Datasets:

vital-ar
/

physics-solver-e4-adversarial-v3

trajectory_id stringlengths 32 43	problem_id stringlengths 11 22	domain stringclasses 6 values	accumulated_directives listlengths 0 2	prompt stringlengths 298 2.52k	target_output stringlengths 441 2.88k	source_candidate_ids listlengths 1 3	dataset_tier stringclasses 1 value	source stringclasses 3 values	instruction stringlengths 298 2.52k	output stringlengths 441 2.88k
E4_solver:lit_em_0001:turn0_sol1	lit_em_0001	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A hydrogen atom consists of an electron and a proton. For our pr...	Derivation: 1. Understand the Problem: We are given that a muon, with a mass 206.77 times greater than an electron, replaces an electron in a hydrogen atom's orbit, resulting in an orbit 206.77 times closer to the proton. We need to find the ratio of the electric forces experienced by the muon and the electron...	[ "lit_em_0001_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A hydrogen atom consists of an electron and a proton. For our pr...	Derivation: 1. Understand the Problem: We are given that a muon, with a mass 206.77 times greater than an electron, replaces an electron in a hydrogen atom's orbit, resulting in an orbit 206.77 times closer to the proton. We need to find the ratio of the electric forces experienced by the muon and the electron...
E4_solver:lit_em_0002:turn0_sol1	lit_em_0002	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: In Fullerton, California, the earth’s magnetic field is in the d...	Derivation: Step 1: Step 2: 1. We are given that the Earth's magnetic field in Fullerton, California, makes an angle of 59 degrees with the horizontal. 2. To find the ratio of the vertical component (By) to the horizontal component (Bx), we can use the tangent function, which relates the opposite side (vertical co...	[ "lit_em_0002_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: In Fullerton, California, the earth’s magnetic field is in the d...	Derivation: Step 1: Step 2: 1. We are given that the Earth's magnetic field in Fullerton, California, makes an angle of 59 degrees with the horizontal. 2. To find the ratio of the vertical component (By) to the horizontal component (Bx), we can use the tangent function, which relates the opposite side (vertical co...
E4_solver:lit_em_0005:turn0_sol1	lit_em_0005	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Consider the following four potentials, which exist in some regi...	Derivation: Step 1: In each case, the electric field is minus the derivative of the potential. Potentials 1 and 2 give the same field,E =−a. Step 2: These are uniform fields. The constant a is the strength of the field, and b is just an arbitrary constant with no physical significance. Step 3: Potentials 3 and 4 bo...	[ "lit_em_0005_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Consider the following four potentials, which exist in some regi...	Derivation: Step 1: In each case, the electric field is minus the derivative of the potential. Potentials 1 and 2 give the same field,E =−a. Step 2: These are uniform fields. The constant a is the strength of the field, and b is just an arbitrary constant with no physical significance. Step 3: Potentials 3 and 4 bo...
E4_solver:lit_em_0007:turn0_sol1	lit_em_0007	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Lightning discharges a cloud during an electrical storm. Suppose...	Derivation: 1. Given: The current in the lightning bolt varies with time as I = bt, where b is a constant. We aim to find the cloud's charge, q, as a function of time. 2. Integrate current with respect to time: To find the charge, we integrate the current with respect to time. This gives us: q = ∫I dt 3....	[ "lit_em_0007_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Lightning discharges a cloud during an electrical storm. Suppose...	Derivation: 1. Given: The current in the lightning bolt varies with time as I = bt, where b is a constant. We aim to find the cloud's charge, q, as a function of time. 2. Integrate current with respect to time: To find the charge, we integrate the current with respect to time. This gives us: q = ∫I dt 3....
E4_solver:lit_em_0011:turn0_sol1	lit_em_0011	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Each of the following circuit elements or combination of element...	Derivation: Step 1: Derivation for an Inductor: Step 2: 1. We are given that the impedance of an inductor is $i\omega L$, where $i$ is the imaginary unit, $\omega$ is the angular frequency, and $L$ is the inductance. 2. The argument (or phase angle) of the impedance is defined as the angle between the voltage a...	[ "lit_em_0011_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Each of the following circuit elements or combination of element...	Derivation: Step 1: Derivation for an Inductor: Step 2: 1. We are given that the impedance of an inductor is $i\omega L$, where $i$ is the imaginary unit, $\omega$ is the angular frequency, and $L$ is the inductance. 2. The argument (or phase angle) of the impedance is defined as the angle between the voltage a...
E4_solver:lit_em_0012:turn0_sol1	lit_em_0012	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose you are holding your hands in front of you, 10 cm apart....	Derivation: 1. Mass of a hand: We are given that the mass of one hand is approximately 0.2 kg. 2. Mass of an oxygen atom: The mass of a proton is 1.7 × 10^-27 kg, and since an oxygen atom is approximately 16 times the mass of a proton, the mass of an oxygen atom is 16 * 1.7 × 10^-27 kg = 27.2 × 10^-27 kg. ...	[ "lit_em_0012_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose you are holding your hands in front of you, 10 cm apart....	Derivation: 1. Mass of a hand: We are given that the mass of one hand is approximately 0.2 kg. 2. Mass of an oxygen atom: The mass of a proton is 1.7 × 10^-27 kg, and since an oxygen atom is approximately 16 times the mass of a proton, the mass of an oxygen atom is 16 * 1.7 × 10^-27 kg = 27.2 × 10^-27 kg. ...
E4_solver:lit_em_0013:turn0_sol1	lit_em_0013	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: In our by-now-familiar neuron, the voltage difference between th...	Derivation: 1. Given: We are provided with the voltage difference between the inner and outer surfaces of the cell membrane, Vout - Vin = -70 mV, and the thickness of the membrane, d = 6.0 nm. 2. Define Variables: Let V be the voltage difference and d be the thickness. We want to find the electric field, E...	[ "lit_em_0013_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: In our by-now-familiar neuron, the voltage difference between th...	Derivation: 1. Given: We are provided with the voltage difference between the inner and outer surfaces of the cell membrane, Vout - Vin = -70 mV, and the thickness of the membrane, d = 6.0 nm. 2. Define Variables: Let V be the voltage difference and d be the thickness. We want to find the electric field, E...
E4_solver:lit_em_0015:turn0_sol1	lit_em_0015	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Show that the equation VL = L ∆I/∆t has the right units.	Derivation: Step 1: Step 2: 1. We start with the given equation: VL = L ∆I/∆t, where V is the voltage, L is the inductance, ∆I/∆t is the rate of change of current with respect to time. 2. We know that 1 henry (H) is defined as 1 joule per ampere squared (J/A²). 3. Expressing L in terms of H, we have L = H. 4. Subs...	[ "lit_em_0015_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Show that the equation VL = L ∆I/∆t has the right units.	Derivation: Step 1: Step 2: 1. We start with the given equation: VL = L ∆I/∆t, where V is the voltage, L is the inductance, ∆I/∆t is the rate of change of current with respect to time. 2. We know that 1 henry (H) is defined as 1 joule per ampere squared (J/A²). 3. Expressing L in terms of H, we have L = H. 4. Subs...
E4_solver:lit_em_0016:turn0_sol1	lit_em_0016	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: You can’t use a light wave to see things that are smaller than t...	Derivation: 1. Understand the Problem: We are given that the size of an atom is approximately $10^{-10}$ meters, and we need to determine if visible light can be used to image these atoms. 2. Define Wavelength of Visible Light: Visible light has wavelengths that range from approximately 400 to 700 nanom...	[ "lit_em_0016_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: You can’t use a light wave to see things that are smaller than t...	Derivation: 1. Understand the Problem: We are given that the size of an atom is approximately $10^{-10}$ meters, and we need to determine if visible light can be used to image these atoms. 2. Define Wavelength of Visible Light: Visible light has wavelengths that range from approximately 400 to 700 nanom...
E4_solver:lit_em_0017:turn0_sol1	lit_em_0017	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose that, as in minilab 1 on p. 38, we have a known ambient ...	Derivation: 1. Define the problem setup: We have two magnetic fields, $B_a$ (ambient field) and $B_m$ (unknown field), that are perpendicular to each other. The unknown field $B_m$ causes a deflection of a compass measuring the total field, which is at an angle $\theta$ from the ambient field $B_a$. 2. **Unde...	[ "lit_em_0017_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose that, as in minilab 1 on p. 38, we have a known ambient ...	Derivation: 1. Define the problem setup: We have two magnetic fields, $B_a$ (ambient field) and $B_m$ (unknown field), that are perpendicular to each other. The unknown field $B_m$ causes a deflection of a compass measuring the total field, which is at an angle $\theta$ from the ambient field $B_a$. 2. **Unde...
E4_solver:lit_em_0018:turn0_sol1	lit_em_0018	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: The gravitational field g of a typical asteroid is so weak that ...	Derivation: 1. Given Data: We are provided with the following values: gravitational field $g = -1.0 \times 10^{-4} \, \text{m/s}^2$, electric field $E = 40 \, \text{V/m}$, mass $m = 1.0 \times 10^{-8} \, \text{kg}$, and charge $q = 1.0 \times 10^{-14} \, \text{C}$. 2. Electric Force Calculation: The elect...	[ "lit_em_0018_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: The gravitational field g of a typical asteroid is so weak that ...	Derivation: 1. Given Data: We are provided with the following values: gravitational field $g = -1.0 \times 10^{-4} \, \text{m/s}^2$, electric field $E = 40 \, \text{V/m}$, mass $m = 1.0 \times 10^{-8} \, \text{kg}$, and charge $q = 1.0 \times 10^{-14} \, \text{C}$. 2. Electric Force Calculation: The elect...
E4_solver:lit_em_0019:turn0_sol1	lit_em_0019	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: The voltage difference between the ends of a AAA battery is 1.5 ...	Derivation: 1. Problem Statement: We are given a AAA battery with a voltage difference of 1.5 V across its ends and a length of 42 mm. We need to find the magnitude of the electric field inside the battery, assuming it is constant. 2. Electric Field Definition: The electric field (E) is defined as the vol...	[ "lit_em_0019_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: The voltage difference between the ends of a AAA battery is 1.5 ...	Derivation: 1. Problem Statement: We are given a AAA battery with a voltage difference of 1.5 V across its ends and a length of 42 mm. We need to find the magnitude of the electric field inside the battery, assuming it is constant. 2. Electric Field Definition: The electric field (E) is defined as the vol...
E4_solver:lit_em_0020:turn0_sol1	lit_em_0020	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose a charged particle is moving through a region of space i...	Derivation: 1. Electric and Magnetic Forces: A charged particle experiences two forces in this setup: an electric force (F_e) and a magnetic force (F_m). The electric force acts in the direction of the electric field (E) and its magnitude depends on the particle's charge (q), the electric field strength (E), a...	[ "lit_em_0020_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose a charged particle is moving through a region of space i...	Derivation: 1. Electric and Magnetic Forces: A charged particle experiences two forces in this setup: an electric force (F_e) and a magnetic force (F_m). The electric force acts in the direction of the electric field (E) and its magnitude depends on the particle's charge (q), the electric field strength (E), a...
E4_solver:lit_em_0021:turn0_sol1	lit_em_0021	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose a charged particle is moving through a region of space i...	Derivation: Step 1: The electric force is either in the same direction as the electric field or in the opposite direction, depending on whether the particles are positively or negatively charged. Step 2: Since the magnetic force is perpendicular to both v and B and the question states that the electric field is per...	[ "lit_em_0021_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose a charged particle is moving through a region of space i...	Derivation: Step 1: The electric force is either in the same direction as the electric field or in the opposite direction, depending on whether the particles are positively or negatively charged. Step 2: Since the magnetic force is perpendicular to both v and B and the question states that the electric field is per...
E4_solver:lit_em_0022:turn0_sol1	lit_em_0022	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: At a particular point in time and space, an electromagnetic plan...	Derivation: Step 1: Since it’s a plane wave, the direction of the magnetic field is perpendicular to the electric field and the direction of propagation, so it must be in the ±y direction. Step 2: What gives E× B in the positive z direction is if the magnetic field is in the +y direction. The Poynting vector is S =...	[ "lit_em_0022_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: At a particular point in time and space, an electromagnetic plan...	Derivation: Step 1: Since it’s a plane wave, the direction of the magnetic field is perpendicular to the electric field and the direction of propagation, so it must be in the ±y direction. Step 2: What gives E× B in the positive z direction is if the magnetic field is in the +y direction. The Poynting vector is S =...
E4_solver:lit_em_0024:turn0_sol1	lit_em_0024	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Deuterium is an isotope of hydrogen in which the nucleus has one...	Derivation of Deuterium's g-Factor: 1. Define the g-Factor: The g-factor (g) of a particle is a dimensionless quantity that relates its magnetic moment (μ) to its angular momentum (L) and mass (m). It is given by the formula: g = (e/m)(\|L\|/\|m\|), where e is the elementary charge. 2. Given Values: We are pr...	[ "lit_em_0024_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Deuterium is an isotope of hydrogen in which the nucleus has one...	Derivation of Deuterium's g-Factor: 1. Define the g-Factor: The g-factor (g) of a particle is a dimensionless quantity that relates its magnetic moment (μ) to its angular momentum (L) and mass (m). It is given by the formula: g = (e/m)(\|L\|/\|m\|), where e is the elementary charge. 2. Given Values: We are pr...
E4_solver:lit_em_0025:turn0_sol1	lit_em_0025	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Find an expression for the energy stored in a capacitor in terms...	Step-by-Step Derivation: 1. Start with the definition of electric potential energy: The energy stored in a capacitor is given by the formula U = q^2 / 2C, where q is the charge and C is the capacitance. 2. Express charge in terms of voltage: We know that the charge on a capacitor is equal to the product o...	[ "lit_em_0025_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Find an expression for the energy stored in a capacitor in terms...	Step-by-Step Derivation: 1. Start with the definition of electric potential energy: The energy stored in a capacitor is given by the formula U = q^2 / 2C, where q is the charge and C is the capacitance. 2. Express charge in terms of voltage: We know that the charge on a capacitor is equal to the product o...
E4_solver:lit_em_0026:turn0_sol1	lit_em_0026	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Each of the following circuit elements or combination of element...	Derivation for a 1 F Capacitor: 1. Given: The impedance of a capacitor is given by Z = -i/ωC, where ω is the angular frequency (ω = 2πf, f = 1 Hz) and C is the capacitance (C = 1 F). 2. Calculate Argument of Impedance: The argument (or phase angle) of the impedance, arg(Z), is the angle between the voltage...	[ "lit_em_0026_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Each of the following circuit elements or combination of element...	Derivation for a 1 F Capacitor: 1. Given: The impedance of a capacitor is given by Z = -i/ωC, where ω is the angular frequency (ω = 2πf, f = 1 Hz) and C is the capacitance (C = 1 F). 2. Calculate Argument of Impedance: The argument (or phase angle) of the impedance, arg(Z), is the angle between the voltage...
E4_solver:lit_em_0027:turn0_sol1	lit_em_0027	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose you are holding your hands in front of you, 10 cm apart....	Derivation: 1. Identify the Problem: We are asked to estimate the total repulsive force between all electrons in one hand and those in the other, held 10 cm (0.1 meters) apart. 2. Recall Coulomb's Law: Coulomb's Law states that the force (F) between two point charges (q1 and q2) separated by a distance (r...	[ "lit_em_0027_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose you are holding your hands in front of you, 10 cm apart....	Derivation: 1. Identify the Problem: We are asked to estimate the total repulsive force between all electrons in one hand and those in the other, held 10 cm (0.1 meters) apart. 2. Recall Coulomb's Law: Coulomb's Law states that the force (F) between two point charges (q1 and q2) separated by a distance (r...
E4_solver:lit_em_0028:turn0_sol1	lit_em_0028	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: The gap between the electrodes in an automobile engine’s spark p...	Derivation: Step 1: ## Derivation for Minimum Voltage Required (a) Step 2: 1. We are given that the required electric field (E) is $3.0 \times 10^6 \, \text{V/m}$. Step 3: 2. The definition of electric field is force per unit charge: $E = \frac{F}{q}$. Step 4: 3. For a uniform field between parallel pla...	[ "lit_em_0028_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: The gap between the electrodes in an automobile engine’s spark p...	Derivation: Step 1: ## Derivation for Minimum Voltage Required (a) Step 2: 1. We are given that the required electric field (E) is $3.0 \times 10^6 \, \text{V/m}$. Step 3: 2. The definition of electric field is force per unit charge: $E = \frac{F}{q}$. Step 4: 3. For a uniform field between parallel pla...
E4_solver:lit_em_0029:turn0_sol1	lit_em_0029	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: If you put four times more current through a solenoid, how many ...	Step-by-step Derivation: 1. Understand the relationship between current and magnetic field: The magnetic field (B) generated by a solenoid is directly proportional to the current (I) passing through it. This relationship is described by Ampere's Law. So, if we increase the current by a factor of 4, the magneti...	[ "lit_em_0029_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: If you put four times more current through a solenoid, how many ...	Step-by-step Derivation: 1. Understand the relationship between current and magnetic field: The magnetic field (B) generated by a solenoid is directly proportional to the current (I) passing through it. This relationship is described by Ampere's Law. So, if we increase the current by a factor of 4, the magneti...
E4_solver:lit_em_0030:turn0_sol1	lit_em_0030	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Find the energy stored in a capacitor in terms of its capacitanc...	Step-by-Step Derivation: 1. Understand the Capacitor's Role: We are dealing with a capacitor, a device that stores energy in an electric field. The energy stored in a capacitor is directly related to its charge and the voltage across it. 2. Define Variables: Let's use: - $C$ as the capacitance of the capa...	[ "lit_em_0030_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Find the energy stored in a capacitor in terms of its capacitanc...	Step-by-Step Derivation: 1. Understand the Capacitor's Role: We are dealing with a capacitor, a device that stores energy in an electric field. The energy stored in a capacitor is directly related to its charge and the voltage across it. 2. Define Variables: Let's use: - $C$ as the capacitance of the capa...
E4_solver:lit_em_0031:turn0_sol1	lit_em_0031	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: If you add a magnetic field with magnitude 1 T to a field of mag...	Derivation: 1. Given Information: We have two magnetic fields, one with a magnitude of 2 T (Tesla) and another with a magnitude of 1 T. We want to find the possible magnitudes of their vector sum. 2. Vector Addition: The vector sum of two magnetic fields B1 and B2 is calculated by adding their mag...	[ "lit_em_0031_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: If you add a magnetic field with magnitude 1 T to a field of mag...	Derivation: 1. Given Information: We have two magnetic fields, one with a magnitude of 2 T (Tesla) and another with a magnitude of 1 T. We want to find the possible magnitudes of their vector sum. 2. Vector Addition: The vector sum of two magnetic fields B1 and B2 is calculated by adding their mag...
E4_solver:lit_em_0032:turn0_sol1	lit_em_0032	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: In an inkjet printer, a series of droplets of ink, each with mas...	Derivation for Inkjet Printer Deflection Angle 1. Coordinate System: We establish a coordinate system with the x-axis along the initial direction of ink droplet motion and the y-axis perpendicular to the electric field E. 2. Independent Motion: Newton's laws imply that the ink droplets' x and y motion are...	[ "lit_em_0032_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: In an inkjet printer, a series of droplets of ink, each with mas...	Derivation for Inkjet Printer Deflection Angle 1. Coordinate System: We establish a coordinate system with the x-axis along the initial direction of ink droplet motion and the y-axis perpendicular to the electric field E. 2. Independent Motion: Newton's laws imply that the ink droplets' x and y motion are...
E4_solver:lit_em_0033:turn0_sol1	lit_em_0033	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A carbon dioxide molecule is structured like O-C-O, with all thr...	DERIVATION: 1. Given Potential: We are given that the potential (V) of a carbon dioxide molecule at a point along its axis is proportional to 1/r³, where r is the distance from the molecule. Mathematically, this is expressed as: V = -b/r³. 2. Electric Field Definition: The electric field (E) at a point is...	[ "lit_em_0033_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A carbon dioxide molecule is structured like O-C-O, with all thr...	DERIVATION: 1. Given Potential: We are given that the potential (V) of a carbon dioxide molecule at a point along its axis is proportional to 1/r³, where r is the distance from the molecule. Mathematically, this is expressed as: V = -b/r³. 2. Electric Field Definition: The electric field (E) at a point is...
E4_solver:lit_em_0034:turn0_sol1	lit_em_0034	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A charged particle is released from rest. We see it start to mov...	Derivation: 1. Understanding the Scenario: We are given a charged particle starting from rest and accelerating due to an unseen force. We need to determine the presence of electric (E) and magnetic (B) fields. 2. Force Analysis: The force on the particle is given by Newton's Second Law: F = ma. Since the ...	[ "lit_em_0034_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A charged particle is released from rest. We see it start to mov...	Derivation: 1. Understanding the Scenario: We are given a charged particle starting from rest and accelerating due to an unseen force. We need to determine the presence of electric (E) and magnetic (B) fields. 2. Force Analysis: The force on the particle is given by Newton's Second Law: F = ma. Since the ...
E4_solver:lit_em_0036:turn0_sol1	lit_em_0036	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A charged particle is released from rest. We see it start to mov...	Derivation: Step 1: There must be either E or B or both, since some force caused the particle to accelerate. It can’t be a pure magnetic field, because F = qv× B is zero for v = 0, so the particle would not have accelerated from rest. Step 2: It can’t be a pure electric field, because a uniform electric field would...	[ "lit_em_0036_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A charged particle is released from rest. We see it start to mov...	Derivation: Step 1: There must be either E or B or both, since some force caused the particle to accelerate. It can’t be a pure magnetic field, because F = qv× B is zero for v = 0, so the particle would not have accelerated from rest. Step 2: It can’t be a pure electric field, because a uniform electric field would...
E4_solver:lit_em_0037:turn0_sol1	lit_em_0037	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: You have a circuit consisting of two unknown resistors in series...	Derivation: 1. Understanding the Problem: We are presented with a circuit containing two unknown resistors in series and another circuit with two unknown resistors in parallel. The key question is what information we can gain about the resistors in the series circuit if the currents through them are equal. 2....	[ "lit_em_0037_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: You have a circuit consisting of two unknown resistors in series...	Derivation: 1. Understanding the Problem: We are presented with a circuit containing two unknown resistors in series and another circuit with two unknown resistors in parallel. The key question is what information we can gain about the resistors in the series circuit if the currents through them are equal. 2....
E4_solver:lit_em_0039:turn0_sol1	lit_em_0039	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose that a parallel-plate capacitor is constructed so that t...	Step-by-Step Derivation 1. Capacitance and Gap Relationship: We start by recognizing that the capacitance (C) of a parallel-plate capacitor is directly proportional to the area (A) of the plates and inversely proportional to the gap (h) between them: C ∝ 1/h. 2. Fixed Voltage (a): If we maintain a constan...	[ "lit_em_0039_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose that a parallel-plate capacitor is constructed so that t...	Step-by-Step Derivation 1. Capacitance and Gap Relationship: We start by recognizing that the capacitance (C) of a parallel-plate capacitor is directly proportional to the area (A) of the plates and inversely proportional to the gap (h) between them: C ∝ 1/h. 2. Fixed Voltage (a): If we maintain a constan...
E4_solver:lit_em_0040:turn0_sol1	lit_em_0040	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A Helmholtz coil is defined as a pair of identical circular coil...	Derivation: Step 1: The field at any point on the axis can be found by adding the contributions to the field from the two loops. The center of the entire apparatus does not, however, coincide with the center of either loop: in each case it is out of that plane by a distance h/2. Step 2: By symmetry, both these fiel...	[ "lit_em_0040_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A Helmholtz coil is defined as a pair of identical circular coil...	Derivation: Step 1: The field at any point on the axis can be found by adding the contributions to the field from the two loops. The center of the entire apparatus does not, however, coincide with the center of either loop: in each case it is out of that plane by a distance h/2. Step 2: By symmetry, both these fiel...
E4_solver:lit_em_0041:turn0_sol1	lit_em_0041	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Show that the magnitude of the electric field produced by a simp...	Derivation: 1. Dipole Definition: Consider a simple two-charge dipole with a positive charge $q$ at a distance $\frac{\mathcal{M}}{2}$ from the origin to the right, and a negative charge $-q$ at $\frac{\mathcal{M}}{2}$ to the left of the origin. 2. Electric Field Expression: The electric field $\mathbf{E}...	[ "lit_em_0041_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Show that the magnitude of the electric field produced by a simp...	Derivation: 1. Dipole Definition: Consider a simple two-charge dipole with a positive charge $q$ at a distance $\frac{\mathcal{M}}{2}$ from the origin to the right, and a negative charge $-q$ at $\frac{\mathcal{M}}{2}$ to the left of the origin. 2. Electric Field Expression: The electric field $\mathbf{E}...
E4_solver:lit_em_0042:turn0_sol1	lit_em_0042	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Consider two solenoids, one of which is smaller so that it can b...	Derivation: 1. Define Variables: Let $B$ be the magnetic field strength inside the larger solenoid, $b$ be the magnetic field strength inside the smaller solenoid, and $x$ be the length of the smaller solenoid inside the larger one. 2. Magnetic Energy Expression: The magnetic energy $U$ of a solenoid is g...	[ "lit_em_0042_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Consider two solenoids, one of which is smaller so that it can b...	Derivation: 1. Define Variables: Let $B$ be the magnetic field strength inside the larger solenoid, $b$ be the magnetic field strength inside the smaller solenoid, and $x$ be the length of the smaller solenoid inside the larger one. 2. Magnetic Energy Expression: The magnetic energy $U$ of a solenoid is g...
E4_solver:lit_em_0043:turn0_sol1	lit_em_0043	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Find the inductance of two identical inductors in parallel.	Step-by-Step Derivation: 1. Define Inductance: We start with the definition of inductance, which is given as U = LI²/2, where U is the energy stored in the inductor, L is the inductance, and I is the current. 2. Parallel Inductors as a Single Inductor: When two identical inductors are connected in paralle...	[ "lit_em_0043_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Find the inductance of two identical inductors in parallel.	Step-by-Step Derivation: 1. Define Inductance: We start with the definition of inductance, which is given as U = LI²/2, where U is the energy stored in the inductor, L is the inductance, and I is the current. 2. Parallel Inductors as a Single Inductor: When two identical inductors are connected in paralle...
E4_solver:lit_em_0045:turn0_sol1	lit_em_0045	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Evaluate your result, in radians, for m = 2.0 × 10−10 kg, u = 20...	Derivation: 1. Convert given values to SI units: - Mass (m) = 2.0 × 10^-10 kg - Velocity (u) = 20 m/s (already in SI) - Charge (q) = 2.0 × 10^-10 C - Electric field (E) = 2.0 × 10^5 V/m - Wavelength (w) = 3 mm = 3 × 10^-3 m 2. Use the formula for angular frequency (ω) in terms of E, q, and w: ω = (q * E) ...	[ "lit_em_0045_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Evaluate your result, in radians, for m = 2.0 × 10−10 kg, u = 20...	Derivation: 1. Convert given values to SI units: - Mass (m) = 2.0 × 10^-10 kg - Velocity (u) = 20 m/s (already in SI) - Charge (q) = 2.0 × 10^-10 C - Electric field (E) = 2.0 × 10^5 V/m - Wavelength (w) = 3 mm = 3 × 10^-3 m 2. Use the formula for angular frequency (ω) in terms of E, q, and w: ω = (q * E) ...
E4_solver:lit_em_0046:turn0_sol1	lit_em_0046	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Find the inductance of two identical inductors in parallel.	Derivation: Step 1: The definition of inductance is U = LI2/2. If we think of the pair of inductors as 91 one big inductor, then applying this definition, I has to be the total current that goes in. Step 2: This gets split in half, so the energy stored in a single inductor is Uone = Lone(I/2)2/2, and the total ener...	[ "lit_em_0046_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Find the inductance of two identical inductors in parallel.	Derivation: Step 1: The definition of inductance is U = LI2/2. If we think of the pair of inductors as 91 one big inductor, then applying this definition, I has to be the total current that goes in. Step 2: This gets split in half, so the energy stored in a single inductor is Uone = Lone(I/2)2/2, and the total ener...
E4_solver:lit_em_0047:turn0_sol1	lit_em_0047	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A charged particle is in a region of space in which there is a u...	Derivation: 1. Given Information: We have a charged particle with an initial velocity $ v_0 $ along the $ z $-axis (denoted as $ \hat{z} $) in a uniform magnetic field $ B $ also along the $ z $-axis (denoted as $ \hat{z} $). 2. Force on the Particle: The force acting on the particle due to th...	[ "lit_em_0047_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A charged particle is in a region of space in which there is a u...	Derivation: 1. Given Information: We have a charged particle with an initial velocity $ v_0 $ along the $ z $-axis (denoted as $ \hat{z} $) in a uniform magnetic field $ B $ also along the $ z $-axis (denoted as $ \hat{z} $). 2. Force on the Particle: The force acting on the particle due to th...
E4_solver:lit_em_0048:turn0_sol1	lit_em_0048	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A resistor has a voltage difference ∆ V across it, causing a cur...	Step-by-Step Derivation 1. Start with the given information: We know that a voltage difference (∆V) causes a current (I) to flow through a resistor. 2. Define power: Power (P) is the rate at which energy is transferred or dissipated. In this case, the energy is being converted into heat. 3. **Relate volt...	[ "lit_em_0048_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: A resistor has a voltage difference ∆ V across it, causing a cur...	Step-by-Step Derivation 1. Start with the given information: We know that a voltage difference (∆V) causes a current (I) to flow through a resistor. 2. Define power: Power (P) is the rate at which energy is transferred or dissipated. In this case, the energy is being converted into heat. 3. **Relate volt...
E4_solver:lit_em_0049:turn0_sol1	lit_em_0049	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: You have a circuit consisting of two unknown resistors in series...	Derivation: 1. Initial Observation: We are given a series circuit with two unknown resistors, and a parallel circuit with two unknown resistors, and we know that the voltage differences across the resistors in the series circuit are equal. 2. Current Equality: Since the resistors in the series circuit are...	[ "lit_em_0049_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: You have a circuit consisting of two unknown resistors in series...	Derivation: 1. Initial Observation: We are given a series circuit with two unknown resistors, and a parallel circuit with two unknown resistors, and we know that the voltage differences across the resistors in the series circuit are equal. 2. Current Equality: Since the resistors in the series circuit are...
E4_solver:lit_em_0050:turn0_sol1	lit_em_0050	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Four long wires are arranged, as shown, so that their crosssecti...	Derivation: 1. Visualize the magnetic fields: Imagine the four wires carrying current, with two wires having current flowing to the right (back wires) and the other two to the left (front wires). This creates four magnetic fields, each at a 45-degree angle to the current direction. 2. **Identify the direction...	[ "lit_em_0050_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Four long wires are arranged, as shown, so that their crosssecti...	Derivation: 1. Visualize the magnetic fields: Imagine the four wires carrying current, with two wires having current flowing to the right (back wires) and the other two to the left (front wires). This creates four magnetic fields, each at a 45-degree angle to the current direction. 2. **Identify the direction...
E4_solver:lit_em_0051:turn0_sol1	lit_em_0051	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: In example 6, p. 101, we analyzed a spherical capacitor. Buildin...	Derivation: 1. Recall the analysis from the example: We know that the potential difference (V) across the spherical capacitor is related to the charge (q) stored on each sphere by $V = \frac{q}{k(1/a - 1/b)}$, where $k$, $a$, and $b$ are constants from the spherical capacitor's geometry. 2. **Express charge i...	[ "lit_em_0051_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: In example 6, p. 101, we analyzed a spherical capacitor. Buildin...	Derivation: 1. Recall the analysis from the example: We know that the potential difference (V) across the spherical capacitor is related to the charge (q) stored on each sphere by $V = \frac{q}{k(1/a - 1/b)}$, where $k$, $a$, and $b$ are constants from the spherical capacitor's geometry. 2. **Express charge i...
E4_solver:lit_em_0052:turn0_sol1	lit_em_0052	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Each of the following circuit elements or combination of element...	Step-by-Step Derivation: 1. Identify Circuit Elements: We have a series combination of a 1 H inductor (L) and a 1 Ω resistor (R). 2. Calculate Impedance: The impedance (Z) of a series R-L circuit is given by Z = √(R^2 + (ωL)^2), where ω is the angular frequency (ω = 2πf, f = 1 Hz). - Z = √(1^2 + (2π(1 Hz)...	[ "lit_em_0052_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Each of the following circuit elements or combination of element...	Step-by-Step Derivation: 1. Identify Circuit Elements: We have a series combination of a 1 H inductor (L) and a 1 Ω resistor (R). 2. Calculate Impedance: The impedance (Z) of a series R-L circuit is given by Z = √(R^2 + (ωL)^2), where ω is the angular frequency (ω = 2πf, f = 1 Hz). - Z = √(1^2 + (2π(1 Hz)...
E4_solver:lit_em_0053:turn0_sol1	lit_em_0053	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: You have a circuit consisting of two unknown resistors in series...	Here's an improved solution based on the provided feedback: 1. Understand the Circuit Configuration: We have two circuits: one with resistors in series and another with resistors in parallel. 2. Equal Currents: Since the currents through each resistor in both circuits are the same (due to their equal voltage d...	[ "lit_em_0053_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: You have a circuit consisting of two unknown resistors in series...	Here's an improved solution based on the provided feedback: 1. Understand the Circuit Configuration: We have two circuits: one with resistors in series and another with resistors in parallel. 2. Equal Currents: Since the currents through each resistor in both circuits are the same (due to their equal voltage d...
E4_solver:lit_em_0055:turn0_sol1	lit_em_0055	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: In example 6, p. 101, we analyzed a spherical capacitor. Buildin...	Derivation: Step 1: Doing this using the relation U = q2/2C would be fairly complicated, since the energy density is not constant. Step 2: We would have to find the energy stored in a spherical shell betweenr andr +dr, and then integrate to findU. It’s much easier to do this using C =q/V , since we already know how...	[ "lit_em_0055_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: In example 6, p. 101, we analyzed a spherical capacitor. Buildin...	Derivation: Step 1: Doing this using the relation U = q2/2C would be fairly complicated, since the energy density is not constant. Step 2: We would have to find the energy stored in a spherical shell betweenr andr +dr, and then integrate to findU. It’s much easier to do this using C =q/V , since we already know how...
E4_solver:lit_em_0056:turn0_sol1	lit_em_0056	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Which of the following expressions make sense, and which are non...	Derivation: Step 1: Step 2: 1. We are given the expression (A × B) × C to analyze. 2. The cross product (A × B) is a vector, as it represents a perpendicular component to both vectors A and B. 3. Applying another cross product to this vector with C will result in a new vector, as the cross product operation is dis...	[ "lit_em_0056_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Which of the following expressions make sense, and which are non...	Derivation: Step 1: Step 2: 1. We are given the expression (A × B) × C to analyze. 2. The cross product (A × B) is a vector, as it represents a perpendicular component to both vectors A and B. 3. Applying another cross product to this vector with C will result in a new vector, as the cross product operation is dis...
E4_solver:lit_em_0057:turn0_sol1	lit_em_0057	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose you are holding your hands in front of you, 10 cm apart....	Derivation: Step 1: A force of 10 26 N is enough to pulverize a planet, so the cancellation is obviously very nearly perfect; even a very tiny percentage difference in charge would lead to a huge force. Step 2: If the fractional difference was ϵ, then the total charge on each hand would be ϵq, where q = 10 26e as e...	[ "lit_em_0057_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Suppose you are holding your hands in front of you, 10 cm apart....	Derivation: Step 1: A force of 10 26 N is enough to pulverize a planet, so the cancellation is obviously very nearly perfect; even a very tiny percentage difference in charge would lead to a huge force. Step 2: If the fractional difference was ϵ, then the total charge on each hand would be ϵq, where q = 10 26e as e...
E4_solver:lit_em_0058:turn0_sol1	lit_em_0058	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Consider the electric field created by a uniform ring of total c...	Derivation: Step 1: Imagine breaking the ring down into many tiny pieces, like a necklace. These pieces all contribute to the field at the point we’re talking about, and their contributions add like vectors. Step 2: The components of these vectors parallel to the plane of the ring all cancel out, but the components...	[ "lit_em_0058_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Consider the electric field created by a uniform ring of total c...	Derivation: Step 1: Imagine breaking the ring down into many tiny pieces, like a necklace. These pieces all contribute to the field at the point we’re talking about, and their contributions add like vectors. Step 2: The components of these vectors parallel to the plane of the ring all cancel out, but the components...
E4_solver:lit_em_0060:turn0_sol1	lit_em_0060	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Express the power dissipated by a resistor in terms of R and ∆V ...	Derivation: 1. Start with the given equation: We begin with the fundamental equation relating power (P), current (I), voltage drop (∆V), and resistance (R) in a circuit: P = I * ∆V. 2. Isolate I: To eliminate I, we rearrange the equation: I = P / ∆V. 3. Substitute I with the expression from step 2: N...	[ "lit_em_0060_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Express the power dissipated by a resistor in terms of R and ∆V ...	Derivation: 1. Start with the given equation: We begin with the fundamental equation relating power (P), current (I), voltage drop (∆V), and resistance (R) in a circuit: P = I * ∆V. 2. Isolate I: To eliminate I, we rearrange the equation: I = P / ∆V. 3. Substitute I with the expression from step 2: N...
E4_solver:lit_em_0062:turn0_sol1	lit_em_0062	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: This problem will lead you through the steps of applying the Bio...	Derivation: 1. Biot-Savart Law Setup: We start with the Biot-Savart law, which relates the magnetic field $B$ to the current $I$, length $c$, and position vector $\mathbf{r}$: $$ B = \frac{\mu_0 I c^2}{4 \pi} \int \frac{d\mathbf{l} \times \mathbf{r}}{\|\mathbf{r}\|^3} $$ 2. **Vector Cross Product Simplification...	[ "lit_em_0062_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: This problem will lead you through the steps of applying the Bio...	Derivation: 1. Biot-Savart Law Setup: We start with the Biot-Savart law, which relates the magnetic field $B$ to the current $I$, length $c$, and position vector $\mathbf{r}$: $$ B = \frac{\mu_0 I c^2}{4 \pi} \int \frac{d\mathbf{l} \times \mathbf{r}}{\|\mathbf{r}\|^3} $$ 2. **Vector Cross Product Simplification...
E4_solver:lit_em_0063:turn0_sol1	lit_em_0063	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: In example 8, p. 59, we found that the field of a uniform, strai...	Derivation: Step 1: V =− E dr =−2kλ lnr + const. This is a little bit of a strange-looking expression, because normally we can’t evaluate a transcendental function on an input that has units. Step 2: However, we can get away with this in the case of a log, because ln(a/b) = lna−lnb, so if we wanted to make the unit...	[ "lit_em_0063_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: In example 8, p. 59, we found that the field of a uniform, strai...	Derivation: Step 1: V =− E dr =−2kλ lnr + const. This is a little bit of a strange-looking expression, because normally we can’t evaluate a transcendental function on an input that has units. Step 2: However, we can get away with this in the case of a log, because ln(a/b) = lna−lnb, so if we wanted to make the unit...
E4_solver:lit_em_0064:turn0_sol1	lit_em_0064	ELECTROMAGNETISM	[]	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Each of the following circuit elements or combination of element...	Derivation for a 2 H inductor in parallel with a 1 F capacitor: 1. Impedance Calculation: The impedance of an inductor (Z_L) is given by Z_L = jωL, and the impedance of a capacitor (Z_C) is Z_C = 1/(jωC). For the given values, Z_L = j(2π)(2) = 4j Ω and Z_C = 1/(j(2π)(1)) = 1/(j2π) Ω. 2. *Parallel Impedance:...	[ "lit_em_0064_sol1" ]	E4_solver_adversarial	literature_curated	Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:. PROBLEM: Each of the following circuit elements or combination of element...	Derivation for a 2 H inductor in parallel with a 1 F capacitor: 1. Impedance Calculation: The impedance of an inductor (Z_L) is given by Z_L = jωL, and the impedance of a capacitor (Z_C) is Z_C = 1/(jωC). For the given values, Z_L = j(2π)(2) = 4j Ω and Z_C = 1/(j(2π)(1)) = 1/(j2π) Ω. 2. *Parallel Impedance:...

End of preview. Expand in Data Studio

Adversarial Physics Solver — Experiment E4 (`solver_e4_adversarial`)

Codebase & Pipeline Repository: vital-ar/adversarial_physics_problems_solver
Hugging Face Dataset ID: vital-ar/physics-solver-e4-adversarial-v3

Dataset Overview

solver_e4_adversarial is Dataset 4A (Model 4A / E4 Solver) in the Lexsi AI (CuratorKIT + AlignTune + AuditKIT) progressive data-curation ablation study for the Adversarial Physics Problem Solver–Verifier system.

solver_e4_adversarial is the final Stage 2 Solver SFT dataset (758 training trajectories) used to train Model A (Solver) to:

Solve university physics problems zero-shot at Turn 0 (622 canonical Reference Solution 1 + 1 verified alternative derivations in minority domains).
Re-solve problems from scratch at Turn 1 (1 hint, 63 rows) and Turn 2 (2 hints, 72 rows) conditioned on self-contained [[CHALLENGE]] directives from Model B (Verifier).

Role in Ablation Ladder: E1 (452 problems) $\rightarrow$ E2 (608 problems) $\rightarrow$ E3 (622 problems) $\rightarrow$ E4 Solver (758 trajectories) / E4 Verifier (720 trajectories).
Target Student Model: CohereLabs/tiny-aya-global (Model A — Solver, fine-tuned via AlignTune).

Splits & Domain Distribution

Split	File	Examples	Composition / Source
`train`	`train.jsonl`	758	`623` Turn-0 + `63` Turn-1 + `72` Turn-2
`validation`	`validation.jsonl`	48	Leak-free (`all-MiniLM-L6-v2` cosine `< 0.80`) held-out validation split
`test`	`test.jsonl`	55	Leak-free (`all-MiniLM-L6-v2` cosine `< 0.80`) held-out test split

Training Split Domain Breakdown (`758` Examples)

Physics Domain	Turn 0 (`Sol 1`)	Turn 0 (`Sol 2`)	Turn 1 (`1 Hint`)	Turn 2 (`2 Hints`)	Total	Share
`MECHANICS`	382	0	18	21	421	55.5%
`ELECTROMAGNETISM`	120	0	0	18	138	18.2%
`MODERN_PHYSICS`	52	0	17	0	69	9.1%
`RELATIVITY`	31	0	12	16	59	7.8%
`OPTICS`	19	1	0	17	37	4.9%
`THERMODYNAMICS`	18	0	16	0	34	4.5%
Total	622	1	63	72	758	100.0%

Multi-Turn Adversarial Trajectory Construction & Validation

Canonical & Minority-Domain Turn 0: Paired all canonical problems with Reference Solution 1 and added verified alternative physical derivations (Solution 2) in OPTICS, RELATIVITY, and THERMODYNAMICS verified against ground-truth answers.
50% / 50% Multi-Turn Hint Split: Partitioned the canonical pool cleanly so half receive 1 non-leaking [[CHALLENGE]] directive (Turn 1) and half receive 2 accumulated distinct directives (Turn 2), always targeting clean Reference Solution 1.
Anti-Leak Directive Validation (validate_verifier_directive): Stripped and rejected any directive leaking the final answer, intermediate target values, or meta evaluation wording.

Export-Time `CuratorKIT` Integrity Gates

curatorkit.normalizers.clean.TextCleaner: Normalized whitespace and Unicode math/control characters across 783 rows.
curatorkit.gates.schema.SchemaGate (min_tokens = 20, max_tokens = 8192): 783 passed, 25 rejected.
curatorkit.normalizers.dedup.ExactDeduplicator: 783 unique records retained (0 exact duplicates removed).
AnswerSentinelGate: 758 passed with canonical ANSWER: sentinel (25 rejected).

Schema

Each record in train.jsonl, validation.jsonl, test.jsonl follows the canonical schema below (with instruction, input, and output convenience columns also included for direct AlignTune SFT loading):

{
  "trajectory_id": "E4_solver:lit_em_0098:turn2_2hints",
  "problem_id": "lit_em_0098",
  "domain": "ELECTROMAGNETISM",
  "turn": 2,
  "accumulated_directives": [
    "You have correctly identified the governing physical equations and the need to relate velocity to the electric field and its integral. However, the integrat...",
    "To correctly derive the inductor's impedance, use the relationship between voltage, current, and inductance in the context of a sinusoidal current, consider..."
  ],
  "prompt": "Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with...",
  "target_output": "**Step-by-Step Solution:**\n\n1. **Starting Point:** The given equation for voltage difference across an inductor is  $V = \\frac{L\\Delta I}{\\Delta t}$ , where  $V$  is the...\n...\nANSWER: The impedance of an inductor is  $Z = L\\omega$ .",
  "source_candidate_ids": [
    "lit_em_0098_tiny_aya",
    "lit_em_0098_err_aya8b",
    "lit_em_0098_tiny_aya"
  ],
  "dataset_tier": "E4_solver_adversarial",
  "source": "literature_curated"
}

Provenance & Integrity Files

manifest.json — CuratorKIT stage gate counts and deduplication report.
lexsi_provenance.json — lexsi.provenance/1 lineage record consumed by AlignTune and AuditKIT.
checksums.txt — SHA-256 digests of all split, metadata, and README.md files.
Project Repository & Codebase: https://github.com/vital-ar/adversarial_physics_problems_solver