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E4_verifier:physgym_0010_err_aya8b | physgym_0010 | ELECTROMAGNETISM | Step 1: **Problem Setup:** We have a point charge $q$ at $(a, b, c)$ in a vacuum region, and a conductor with surfaces at $x=0$ and $y=0$. We need to find the electric field $E(x, 0, z)$ on the $xOz$ surface.
Step 2: **Gauss's Law Application:** Since the conductor is in equilibrium, Gauss's Law tells us that the flux ... | To correctly determine the electric field intensity on the $xOz$ plane, you must use Gauss's Law, which states that the electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space $\varepsilon_0$. Additionally, you need to consider the contributions from image charge... | [[CHALLENGE]] | incorrect_physical_model | part_2_synthetic_aya8b_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correctly determine the electric field intensity on the $xOz$ plane, you must use Gauss's Law, which states that the electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space $\varepsilon_0$. Additionally, you need to consider the contributions from image charge... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correctly determine the electric field intensity on the $xOz$ plane, you must use Gauss's Law, which states that the electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space $\varepsilon_0$. Additionally, you need to consider the contributions from image charge... | |
E4_verifier:physgym_0010_tiny_aya | physgym_0010 | ELECTROMAGNETISM | Step 1: Identify the problem setup and governing equations
We are given a vacuum region where \(x > 0\) and \(y > 0\), and a conductor region bounded by the \(xOz\) and \(yOz\) planes. A point charge \(q\) is fixed at \((a, b, c)\) in the vacuum, and the system is in electrostatic equilibrium. We need to find the elect... | To correctly determine the electric field intensity on the surface of the conductor at the \(xOz\) plane, you must apply Gauss's Law to a Gaussian surface that encloses the point charge and consider the symmetry of the problem. | [[CHALLENGE]] | real_model_a_failure | part_3_real_tiny_aya_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correctly determine the electric field intensity on the surface of the conductor at the \(xOz\) plane, you must apply Gauss's Law to a Gaussian surface that encloses the point charge and consider the symmetry of the problem.
[[CHALLENGE]] | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correctly determine the electric field intensity on the surface of the conductor at the \(xOz\) plane, you must apply Gauss's Law to a Gaussian surface that encloses the point charge and consider the symmetry of the problem.
[[CHALLENGE]] | |
E4_verifier:physgym_0043_err_aya8b | physgym_0043 | ELECTROMAGNETISM | Step 1: **Electric Field Intensity:** We start by equating the electric field intensities in the two media, ignoring the effect of the dielectric constants: $E_1h_1 + E_2h_2 = U_0$. This is a simplified version of the actual relationship.
Step 2: **Current Density:** Correctly, we have $j_1 = \sigma_1 E_1$ and $j_2 = \... | To correctly determine the time-dependent current $I(t)$, you must use the relationship between the electric fields, charge densities, and conductivities in each medium, taking into account the dielectric constants. The key equations to use are: - Electric Field Intensity:$E_1h_1 + E_2h_2 = U_0$ (but with the correct r... | [[CHALLENGE]] | incorrect_formula | part_2_synthetic_aya8b_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correctly determine the time-dependent current $I(t)$, you must use the relationship between the electric fields, charge densities, and conductivities in each medium, taking into account the dielectric constants. The key equations to use are: - Electric Field Intensity:$E_1h_1 + E_2h_2 = U_0$ (but with the correct r... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correctly determine the time-dependent current $I(t)$, you must use the relationship between the electric fields, charge densities, and conductivities in each medium, taking into account the dielectric constants. The key equations to use are: - Electric Field Intensity:$E_1h_1 + E_2h_2 = U_0$ (but with the correct r... | |
E4_verifier:physgym_0043_tiny_aya | physgym_0043 | ELECTROMAGNETISM | Step 1: **Governing Equations:
- **Capacitance:** The capacitance \( C \) of a parallel-plate capacitor with dielectric layers is given by:
\[
C = \frac{A}{d} \left( \varepsilon_1 \frac{h_1}{h_1 + h_2} + \varepsilon_2 \frac{h_2}{h_1 + h_2} \right)
\]
- **Power Source:** The power source provides a voltage \( U_0 \), so... | To correctly determine the time-dependent relationship of the current $I(t)$, you must consider the coupled electric fields and charge densities within both dielectric materials. Utilize the principles of capacitance, electric field distribution, and the relationship between charge density and voltage. The governing eq... | [[CHALLENGE]] | real_model_a_failure | part_3_real_tiny_aya_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correctly determine the time-dependent relationship of the current $I(t)$, you must consider the coupled electric fields and charge densities within both dielectric materials. Utilize the principles of capacitance, electric field distribution, and the relationship between charge density and voltage. The governing eq... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correctly determine the time-dependent relationship of the current $I(t)$, you must consider the coupled electric fields and charge densities within both dielectric materials. Utilize the principles of capacitance, electric field distribution, and the relationship between charge density and voltage. The governing eq... | |
E4_verifier:physgym_0250_err_aya8b | physgym_0250 | ELECTROMAGNETISM | Step 1: Potential Difference and Dielectric Constant:
We start with a potential difference $u$ across the two dielectrics, and we define $\lambda = \frac{2\pi(\varepsilon_{1}+\varepsilon_{2})u}{2\xi_{+}}$, where $\xi_{+}$ is a length related to the cylindrical holes.
Step 2: Gauss's Theorem Application:
Using Gauss's t... | Use the correct relationship between current density $J$, electric field $\vec{E}$, and conductivity $\sigma$ in Gauss's theorem: $$ J = \sigma\vec{E} $$ Correct Current Density Calculation:Integrate the product of conductivity, electric field, and surface area to find the total current: $$ I = \iint\sigma\vec{E}\cdot ... | [[CHALLENGE]] | algebraic_error | part_2_synthetic_aya8b_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | Use the correct relationship between current density $J$, electric field $\vec{E}$, and conductivity $\sigma$ in Gauss's theorem: $$ J = \sigma\vec{E} $$ Correct Current Density Calculation:Integrate the product of conductivity, electric field, and surface area to find the total current: $$ I = \iint\sigma\vec{E}\cdot ... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | Use the correct relationship between current density $J$, electric field $\vec{E}$, and conductivity $\sigma$ in Gauss's theorem: $$ J = \sigma\vec{E} $$ Correct Current Density Calculation:Integrate the product of conductivity, electric field, and surface area to find the total current: $$ I = \iint\sigma\vec{E}\cdot ... | |
E4_verifier:physgym_0250_tiny_aya | physgym_0250 | ELECTROMAGNETISM | Step 1: **Governing Equations**: The problem involves an electrolyte with two dielectrics separated by an interface. The conductivity and dielectric constant of each dielectric affect the electric field and potential distribution. The power source provides a voltage \( U \) and current \( I_0 \) at \( t = 0 \).
Step 2:... | The problem involves the charging of a system with two dielectrics separated by an interface, and the correct relationship between current and time is derived using the principles of electrostatics and circuit analysis. The key physical laws to consider are: - Ohm's Law:$I = \frac{U}{r_0}$ (initial condition). - Gauss'... | [[CHALLENGE]] | real_model_a_failure | part_3_real_tiny_aya_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | The problem involves the charging of a system with two dielectrics separated by an interface, and the correct relationship between current and time is derived using the principles of electrostatics and circuit analysis. The key physical laws to consider are: - Ohm's Law:$I = \frac{U}{r_0}$ (initial condition). - Gauss'... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | The problem involves the charging of a system with two dielectrics separated by an interface, and the correct relationship between current and time is derived using the principles of electrostatics and circuit analysis. The key physical laws to consider are: - Ohm's Law:$I = \frac{U}{r_0}$ (initial condition). - Gauss'... | |
E4_verifier:physgym_0636_err_aya8b | physgym_0636 | ELECTROMAGNETISM | Step 1: **Potential Energy Setup:** We begin by considering a single circular loop with radius $R$ and charge $Q$ in the Cartesian coordinate system $Oxyz$. The potential energy $V$ at a point $(r, \theta)$ due to this loop is given by:
$$ V = \frac{kQ}{r} - \frac{kR^2}{r^2} $$
where $k$ is a constant we will ignore fo... | You have made several errors, primarily in ignoring crucial terms and constants. The correct approach involves considering the full potential energy expression from the superposition principle, including the $1/r$ and $1/r^3$ terms, and applying the chain rule to find the angular and radial frequencies. The period diff... | [[CHALLENGE]] | dimensional_unit_error | part_2_synthetic_aya8b_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | You have made several errors, primarily in ignoring crucial terms and constants. The correct approach involves considering the full potential energy expression from the superposition principle, including the $1/r$ and $1/r^3$ terms, and applying the chain rule to find the angular and radial frequencies. The period diff... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | You have made several errors, primarily in ignoring crucial terms and constants. The correct approach involves considering the full potential energy expression from the superposition principle, including the $1/r$ and $1/r^3$ terms, and applying the chain rule to find the angular and radial frequencies. The period diff... | |
E4_verifier:physgym_0636_tiny_aya | physgym_0636 | ELECTROMAGNETISM | Step 1: **Coordinate System and Setup:
- Establish a Cartesian coordinate system \( Oxyz \) with the origin at the center of the sphere.
- The sphere has radius \( R \) and is stationary.
- \( n \) rings of radius \( R \) are placed along the meridional circles, passing through \( (0, 0, R) \) and \( (0, 0, -R) \).
- T... | To correct the derivation, we must leverage the principles of electrostatics and mechanics. The key physical laws and equations required are: - Coulomb's Lawfor the electric field between charged objects. | [[CHALLENGE]] | real_model_a_failure | part_3_real_tiny_aya_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correct the derivation, we must leverage the principles of electrostatics and mechanics. The key physical laws and equations required are: - Coulomb's Lawfor the electric field between charged objects.
[[CHALLENGE]] | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correct the derivation, we must leverage the principles of electrostatics and mechanics. The key physical laws and equations required are: - Coulomb's Lawfor the electric field between charged objects.
[[CHALLENGE]] | |
E4_verifier:physgym_0649_err_aya8b | physgym_0649 | ELECTROMAGNETISM | Step 1: **Magnetic Field Calculation:** We begin by calculating the magnetic field $\vec{B_z}$ at a small displacement $z$ along the $z$-axis due to the charged rings, following the Biot-Savart law:
$$
\vec{B_z}=\frac{\mu_{0}Q\Omega}{2\pi}\frac{R^{2}}{(R^{2}+l^{2})^{\frac{3}{2}}}\hat{z}
$$
Step 2: **Uniform Field Appro... | To correct the derivation, the Solver should use the angular momentum conservationand effective potential energyconcepts . The key error lies in the sign of the $r^2$ term in the effective potential energy. The correct expression for the effective potential energy $V_{eff}$ should include a positive coefficient for the... | [[CHALLENGE]] | arithmetic_sign_error | part_2_synthetic_aya8b_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correct the derivation, the Solver should use the angular momentum conservationand effective potential energyconcepts . The key error lies in the sign of the $r^2$ term in the effective potential energy. The correct expression for the effective potential energy $V_{eff}$ should include a positive coefficient for the... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correct the derivation, the Solver should use the angular momentum conservationand effective potential energyconcepts . The key error lies in the sign of the $r^2$ term in the effective potential energy. The correct expression for the effective potential energy $V_{eff}$ should include a positive coefficient for the... | |
E4_verifier:physgym_0649_tiny_aya | physgym_0649 | ELECTROMAGNETISM | Step 1: Define the Problem
- **Setup**: Two uniformly charged rings (radius $R$, charge $Q$) are placed on planes $z = \pm l$, perpendicular to the $z$-axis. A point charge $-q$ and mass $m$ is at the origin.
- **Objective**: Find the minimum $\Omega$ such that the point charge remains stable in all directions after a ... | To solve this problem correctly, you must consider the torqueexperienced by the point charge due to the rotating rings and balance it against the centripetal force required to keep it in its initial plane. The governing equation is Torque = Angular Momentum, or: $$ \tau = L = mvr $$ where $\tau$ is the torque, $L$ is a... | [[CHALLENGE]] | real_model_a_failure | part_3_real_tiny_aya_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To solve this problem correctly, you must consider the torqueexperienced by the point charge due to the rotating rings and balance it against the centripetal force required to keep it in its initial plane. The governing equation is Torque = Angular Momentum, or: $$ \tau = L = mvr $$ where $\tau$ is the torque, $L$ is a... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To solve this problem correctly, you must consider the torqueexperienced by the point charge due to the rotating rings and balance it against the centripetal force required to keep it in its initial plane. The governing equation is Torque = Angular Momentum, or: $$ \tau = L = mvr $$ where $\tau$ is the torque, $L$ is a... | |
E4_verifier:physgym_0653_err_aya8b | physgym_0653 | ELECTROMAGNETISM | Step 1: **Initial Setup:** As described in the problem, we have a simple pendulum system with a massless string and a conductor ring suspended at its end. The ring's initial horizontal velocity *v* is due to a uniform magnetic field *B*.
Step 2: **Superconducting Properties:** The magnetic flux *Φ* through the ring rem... | To correctly solve for the minimum initial velocity $v_0$, you must consider the dynamic equation of the pendulum, including both the gravitational torque and the torque from the Ampere force. The governing equation is derived from the conservation of energy, relating the initial potential energy to the final kinetic e... | [[CHALLENGE]] | incorrect_physical_model | part_2_synthetic_aya8b_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correctly solve for the minimum initial velocity $v_0$, you must consider the dynamic equation of the pendulum, including both the gravitational torque and the torque from the Ampere force. The governing equation is derived from the conservation of energy, relating the initial potential energy to the final kinetic e... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correctly solve for the minimum initial velocity $v_0$, you must consider the dynamic equation of the pendulum, including both the gravitational torque and the torque from the Ampere force. The governing equation is derived from the conservation of energy, relating the initial potential energy to the final kinetic e... | |
E4_verifier:physgym_0653_tiny_aya | physgym_0653 | ELECTROMAGNETISM | Step 1: **Governing Equations:
- Gravitational force: \( F_g = mg \sin \theta \)
- Magnetic force: \( F_m = I \frac{d}{dt} (L B) \)
- Torque equation: \( \tau = I \omega \)
- Angular momentum conservation: \( L = m r^2 \omega \)
Step 2: **Initial Conditions:
- The ring is initially horizontal, so \( \theta = 0 \) and \... | To solve this problem correctly, you must consider the conservation of angular momentum and the relationship between the magnetic force, torque, and the pendulum's motion. The governing equation for the torque due to the Ampere force on the ring is τ = I * B * l * a, where *l* is the length of the current-carrying part... | [[CHALLENGE]] | real_model_a_failure | part_3_real_tiny_aya_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To solve this problem correctly, you must consider the conservation of angular momentum and the relationship between the magnetic force, torque, and the pendulum's motion. The governing equation for the torque due to the Ampere force on the ring is τ = I * B * l * a, where *l* is the length of the current-carrying part... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To solve this problem correctly, you must consider the conservation of angular momentum and the relationship between the magnetic force, torque, and the pendulum's motion. The governing equation for the torque due to the Ampere force on the ring is τ = I * B * l * a, where *l* is the length of the current-carrying part... | |
E4_verifier:physgym_0702_err_aya8b | physgym_0702 | ELECTROMAGNETISM | Step 1: We start by recognizing that the system consists of four infinite, parallel charged lines, creating a uniform electric field in the $xy$-plane. The electric field due to each line can be calculated using the formula for the electric field of a charged line: $E = \frac{\lambda}{2\pi\varepsilon_{0}a}$, where $\la... | Correct the derivation by utilizing the principle of centripetal force and the concept of angular velocity in a rotating frame. The minimum angular velocity $\Omega_{\min}$ required to maintain stable equilibrium is achieved when the centripetal force balances the ion's weight. Centripetal force ($F_{\text{centripetal}... | [[CHALLENGE]] | incorrect_formula | part_2_synthetic_aya8b_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | Correct the derivation by utilizing the principle of centripetal force and the concept of angular velocity in a rotating frame. The minimum angular velocity $\Omega_{\min}$ required to maintain stable equilibrium is achieved when the centripetal force balances the ion's weight. Centripetal force ($F_{\text{centripetal}... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | Correct the derivation by utilizing the principle of centripetal force and the concept of angular velocity in a rotating frame. The minimum angular velocity $\Omega_{\min}$ required to maintain stable equilibrium is achieved when the centripetal force balances the ion's weight. Centripetal force ($F_{\text{centripetal}... | |
E4_verifier:physgym_0702_tiny_aya | physgym_0702 | ELECTROMAGNETISM | Step 1: **Electric Field Contributions:
- The electric field due to a charged line at a distance \( r \) is given by \( E = \frac{\lambda}{2\pi\varepsilon_0 r} \) for a line along the \( z \)-axis.
- For a line at \( (a, 0) \), the electric field at \( (0, a) \) is \( E_1 = \frac{\lambda}{2\pi\varepsilon_0 a} \).
- For... | To find the minimum angular velocity $\Omega_{\min}$ required for the ion's stable equilibrium, apply the concept of centripetal force balance. The ion experiences a centripetal force due to the rotating charged lines, which must equal the force due to the electrostatic potential gradient. Use the equation: $$\omega^2 ... | [[CHALLENGE]] | real_model_a_failure | part_3_real_tiny_aya_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To find the minimum angular velocity $\Omega_{\min}$ required for the ion's stable equilibrium, apply the concept of centripetal force balance. The ion experiences a centripetal force due to the rotating charged lines, which must equal the force due to the electrostatic potential gradient. Use the equation: $$\omega^2 ... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To find the minimum angular velocity $\Omega_{\min}$ required for the ion's stable equilibrium, apply the concept of centripetal force balance. The ion experiences a centripetal force due to the rotating charged lines, which must equal the force due to the electrostatic potential gradient. Use the equation: $$\omega^2 ... | |
E4_verifier:physgym_0039_err_aya8b | physgym_0039 | MECHANICS | Step 1: **Coordinate System and Forces:
We set up a coordinate system with the origin at the top of the block. Let $y$ be the vertical displacement from its equilibrium position. The net force on the block, when pushed and released, includes the buoyant force and gravity.
Step 2: **Buoyant Force:
The buoyant force, $F_... | Correct the derivation by using the correct relationship between the spring constant $k$, mass $m$, and the period $T$ in simple harmonic motion: the equation $\ddot{y} + \frac{k}{m}y = 0$ implies $T = 2\pi\sqrt{\frac{m}{k}}$. | [[CHALLENGE]] | algebraic_error | part_2_synthetic_aya8b_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | Correct the derivation by using the correct relationship between the spring constant $k$, mass $m$, and the period $T$ in simple harmonic motion: the equation $\ddot{y} + \frac{k}{m}y = 0$ implies $T = 2\pi\sqrt{\frac{m}{k}}$.
[[CHALLENGE]] | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | Correct the derivation by using the correct relationship between the spring constant $k$, mass $m$, and the period $T$ in simple harmonic motion: the equation $\ddot{y} + \frac{k}{m}y = 0$ implies $T = 2\pi\sqrt{\frac{m}{k}}$.
[[CHALLENGE]] | |
E4_verifier:physgym_0039_tiny_aya | physgym_0039 | MECHANICS | Step 1: Governing Physical Equations
The motion of the wooden block can be described by Newton's second law of motion, which relates force to mass and acceleration. The force acting on the block is the difference between the upward force from the water and the downward force due to gravity.
Step 2: Forces Acting on the... | You have made several errors particularly in identifying the governing equations and applying them correctly. The correct approach is to recognize that the block undergoes simple harmonic motion due to the restoring force provided by the buoyant force. This leads to the equation of motion $\rho_{1}H S \ddot{y} + \rho_{... | [[CHALLENGE]] | real_model_a_failure | part_3_real_tiny_aya_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | You have made several errors particularly in identifying the governing equations and applying them correctly. The correct approach is to recognize that the block undergoes simple harmonic motion due to the restoring force provided by the buoyant force. This leads to the equation of motion $\rho_{1}H S \ddot{y} + \rho_{... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | You have made several errors particularly in identifying the governing equations and applying them correctly. The correct approach is to recognize that the block undergoes simple harmonic motion due to the restoring force provided by the buoyant force. This leads to the equation of motion $\rho_{1}H S \ddot{y} + \rho_{... | |
E4_verifier:physgym_0041_err_aya8b | physgym_0041 | MECHANICS | Step 1: **Initial Conditions:** At liftoff, the net force on the tester is $N_1 = (1 + k_1)mg_0$, as stated, assuming the correct units for mass $m$ (kg) and acceleration $g_0$ (m/s²).
Step 2: **Gravitational Acceleration at Height $h$:** Following the reference, we adjust the gravitational acceleration formula to $g =... | To resolve the flaws you must: 1. Ensure Consistent Units:Convert the Earth's radius $R$ from centimeters to meters. 2. Correct Discriminant Calculation:Use the correct values for $a$, $b$, and $c$ in the quadratic formula's discriminant, accounting for the dimensional consistency. 3. Apply the Correct Constraint:The d... | [[CHALLENGE]] | dimensional_unit_error | part_2_synthetic_aya8b_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To resolve the flaws you must: 1. Ensure Consistent Units:Convert the Earth's radius $R$ from centimeters to meters. 2. Correct Discriminant Calculation:Use the correct values for $a$, $b$, and $c$ in the quadratic formula's discriminant, accounting for the dimensional consistency. 3. Apply the Correct Constraint:The d... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To resolve the flaws you must: 1. Ensure Consistent Units:Convert the Earth's radius $R$ from centimeters to meters. 2. Correct Discriminant Calculation:Use the correct values for $a$, $b$, and $c$ in the quadratic formula's discriminant, accounting for the dimensional consistency. 3. Apply the Correct Constraint:The d... | |
E4_verifier:physgym_0041_tiny_aya | physgym_0041 | MECHANICS | Step 1: Governing Physical Equations
The problem involves a rocket ascending vertically with constant acceleration \( a = k_1 g_0 \), where \( g_0 \) is the gravitational acceleration at Earth's surface. The pressure exerted by the instrument on the bottom of the groove is \( k_2 \) times the pressure at liftoff. We ne... | To correctly solve this problem, you must use the quadratic formula derived from the relationship between the pressure at height $h$ and the initial pressure, along with the constraint on the discriminant. The governing equation is: $$h^2 + 2k_1Rh + k_1R^2(k_2 - 1) = 0$$ This quadratic equation must have real solutions... | [[CHALLENGE]] | real_model_a_failure | part_3_real_tiny_aya_challenged | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correctly solve this problem, you must use the quadratic formula derived from the relationship between the pressure at height $h$ and the initial pressure, along with the constraint on the discriminant. The governing equation is: $$h^2 + 2k_1Rh + k_1R^2(k_2 - 1) = 0$$ This quadratic equation must have real solutions... | Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[... | To correctly solve this problem, you must use the quadratic formula derived from the relationship between the pressure at height $h$ and the initial pressure, along with the constraint on the discriminant. The governing equation is: $$h^2 + 2k_1Rh + k_1R^2(k_2 - 1) = 0$$ This quadratic equation must have real solutions... |
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Adversarial Physics Verifier — Experiment E4 (verifier_e4_curated)
Codebase & Pipeline Repository: vital-ar/adversarial_physics_problems_solver
Dataset Overview
verifier_e4_curated is the Stage 2 Verifier SFT dataset used to train Model B (Verifier) (CohereLabs/tiny-aya-global via AlignTune QLoRA) in the Adversarial Physics Problem Solver–Verifier architecture.
Given a physics problem_statement and a candidate proposed_solution, Model B learns to:
- Detect physical, mathematical, algebraic, boundary-condition, and dimensional errors across both synthetic controlled errors (
CohereForAI/aya-expanse-8b) and real zero-shot student failures (CohereLabs/tiny-aya-global), emitting a concise (1–2sentence,<= 192token), strictly self-contained corrective directive terminated by[[CHALLENGE]]. - Recognize physically and mathematically valid derivations across all 6 physics domains without false-positive nitpicking, emitting a brief confirmation terminated by
[[APPROVE]].
Dataset Composition & CuratorKIT Curation Pipeline
| Part | Verdict | Examples | Source & Curation Details |
|---|---|---|---|
| Part 1 — Domain-Balanced Approvals | [[APPROVE]] |
180 | Equally balanced across all 6 physics domains (30 per domain = 26.4% of the 681 canonical problems), drawn from Canonical Solution 1 (681) and Minority-Domain Alternative Solution 2 (82). |
| Part 2 — Controlled Physics Error Challenges | [[CHALLENGE]] |
681 | 1 synthetic flawed derivation per canonical problem generated by CohereForAI/aya-expanse-8b across 5 physics error categories (incorrect_formula, algebraic_error, sign_or_direction_error, boundary_condition_error, dimensional_unit_error), sanitized so no meta-headers ("Flawed Derivation") give away the label. |
| Part 3 — Real Student Zero-Shot Challenges | [[CHALLENGE]] |
650 | Real zero-shot CohereLabs/tiny-aya-global failures that survived deterministic ANSWER: mismatch filtering (28 accidentally-correct solutions removed) and CuratorKIT trajectory deduplication (3 near-duplicate trajectories removed before directive generation). |
| Total | [[CHALLENGE]] / [[APPROVE]] |
1,511 | 100% verified by validate_verifier_directive (zero final-answer leaks, zero step/line number references, zero relative references) and CuratorKIT TextCleaner, SchemaGate, ExactDeduplicator, and MinHashDeduplicator. |
Domain Breakdown of [[APPROVE]] vs. [[CHALLENGE]]
| Physics Domain | [[APPROVE]] (Part 1) |
[[CHALLENGE]] (Parts 2 & 3) |
Total |
|---|---|---|---|
ELECTROMAGNETISM |
30 | 294 | 324 |
MECHANICS |
30 | 752 | 782 |
MODERN_PHYSICS |
30 | 123 | 153 |
OPTICS |
30 | 48 | 78 |
RELATIVITY |
30 | 70 | 100 |
THERMODYNAMICS |
30 | 44 | 74 |
| Total | 180 | 1,331 | 1,511 |
Schema
Each record in train.jsonl follows the canonical VerifierTrajectory JSON schema:
{
"trajectory_id": "E4_verifier:physgym_0010_tiny_aya",
"problem_id": "physgym_0010",
"domain": "ELECTROMAGNETISM",
"proposed_solution": "Step 1: ...\nANSWER: ...",
"directive_text": "To correctly determine the electric field intensity on the surface of the conductor at the xOz plane, you must apply Gauss's Law to a Gaussian surface that encloses the point charge and consider the symmetry of the problem.",
"verdict": "[[CHALLENGE]]",
"error_type": "real_model_a_failure",
"part": "part_3_real_tiny_aya_challenged",
"prompt": "### Physics Problem:\n...\n\n### Proposed Solution:\n...\n\n### Instructions:\nVerify the proposed solution. Provide a 1-2 sentence self-contained physical assessment (without referencing step/line numbers or leaking the final answer) and end on a new line with `[[APPROVE]]` or `[[CHALLENGE]]`.",
"target_output": "To correctly determine the electric field intensity on the surface of the conductor at the xOz plane, you must apply Gauss's Law to a Gaussian surface that encloses the point charge and consider the symmetry of the problem.\n[[CHALLENGE]]"
}
Provenance & Integrity Files
manifest.json—CuratorKITstage gate counts and deduplication statistics.lexsi_provenance.json—lexsi.provenance/1lineage record consumed byAlignTuneandAuditKIT.checksums.txt— SHA-256 digests of all dataset files.- Project Repository & Codebase: https://github.com/vital-ar/adversarial_physics_problems_solver
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