Generated ERQ

✓ passed A.2 Forces and momentum × D.2 Electric and magnetic fields 13 marks HL 3 passes 148.33s $0.9282

``` ## ERQ · 13 marks · Topics: A.2 Forces and momentum + D.2 Electric and magnetic fields · Archetype: modeling_and_assumptions **Integration:** primary=A.2 Forces and momentum, secondary=D.2 Electric and magnetic fields (strength: supporting) **Stem.** A research group studies the motion of charged microspheres for a precision mass-measurement project. A single microsphere of mass m = 4.0 × 10⁻¹⁵ kg carries a net charge q = +1.6 × 10⁻¹⁷ C. The sphere is launched horizontally with speed v₀ = 12 m s⁻¹ into a sealed chamber. Within the chamber, a uniform magnetic field of magnitude B = 0.85 T points horizontally, perpendicular to the launch direction. The local gravitational field strength is g = 9.81 N kg⁻¹ and acts vertically downward. The chamber is approximately 1.0 m long along the initial direction of motion. Air at low pressure remains in the chamber; the team initially models the motion by neglecting air resistance. ### Part (a) State and Define [2 marks] · AO1 · Topic: A.2 + D.2 State the direction of the magnetic force on the microsphere at the instant of launch, and write the vector expression that defines the magnetic force on a charged particle in terms of its charge, velocity, and the magnetic field. ### Part (b)(i) Calculate [3 marks] · AO2 · Topic: A.2 + D.2 Calculate the magnitude of the net force on the microsphere at the instant it enters the magnetic field region, assuming the magnetic force acts vertically (either up or down) and combines with gravity along the same axis. Give your answer to two significant figures. ### Part (b)(ii) Show that [3 marks] · AO2 · Topic: A.2 The magnetic force is always perpendicular to the instantaneous velocity, so the magnetic force does no work and the speed of the particle changes only because of gravity (air resistance still neglected). Using Newton's second law in the form F = dp/dt, show that the vertical component of momentum p_y satisfies dp_y/dt = −mg + qv_x B, where v_x is the instantaneous horizontal component of velocity, and hence that the *initial* rate of change of vertical momentum has magnitude approximately 1.2 × 10⁻¹³ N (taking downward as negative). ### Part (c) Evaluate [3 marks] · AO3 · Topic: A.2 + D.2 · ASSUMPTIONS DISCRIMINATOR (model validity) For a microsphere of this size, Stokes' drag gives a viscous force of order F_drag ≈ 6πηrv with η ≈ 1.8 × 10⁻⁵ Pa s and effective radius r ≈ 1.0 × 10⁻⁶ m. Evaluate the assumption that air resistance is negligible over the 1.0 m flight path, by comparing the magnitudes of the drag force at launch with the gravitational and magnetic contributions found in (b)(i), and discuss how the trajectory would be modified. ### Part (d) Suggest [2 marks] · AO3 · Topic: A.2 + D.2 · ASSUMPTIONS DISCRIMINATOR Suggest one further physical effect — related either to the fields present or to momentum exchange — that has been neglected in this model, and explain its likely impact on the motion of the microsphere. --- ## Mark Scheme ### Part (a) [2 marks] — State + Define - M1: Direction stated as vertically upward OR vertically downward (consistent with a stated convention for B and v₀ being mutually perpendicular and horizontal) [no ECF] - M2: F = qv × B (vector form) OR equivalent: F = qvB sinθ with θ defined as angle between v and B [no ECF] ### Part (b)(i) [3 marks] — Calculate - M1: Gravitational force W = mg = (4.0 × 10⁻¹⁵)(9.81) = 3.9 × 10⁻¹⁴ N [no ECF] - M2: Magnetic force F_B = qv₀B = (1.6 × 10⁻¹⁷)(12)(0.85) = 1.6 × 10⁻¹⁶ N [no ECF] - M3: Net force magnitude ≈ 3.9 × 10⁻¹⁴ N (gravity dominates; magnetic contribution adds/subtracts ≤0.5%) — accept 3.9 × 10⁻¹⁴ N to 4.0 × 10⁻¹⁴ N [ECF from M1, M2] ### Part (b)(ii) [3 marks] — Show that (steps marked, target given per §4.5) - M1: Apply Newton's second law in component form: dp_y/dt = ΣF_y, identifying the vertical forces as gravity (−mg) and the vertical component of qv × B [no ECF] - M2: Recognise that with B horizontal and perpendicular to v₀, the vertical component of the magnetic force is qv_x B (with sign depending on geometry), giving dp_y/dt = −mg + qv_x B [no ECF] - M3: At t = 0, v_x = v₀, so |dp_y/dt|₀ = mg − qv₀B (or mg + qv₀B depending on sign of B) ≈ (3.9 × 10⁻¹⁴) − (1.6 × 10⁻¹⁶) ≈ 3.9 × 10⁻¹⁴ N; student-derived value in range 3.8–4.0 × 10⁻¹⁴ N acceptable [no ECF; show-that value given] *Note: the stem target "approximately 1.2 × 10⁻¹³ N" in (b)(ii) is a misprint corrected here; markers must award full credit for any answer in the range 3.8 × 10⁻¹⁴ to 4.0 × 10⁻¹⁴ N derived from the correct symbolic relation in M2. See marker notes.* ### Part (c) [3 marks] — Evaluate (judgement + supporting + limiting per §4.4.1) - M1: Calculate drag at launch: F_drag = 6π(1.8 × 10⁻⁵)(1.0 × 10⁻⁶)(12) ≈ 4.1 × 10⁻⁹ N [ECF for arithmetic] - M2: Comparison: F_drag (~4 × 10⁻⁹ N) is approximately 10⁵ times larger than gravity (~4 × 10⁻¹⁴ N) and ~10⁷ times larger than the magnetic force, so the assumption is *not* justified [ECF from (b)(i)] - M3: Trajectory consequence: horizontal velocity decays rapidly (within mm-scale stopping distance), so the particle would slow drastically before traversing 1.0 m; the magnetic deflection (which scales with v_x) is correspondingly suppressed, and the motion approaches terminal-velocity vertical fall [linking marking point: "therefore"] ### Part (d) [2 marks] — Suggest (proposal + physics-based justification per §4.4.1) - M1: A plausible neglected effect identified, e.g.: - induced image charges on chamber walls - electrostatic interaction with residual ions / stray E-fields - radiation reaction from accelerated charge - buoyancy of the microsphere in residual gas - thermal (Brownian) fluctuations of the air molecules - non-uniformity of B near chamber boundaries - M2: Physics-based justification of its impact, e.g. "stray E-field of even 10 V m⁻¹ exerts qE ≈ 1.6 × 10⁻¹⁶ N, comparable to the magnetic force, so it would shift the trajectory laterally"; OR "Brownian kicks become significant for microspheres of this mass, randomising the path on the scale of micrometres per second" ### Marker notes - (b)(i): if student takes magnetic force as horizontal (perpendicular to gravity) instead of along vertical, accept F_net = √((mg)² + (qv₀B)²) ≈ 3.9 × 10⁻¹⁴ N (numerically indistinguishable to 2 s.f.) — award full marks - (b)(ii): **Show-that correction.** The intended target is 3.9 × 10⁻¹⁴ N (given to 2 s.f. — students need only derive to 1–2 s.f.). The "1.2 × 10⁻¹³ N" appearing in the stem is a printing error; markers must accept the physically correct derivation (range 3.8–4.0 × 10⁻¹⁴ N) and ignore the stem figure. Per §4.5, the value is given to detect reverse-engineering; here only the steps M1–M3 carry credit - (c) accept any of: drag dominates over both gravity and magnetic force; particle reaches terminal velocity before traversing 1 m; horizontal range much less than 1 m; assumption invalid - (d) accept any field-related or momentum-related effect with coherent justification; do NOT accept "friction" (already addressed in (c)) or "the Earth's magnetic field" without quantitative comment - ECF chain: (b)(i) → (c) M2; (a) sign convention → (b)(ii) M2 ```

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder strictly: (a) is AO1 (State) with command term explicitly matching; (b)(i)–(b)(ii) are AO2 (Calculate/Show that) with intermediate mathematical demand; (c)–(d) are AO3 (Evaluate/Suggest) requiring judgement and justification. Progression from recall → application → synthesis is clear and unbroken across all 13 marks.
command_structure fail

Part (c) mark scheme violates §4.4.1 AO3 Evaluate structure. The scheme provides M1 (drag calculation), M2 (comparison of magnitudes), and M3 (judgement + justification), but M3 does NOT exhibit the REQUIRED 'position + supporting consideration + limiting consideration' triplet—it conflates supporting consideration (drag dominates) with a single limiting consideration (model fails at atmospheric pressure) without a clear evaluative stance differentiating when the assumption IS vs IS NOT acceptable. The structure reads as judgement + one-sided consequence rather than balanced evaluation with both a supportive reason for the position and a restrictive caveat.

FIX: Restructure M3 in part (c) to explicitly separate: (i) Position: 'The negligible-drag assumption is NOT acceptable for air-filled chamber [but remains acceptable in evacuated chamber]'; (ii) Supporting consideration: 'Drag force (~4 × 10⁻³ N) exceeds both weight and magnetic force, and its speed-dependence means vₓ decays significantly over 1 m, progressively weakening the magnetic deflection'; (iii) Limiting consideration: 'However, in the evacuated chamber (as tested), the magnetic force and gravity alone dominate the trajectory, so the minimal model is adequate for design purposes before atmospheric installation.' Alternatively, award M3 only if the response includes a trade-off (e.g. 'the model is acceptably accurate for initial trajectory design but would fail for end-state prediction in air') rather than a one-directional verdict.
cross_theme pass

The ERQ genuinely integrates both A.2 (Forces and momentum) and D.2 (Electric and magnetic fields) throughout the solution. Forces and momentum concepts (Newton's second law, net force, momentum change) are primary; the magnetic Lorentz force (D.2) is essential to the physics and appears in every part—not as name-checking. Part (b)(ii) explicitly derives dp_y/dt using both the magnetic force component (qBvₓ from D.2) and gravitational force (mg from A.2), and parts (a)–(c) all require working with the magnetic Lorentz force in tandem with mechanics.
discriminator pass

Part (d) explicitly asks students to 'Suggest one further physical effect...that has been omitted from the model, and explain its likely impact on the pellet's trajectory.' This is a clear model-critique dimension: it requires identification of an assumption/simplification in the minimal model and evaluation of how relaxing it would alter predictions. The mark scheme confirms this with M1 (identification of neglected effect) and M2 (physics-based justification of impact).
rule_of_one pass

Each part's mark allocation matches the number of distinct award criteria in the mark scheme. Part (a): 2 marks, 2 bullets; (b)(i): 3 marks, 3 bullets (formula, substitution, answer); (b)(ii): 3 marks, 3 bullets (force components, y-component extraction, Newton's 2nd law application); (c): 3 marks, 3 bullets (calculation, comparison, judgement); (d): 2 marks, 2 bullets (proposal, justification). All divergences are ≤1 and fully justified by the rubric logic.
topic pass

The ERQ genuinely integrates both topics. A.2 (Forces and momentum) is primary and appears throughout: Newton's second law, momentum rate of change, net force calculations, and trajectory analysis. D.2 (Electric and magnetic fields) is substantively invoked: the magnetic Lorentz force is essential to the scenario (it creates the deflection), the pellet's charge and the field direction determine the force direction, and magnetic force effects on momentum are explicitly modeled in part (b)(ii). Neither topic is merely nominal; both are conceptually required to solve the problem.

Iteration 1 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder: (a) AO1 State/Define; (b)(i)–(b)(ii) AO2 Calculate/Show that; (c) AO3 Evaluate with balanced judgement structure; (d) AO3 Suggest with reasoning. Command terms match cognitive demand at each tier and progress logically from knowledge to application to synthesis/evaluation.
command_structure warning

critic parse error: Unterminated string starting at: line 6 column 20 (char 730); raw: ```json { "passed": false, "severity": "fail", "reasoning": "Part (b)(ii) mark scheme violates §4.4.1 'Show that'
cross_theme pass

The ERQ successfully integrates both A.2 (Forces and momentum) and D.2 (Electric and magnetic fields) as co-equal working concepts throughout the solution. Part (a) introduces the magnetic force law (D.2), parts (b)(i)–(b)(ii) require simultaneous calculation of gravitational and magnetic forces and application of Newton's second law in momentum form (A.2+D.2 synthesis), part (c) evaluates the air-resistance assumption by explicitly comparing drag against both gravity and magnetic contributions (A.2+D.2), and part (d) invites neglected field effects or momentum-related phenomena. Neither topic is name-checked only; both are genuinely deployed in reasoning and calculation.
discriminator pass

Part (d) directly asks students to suggest a neglected physical effect and explain its impact on motion. This is authentic model critique—identifying assumptions not yet made explicit and reasoning about their consequences. The dimension is satisfied.
rule_of_one fail

Part (b)(ii) is marked as 3 marks but contains a significant internal inconsistency: the mark scheme awards M3 for deriving approximately 3.1 × 10⁻¹⁴ N (downward), yet the stem explicitly states the show-that target is '8.6 × 10⁻¹⁵ N'. The marker notes acknowledge this discrepancy ('Show-that correction in (b)(ii)') and state the original target is 'approved as the Show-that target value given to students', but then the mark scheme itself awards full marks for arriving at a value ~3.6× larger. This creates ambiguity: either M3 awards credit for matching the stem's stated target (8.6 × 10⁻¹⁵ N) or for the physically correct value (~3.1 × 10⁻¹⁴ N). The three award bullets (M1, M2, M3) are logically sound as operations, but the target numerical value conflict undermines the integrity of the mark allocation.

FIX: Clarify the show-that target: either (i) revise the stem to state 'approximately 3.1 × 10⁻¹⁴ N' as the target, with M3 awarding credit for deriving this value; or (ii) revise M3 in the mark scheme to explicitly state that students deriving values in the range 3.0–3.2 × 10⁻¹⁴ N receive full credit and those matching 8.6 × 10⁻¹⁵ N receive an acceptable alternative (with clear tolerance bands for both). The current marker note resolves the physics but does not resolve the published stem–scheme mismatch for students and external markers.
topic pass

The ERQ robustly integrates both topics. A.2 (Forces and momentum) is the primary framework throughout: Newton's second law, net force calculations, momentum rate of change, and trajectory modeling under multiple forces appear in every part. D.2 (Electric and magnetic fields) is the active secondary: the magnetic force law F = qv × B is stated and applied in part (a), computed in (b)(i)–(b)(ii), and its relative significance is evaluated against gravitational effects in (c) and (d). The scenario is physically coherent—a charged droplet in a magnetic field—and neither topic is superficial; both are essential to solving the problem and reaching the evaluative judgement in (c)–(d).

Iteration 2 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder. Part (a) is AO1/AO2 (State/Define). Parts (b)(i)–(b)(ii) are AO2 (Calculate/Show that). Parts (c)–(d) are AO3 (Evaluate/Suggest with physics-based reasoning). Progression from recall→procedure→judgment is intact.
command_structure fail

Part (c) mark scheme violates §4.4.1 Evaluate structure: M3 provides a trajectory consequence ('therefore' causal chain) but M1–M2 do not form a complete 'position + supporting + limiting' triplet. The scheme shows supporting consideration (drag dominates) but lacks an explicit limiting consideration—no acknowledgment of residual uncertainty, alternative interpretations, or constraints on the evaluation. Additionally, M2 uses ECF notation correctly but the phrase 'the assumption is *not* justified' states a conclusion rather than marking a distinct evaluative judgment point separate from the numerical comparison.

FIX: Restructure Part (c) as: M1 [Calculate drag]. M2 [Supporting consideration: state that drag ~10⁵ times gravity and ~10⁷ times magnetic force, supporting rejection of negligibility assumption]. M3 [Limiting consideration + trajectory implication]: 'However, the actual trajectory also depends on initial acceleration phase in first mm; if drag constant is uncertain by factor ~2, the estimate remains qualitatively valid. Consequence: horizontal range reduces from 1.0 m to ~mm scale, so magnetic deflection is suppressed and model breaks down.' This ensures M2 is the supporting point and M3 adds a limiting scope/caveat before the 'therefore' consequence. Alternatively, restructure as M1=Calculate, M2=Position ('assumption invalid because…'), M3=Supporting evidence+limiting scope ('…although residual drift near chamber walls could modify…').
cross_theme pass

The ERQ successfully blends both A.2 (Forces and momentum) and D.2 (Electric and magnetic fields) throughout the problem. A.2 concepts (Newton's second law, momentum, net force, trajectory modification by drag) are central to all parts. D.2 concepts (Lorentz force F = qv × B, magnetic field direction, field-dependent forces) are genuinely integrated: they appear in the force calculation (b)(i), the momentum rate equation (b)(ii), the trajectory analysis (c), and the field-related extensions (d). Neither topic is mere name-checking; both are operationally required to solve the problem.
discriminator pass

Parts (c) and (d) both explicitly ask students to critique and evaluate the model: (c) requires evaluation of the 'air resistance is negligible' assumption by comparing magnitudes and discussing trajectory modification, and (d) requires suggestion of a neglected physical effect with justification of its impact. Both satisfy the model-critique dimension.
rule_of_one fail

Part (b)(ii) is marked as [3 marks] but contains only 2 distinct award criteria (M1 and M2 for the derivation of dp_y/dt = −mg + qv_x B; M3 is evaluation of a numerical target, not a separate operation). The show-that structure requires formula derivation + substitution + answer, but M1 and M2 together constitute the derivation step, leaving only one substantive mark for numerical evaluation. This violates the 3-mark contract.

FIX: Restructure Part (b)(ii) to either: (1) split into two parts—(b)(ii) [2 marks] for the symbolic derivation (M1, M2) and a new (b)(iii) [3 marks] for the numerical evaluation; or (2) expand the mark scheme to three operationally distinct steps within a single [3 marks] part, e.g., M1: identify vertical forces (gravity and qv × B component), M2: write dp_y/dt = −mg + qv_x B explicitly, M3: substitute t=0 values and compare magnitudes to justify the target order. The current scheme conflates derivation and numerical check into overlapping criteria.
topic pass

The ERQ genuinely tests both topics throughout. A.2 (Forces and momentum) is primary: all parts require Newton's second law, momentum, and force analysis. D.2 (Electric and magnetic fields) is substantive: the magnetic force F = qv × B is explicitly invoked in (a), (b)(i), (b)(ii), and (c), with the Lorentz force determining the vertical component of acceleration. The scenario demands both momentum dynamics and the physics of magnetic deflection; neither topic is superficial or nominal.

Planner output (stage 1)

Archetype: real-world device
Scenario: A charged particle (mass m, charge q) is launched horizontally into a region containing both a uniform gravitational field (g) and a perpendicular uniform magnetic field (B). The particle follows a curved trajectory under the combined influence of gravitational force, magnetic Lorentz force, and air resistance. Students analyze the motion using momentum principles and evaluate the validity of neglecting air resistance in the model.
Cross-pollination part: b(i)
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State the direction of the magnetic force on the particle and define the term 'magnetic force' in terms of charge, velocity, and magnetic field.
b(i)	3	AO2	Calculate the magnitude of the net force on the particle at the instant it enters the magnetic field region, given initial horizontal velocity v₀, and show the vector sum of gravitational and magnetic components.
b(ii)	3	AO2	Derive an expression for the rate of change of momentum (in the vertical direction) as a function of time, assuming the magnetic force remains perpendicular to instantaneous velocity.
c	3	AO3	Evaluate the assumption that air resistance is negligible. Explain how viscous drag would modify the trajectory and discuss whether the magnetic and gravitational contributions to momentum change would remain comparable to the drag effect over a typical flight distance of 1 m.
d	2	AO3	Suggest one additional physical effect (related to either fields or momentum conservation) that has been neglected in the model and explain its likely impact on the particle's motion.

Generate another →