibrium, derive an expression of the rate of change of pressure with respect to height, $d p / d z$ in terms of $g$ and $\rho(z)$. | [["Award 0.1 pt if the answer states that the sum of forces equals zero in hydrostatic equilibrium. Otherwise, award 0 pt.", "Award 0.1 pt if the answer correctly identifies the pressure force contributions above and below the thin layer, i.e. $p(z)S = p(z + \\mathrm{d}z)S + \\rho(z) g S \\mathrm{d}z$. Otherwise, award... | ["\\boxed{$\\frac{dp}{dz} = -\\rho(z) g$}"] | ["Expression"] | [null] | [0.3] | text+illustration figure | Mechanics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAI5AncDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
35 | APhO_2025_3_C_2 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Express $d p / d z$ in terms of $\mu_{\text{air}}, g, p(z)$ and $T(z)$, the temperature at height $z$ and fundamental constants. | [["Award 0.1 pt if the answer correctly uses the ideal gas law $pV = nRT$ and rewrites it as $\\rho(z) = \\frac{p(z) \\mu_{air}}{R T(z)}$, where $\\mu_{air}$ is the molar mass of air and $R$ is the gas constant. Otherwise, award 0 pt.", "Award 0.1 pt if the answer gives the correct final hydrostatic equilibrium express... | ["\\boxed{$\\frac{dp}{dz} = -\\frac{\\mu_{\\text{air}} p(z)}{R T(z)} g$}"] | ["Expression"] | [null] | [0.2] | text+illustration figure | Thermodynamics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAI5AncDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
36 | APhO_2025_3_C_3 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Assuming an isothermal atmosphere, $T(z) = T$, find an expression for $p(z)$ in terms of $z, \mu_{\text{air}}, g, p_{o}, T$ and fundamental constants. | [["Award 0.1 pt if the answer recognizes and correctly rewrites the equation as a separable differential equation $\\frac{dp}{p} = - \\frac{\\mu_{air}}{R T} g dz$. Otherwise, award 0 pt.", "Award 0.1 pt if the answer gives the correct final solution $p(z) = p_0 \\exp \\left( - \\frac{\\mu_{air}}{R T} g z \\right)$, whe... | ["\\boxed{$p(z) = p_0 \\exp\\left(-\\frac{\\mu_{\\text{air}}}{RT} gz\\right)$}"] | ["Expression"] | [null] | [0.2] | text+illustration figure | Thermodynamics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAI5AncDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
37 | APhO_2025_3_C_4 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | For the adiabatically rising air mass, find the adiabatic lapse rate $\Gamma_{a}$ in terms of $c_{p}$, the molar specific heat at constant pressure, $\mu_{\text{air}}$ and $g$. | [["Award 0.1 pt if the answer writes the adiabatic relation in any correct form for an ideal gas, e.g., $p V^{\\gamma} = \\text{const.}$ or equivalently $p^{1-\\gamma} T^{\\gamma} = \\text{const.}$ where $\\gamma = c_p/c_v$, $c_p$ and $c_v$ are the molar specific heats at constant pressure and volume. Otherwise, award ... | ["\\boxed{$\\Gamma_a = \\frac{\\mu_{\\text{air}}}{c_p} g$}"] | ["Expression"] | [null] | [0.6] | text+illustration figure | Thermodynamics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAI5AncDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
38 | APhO_2025_3_C_5 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | (1) Find the equation of motion for $\delta z$, the instantaneous vertical displacement.
(2) Under what condition is the equilibrium at $z$ stable?
(3) What is the angular frequency $\omega$ of small oscillation? Express your answers in terms of $T, \Gamma, g, \mu_{\text{air}}$ and $c_{p}$. | [["Award 0.2 pt if the answer obtains the gravitational force with parcel density correctly as $\\delta m g = \\rho_p \\delta V g$, where $\\delta m$ is the mass of the parcel, $\\rho_p$ is the density of the parcel, and $\\delta V$ is its volume. Otherwise, award 0 pt.", "Award 0.3 pt if the answer obtains the buoyanc... | ["\\boxed{$\\frac{d^2 z}{d t^2} = \\frac{\\Gamma - \\mu_{\\text{air}} g/c_p}{T} g \\delta_z$}", "\\boxed{$\\mu_{\\text{air}} g/c_p > \\Gamma$}", "\\boxed{$\\omega = \\sqrt{\\frac{\\mu_{\\text{air}} g/c_p - \\Gamma}{T} g}$}"] | ["Equation", "Inequality", "Expression"] | [null, null, null] | [1.1, 0.1, 0.2] | text+illustration figure | Mechanics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAI5AncDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
39 | APhO_2025_3_D_1 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Express $d p_{s} / d T$ for the water liquid-vapor coexistence curve in terms of the water latent heat of evaporation $L, \mu_{\text{H_2O}}, p_{s}, T$ and fundamental constants. | [["Award 0.2 pt if the answer obtains the correct entropy change as $\\Delta S = \\frac{L m}{T}$, where $L$ is the latent heat of evaporation, $m$ is the mass of liquid water, and $T$ is the temperature. Otherwise, award 0 pt.", "Award 0.2 pt if the answer correctly states that $V_{vapor} \\gg V_{liquid}$ and therefore... | ["\\boxed{$\\frac{d p_s}{d T} = \\frac{\\mu_{\\text{H_2O}} L p_s}{R T^2}$}"] | ["Expression"] | [null] | [0.5] | text+illustration figure | Thermodynamics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAJfAqIDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
40 | APhO_2025_3_D_2 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | If for some reference temperature $T_{o}$, $p_{s} = p_{s o}$, find an expression for $p_{s}(T)$ in terms of $p_{s o}, \mu_{\text{H_2O}}, L, T, T_{o}$ and fundamental constants. | [["Award 0.1 pt if the answer recognizes a separable differential equation and obtains $\\ln [\\frac{p_s(T)}{p_{so}}] = -\\frac{\\mu_{\\text{H_2O}} L}{R} (\\frac{1}{T} - \\frac{1}{T_o})$, where $p_s$ is the saturation vapor pressure, $\\mu_{\\text{H_2O}}$ is the molar mass of water, $L$ is latent heat, $R$ is the gas c... | ["\\boxed{$p_s(T) = p_{s0} \\exp\\left[-\\frac{\\mu_{\\text{H_2O}} L}{R} \\left(\\frac{1}{T} - \\frac{1}{T_0}\\right)\\right]$}"] | ["Expression"] | [null] | [0.2] | text+illustration figure | Thermodynamics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAJfAqIDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
41 | APhO_2025_3_D_3 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Assuming that the air mass starts at $T_{i} = 17.0^{\circ} \mathrm{C}$ and $p_{i} = 10^{5} \mathrm{Pa}$. Find the temperature $T_{l}$ at which liquid water starts forming in it if $\phi = 10^{-2}$. Assume that the water content in the air mass stays constant during the rise. Use $L = 2460 \mathrm{kJ} / \mathrm{kg}$ and... | [["Award 0.4 pt if the answer uses Dalton's law to correctly express the partial pressure of water vapor as $p_w = \\frac{n_{H_2O}}{n_{air}} p = \\frac{m_{H_2O}/\\mu_{H_2O}}{m_{air}/\\mu_{air}} p = \\phi \\frac{\\mu_{air}}{\\mu_{H_2O}} p$, where $n$ is number of moles, $m$ is mass, $\\mu$ is molar mass, $p$ is total pr... | ["\\boxed{286.8}"] | ["Numerical Value"] | ["K"] | [2.0] | text+illustration figure | Thermodynamics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAJfAqIDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
42 | APhO_2025_3_E_1 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Consider a simple prism with an apex angle of $\varphi$ and direct a light ray onto it at an incidence angle $\alpha$, as shown in Figure E.1. Let the refractive index of the prism be $n$. Express the angle of deviation $\delta$ of the light ray after passing through the prism in terms of $\alpha, n$ and $\varphi$. | [["Award 0.1 pt if the answer writes Snell's law correctly for the first refraction as $\\frac{\\sin \\alpha}{\\sin \\alpha'} = n$, where $\\alpha$ is the angle of incidence, $\\alpha'$ is the refracted angle, and $n$ is the refractive index. Otherwise, award 0 pt.", "Award 0.1 pt if the answer writes Snell's law corre... | ["\\boxed{$\\delta = \\alpha + \\arcsin\\{n \\sin[\\varphi - \\arcsin(\\frac{\\sin\\alpha}{n})]\\} - \\varphi$}"] | ["Expression"] | [null] | [0.8] | text+variable figure | Optics | APhO_2025 | /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAJaBxcDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIh... | None. | |
43 | APhO_2025_3_E_2 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Calculate the deviation angle $\delta$ for incidence angles $\alpha = 20^{\circ}, 30^{\circ}, 40^{\circ}, 50^{\circ}, 60^{\circ}, 70^{\circ}$, in that order. Output the six values in degrees ($^{\circ}$), each with three significant figures, listed sequentially and separately. The refractive index of ice is $n = 1.31$. | [["Award 0.2 pt if the answer gives all six correct values (when \\alpha = 20^{\\circ}, \\delta = 27.5^{\\circ}; when \\alpha = 30^{\\circ}, \\delta = 23.0^{\\circ}; when \\alpha = 40^{\\circ}, \\delta = 21.8^{\\circ}; when \\alpha = 50^{\\circ}, \\delta = 22.5^{\\circ}; when \\alpha = 60^{\\circ}, \\delta = 24.7^{\\ci... | ["\\boxed{$\\delta = 27.5^{\\circ}$ when $\\alpha = 20^{\\circ}$}", "\\boxed{$\\delta = 23.0^{\\circ}$ when $\\alpha = 30^{\\circ}$}", "\\boxed{$\\delta = 21.8^{\\circ}$ when $\\alpha = 40^{\\circ}$}", "\\boxed{$\\delta = 22.5^{\\circ}$ when $\\alpha = 50^{\\circ}$}", "\\boxed{$\\delta = 24.7^{\\circ}$ when $\\alpha = ... | ["Numerical Value", "Numerical Value", "Numerical Value", "Numerical Value", "Numerical Value", "Numerical Value"] | ["degrees", "degrees", "degrees", "degrees", "degrees", "degrees"] | [0.1, 0.1, 0.1, 0.1, 0.1, 0.1] | text+variable figure | Optics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAJaBxcDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. | |
44 | APhO_2025_3_E_3 | [Atmospheric Physics]
The Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understan... | Using the numerical results from the previous question (E.2), determine at what angle the halo appears the brightest relative to the direction of the Sun. Express your answer in degrees ($^{\circ}$) with three significant figures. | [["Award 0.1 pt if the answer correctly reads and states the minimal value of $\\delta$ as approximately $21.8^\\circ$. Otherwise, award 0 pt.", "Award 0.1 pt if the answer concludes that the angular size of the halo corresponds to this minimal value of $\\delta$, i.e. about $21.8^\\circ$. Otherwise, award 0 pt."]] | ["\\boxed{21.8}"] | ["Numerical Value"] | ["degrees"] | [0.2] | text+variable figure | Optics | APhO_2025 | ["/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAJaBxcDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBR... | None. |
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