Generated ERQ

✓ passed B.5 Current and circuits × D.4 Induction (HL) 14 marks HL 3 passes 143.07s $0.8421

## ERQ · 14 marks · Topics: B.5 Current and circuits + D.4 Induction (HL) · Archetype: data_response **Integration:** primary=B.5 Current and circuits, secondary=D.4 Induction (HL) (strength: supporting) **Stem.** A student designs a regenerative bicycle brake. A diametrically-magnetised cylindrical permanent magnet of strength B = 0.45 T is fixed to the wheel hub and rotates at angular velocity ω (in rad s⁻¹). A flat sensing coil of N = 240 turns and cross-sectional area A = 1.8 × 10⁻³ m² is mounted close to the magnet so that the flux linkage is well-approximated by Φ_link(t) = NBA cos(ωt). The coil (internal resistance r = 2.5 Ω) is connected to a small heating element of resistance R = 6.0 Ω. As the wheel slows, the student records the rms current I_rms in the circuit at several angular velocities; the data are shown below. | ω / rad s⁻¹ | 5.0 | 10.0 | 15.0 | 20.0 | 25.0 | |-------------|------|------|------|------|------| | I_rms / A | 0.32 | 0.65 | 0.96 | 1.28 | 1.59 | | uncertainty in I_rms: ±0.03 A at each point | The student wishes to extrapolate the data to predict the braking performance at ω = 40 rad s⁻¹ (typical cruising cadence). ### Part (a) State [2 marks] · AO1 · Topic: B.5 State the direction of energy transfer in the circuit, and identify the component in which electrical energy is dissipated as thermal energy at the greatest rate. Justify your identification by reference to resistance values. ### Part (b)(i) Calculate [3 marks] · AO2 · Topic: D.4 + B.5 Using the data point at ω = 20.0 rad s⁻¹, calculate the peak emf induced in the coil and compare it with the value predicted from ε₀ = NBAω. ### Part (b)(ii) Determine [3 marks] · AO2 · Topic: B.5 Using the data table, determine the rms power dissipated in the heating element at ω = 20.0 rad s⁻¹, and state the absolute uncertainty in this power. ### Part (c) Deduce [3 marks] · AO3 · Topic: B.5 + D.4 By identifying how I_rms scales with ω from the data, deduce the angular velocity at which the power dissipated in the heating element first exceeds 5.0 W. ### Part (d) Evaluate [3 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The student linearly extrapolates the I_rms-vs-ω trend to ω = 40 rad s⁻¹ and predicts the heating-element power on that basis. Evaluate whether this extrapolation is reliable, referring to one specific physical effect that the simple model Φ_link = NBA cos(ωt) neglects. --- ## Mark Scheme ### Part (a) [2 marks] — State - M1: Energy is transferred from the rotational kinetic energy of the wheel/magnet → (via induced emf) → electrical energy in the circuit → thermal energy in R and r [no ECF] - M2: Heating element R dissipates the greater rate because R (6.0 Ω) > r (2.5 Ω) and the same current flows through both in series, so P = I²R is larger for R [no ECF] ### Part (b)(i) [3 marks] — Calculate - M1: ε_rms = I_rms × (R + r) = 1.28 × 8.5 = 10.88 V, so ε₀ = √2 × ε_rms ≈ 15.4 V [no ECF] - M2: Predicted ε₀ = NBAω = 240 × 0.45 × 1.8 × 10⁻³ × 20.0 = 3.89 V [no ECF] - M3: Measured peak emf (≈ 15.4 V) is much larger than predicted (≈ 3.89 V) — by a factor of ~4 — indicating the simple flat-coil flux-linkage model under-estimates the actual coupling (or the effective NBA product is larger) [ECF from M1, M2] ### Part (b)(ii) [3 marks] — Determine - M1: P_R = I_rms² × R = (1.28)² × 6.0 = 9.83 W [no ECF] - M2: Fractional uncertainty in P from ΔP/P = 2ΔI/I = 2 × (0.03/1.28) = 0.0469 [no ECF] - M3: Absolute uncertainty ΔP ≈ 0.047 × 9.83 ≈ 0.46 W, so P_R = (9.8 ± 0.5) W [ECF from M1] ### Part (c) [3 marks] — Deduce (causal chain per §4.4.1) - M1: From Faraday's law ε = −dΦ_link/dt = NBAω sin(ωt), therefore ε_rms ∝ ω; data confirm I_rms/ω ≈ 0.064 A s rad⁻¹ is constant across the table, so I_rms ∝ ω [no ECF] - M2: Therefore power in R scales as P_R = I_rms² R ∝ ω², so P_R(ω) = (0.064 ω)² × 6.0 = 0.0246 ω² (in W, with ω in rad s⁻¹) [ECF from M1] - M3: Setting 0.0246 ω² = 5.0 gives ω = √(5.0/0.0246) ≈ 14.3 rad s⁻¹, therefore the heating-element power first exceeds 5.0 W at ω ≈ 14 rad s⁻¹ (consistent with the data point at ω = 15.0 where P ≈ 5.5 W) [ECF from M2] ### Part (d) [3 marks] — Evaluate (position + supporting + limiting per §4.4.1) - M1 (prediction from linear extrapolation): At ω = 40 rad s⁻¹, linear extrapolation gives I_rms ≈ 0.064 × 40 ≈ 2.56 A and P_R ≈ I²R ≈ 39 W [ECF from (c) M1] - M2 (limiting consideration — physical effect neglected): The simple model ignores the self-inductance L of the 240-turn coil; at high ω the inductive reactance X_L = ωL becomes comparable to (R + r), so the circuit impedance Z = √((R+r)² + (ωL)²) > (R+r), reducing I_rms below the linear prediction [no ECF] - M3 (evaluative judgement): Therefore the extrapolation is unreliable at ω = 40 rad s⁻¹: the actual power will be significantly less than 39 W, and the brake will under-perform relative to the student's design target [ECF from M1, M2] ### Marker notes - (b)(i) alternative: accept comparison via ratio (≈ 3.96×) provided both values are computed. - (b)(ii) accept (9.8 ± 0.5) W or (9.83 ± 0.46) W; do not penalise sig figs here. - (c) accept any method showing the ω² scaling and solving for ω; values in range 14.0–14.5 rad s⁻¹ acceptable. - (d) accept alternative neglected effects for M2: (i) back-emf from eddy currents in the magnet/coil former altering effective B; (ii) temperature rise of R increasing its resistance with current; (iii) magnetic saturation / non-sinusoidal flux geometry at high ω. Award M2 for any one such effect with a physics-based mechanism, and M3 for the consistent evaluative judgement that the linear extrapolation over-predicts P_R.

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ correctly follows the cognitive ladder: (a) AO1 with State command; (b)(i)–(ii) AO2 with Determine/Calculate; (c) AO3 with Derive + Comment; (d) AO3 with Evaluate. Each tier is appropriately sequenced and command terms align with assessment objectives.
command_structure fail

Part (d) mark scheme violates §4.4.1 Evaluate structure: M1 and M2 identify limitations but lack explicit position + supporting consideration + limiting consideration. The scheme conflates identification of issues with evaluation of their impact. M3 provides only a single limiting consideration without a supporting counterargument or trade-off perspective. The Evaluate command requires weighing competing factors or constraints; the current scheme merely lists problems without evaluative balance.

FIX: Restructure Part (d) M1–M3 to follow the Evaluate rubric: (M1) Position: 'The simple model assumes ideal sinusoidal flux and constant resistance, but at higher ω two physical effects will degrade performance.' (M2) Supporting consideration: 'Coil self-inductance introduces inductive reactance X_L = ωL that increases with ω, reducing the effective current below ε₀/(R + r) and phase-shifting it; this directly contradicts the model's assumption of ohmic behaviour.' (M3) Limiting consideration: 'Conversely, the non-sinusoidal flux profile from the magnet geometry produces harmonics that may partially compensate for inductive losses, and the scaling to a larger coil may reduce per-unit resistance; thus the actual emf shortfall could be less severe than the worst-case inductance estimate predicts.' This structure explicitly weighs the dominant inductive limitation against mitigating factors, fulfilling §4.4.1 Evaluate requirements.
cross_theme pass

The question genuinely integrates both B.5 (Current and circuits) and D.4 (Induction, HL). B.5 provides the circuit analysis framework (emf, internal/external resistance, power dissipation), while D.4 supplies the core physics: Faraday's law (ε = −dΦ/dt, manifested as ε₀ = NB₀Aω) and the sinusoidal time-dependence of induced emf. Both topics are essential throughout: (b)(i) uses the induction formula directly to extract B₀; (b)(ii) applies circuit laws to find current and Joule heating; (c) derives the time-averaged power by combining the sinusoidal emf from induction with circuit resistance; (d) evaluates physical limitations that affect induction (self-inductance, flux non-linearity) and their circuit consequences. Neither topic is relegated to a final assumption or name-checked only—both are active in the solution.
discriminator pass

Part (d) explicitly asks students to 'Evaluate two physical limitations of the simple model used in parts (b)–(c) that would cause the actual induced emf and braking power of the scaled-up device to deviate from the predicted values.' This is a clear model-critique question requiring identification and discussion of assumptions/limitations and their consequences, not a calculation.
rule_of_one fail

Part (d) is marked as [3 marks] but the mark scheme lists only 3 award criteria (M1, M2, M3), which aligns numerically. However, the rubric describes M3 as a 'judgement linking the limitations'—a synthesis step distinct from identifying the second limitation (M2). The dimensional contract requires each mark to correspond to one distinct operation. Here, M2 asks students to 'identify a second limitation' with an 'OR' offering multiple options, making it structurally unclear whether this counts as one criterion or multiple. More critically, M1 and M2 are both pure identifications with no calculation, substitution, or quantitative anchor, while M3 is a qualitative judgement. This violates the principle that each mark should correspond to a discrete, measurable operation. The part should either be restructured with clearer operational boundaries (e.g., M1: identify limitation A with mechanism; M2: identify limitation B with mechanism; M3: link to scaled-up prediction) or reduced to 2 marks if only identification + synthesis are required.

FIX: Reframe Part (d) as [2 marks] with M1 = identify and explain two distinct limitations (e.g. self-inductance AND temperature-dependent R, or flux non-sinusoidality AND magnetic saturation), and M2 = judge how these cause actual performance to deviate from the ω²-scaling prediction for the scaled-up device. Alternatively, keep [3 marks] and split as: M1 identify limitation A with physical justification; M2 identify limitation B with physical justification; M3 synthesize both to evaluate feasibility of the >5 V target at ω = 40 rad s⁻¹.
topic pass

The ERQ effectively integrates both topics throughout. B.5 (Current and circuits) is tested via emf/p.d. distinction (a), Ohm's law and power calculations (b)(ii), and circuit analysis (c). D.4 (Induction HL) is tested via Faraday's law (deriving ε₀ = NB₀Aω in b(i)), magnetic flux variation (b–c), and induced emf analysis (c). The scenario requires both the electromagnetic induction mechanism (D.4) to generate the emf AND circuit analysis (B.5) to determine currents and power dissipation; neither topic is superficial or omissible.

Iteration 1 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder. Part (a) is AO1 (State Faraday's law), parts (b)(i)–(b)(ii) are AO2 (Calculate/Determine with command terms), part (c) is AO3 (Deduce causal scaling relationship), and part (d) is AO3 (Evaluate feasibility with assumptions). Progression is strict and appropriate for HL with 14 marks.
command_structure fail

Part (c) fails the AO3 'Deduce' command-term structure. Per §4.4.1, Deduce requires an explicit causal chain ('X therefore Y'), but M1–M3 present logical steps without the required explicit linking warrant. M2 correctly identifies the ω² scaling, but M3 states a consequence without explicitly justifying *why* the low-speed power is 'impractical' via a physics-based threshold or design constraint. Additionally, part (d) 'Evaluate' structure is borderline: M1 states position and notes two effects will reduce output, but does not clearly separate 'supporting consideration' (inductive reactance reduces I) from 'limiting consideration' as distinct marks—M2/M3 conflate them. M3 hedges the conclusion ('plausibly achievable but unreliable') without a clear final evaluative stance required by §4.4.1.

FIX: Part (c): Revise M3 to explicitly state 'Therefore, braking power P_brake ≈ 0.02 × 0.33 W ≈ 0.007 W at ω = 3 rad s⁻¹, which is below the threshold (~0.1 W) for practical braking force, making the system impractical for low-speed control.' Part (d): Clearly split M2 and M3: M2 (supporting) = 'Self-inductance X_L = ωL increases impedance Z above (R+r), reducing I_rms and power below ω² prediction'; M3 (limiting) = 'However, scaling N and A increases geometric emf gain and can offset reactance if ωL is engineered small relative to (R+r); scaling is therefore feasible at 40 rad s⁻¹ provided inductive design is optimized.' This separates the two perspectives required for Evaluate.
cross_theme pass

The ERQ excellently integrates B.5 (Current and circuits) and D.4 (Induction HL) throughout. Part (a) requires stating Faraday's law and deriving peak emf using induction principles. Parts (b)–(c) apply the induced-emf expression from D.4 directly in Ohm's law and power calculations from B.5, creating genuine causal chains: ε_rms ∝ ω (induction) → I_rms ∝ ω (circuits) → P ∝ ω² (both topics combined). Part (d) evaluates model validity by invoking D.4 effects (self-inductance, inductive reactance) that constrain the B.5 circuit analysis. Neither topic is name-checked only; both are actively used in solving the problem.
discriminator pass

Part (d) explicitly requires the student to 'Evaluate the feasibility of this target by considering the physical assumptions of the model that may break down at higher angular speeds.' The mark scheme confirms that M2 and M3 demand critique of model assumptions (self-inductance, temperature-dependent resistance, saturation, eddy currents, etc.). This is genuine model critique, not merely computation.
rule_of_one fail

Part (c) is marked as [3 marks] but contains 3 award criteria that form a causal chain rather than 3 distinct operations. More critically, part (d) is marked [3 marks] with M1 being a compound statement (position + two embedded physical effects), M2 and M3 being alternative limiting considerations rather than sequential operations. The mark scheme conflates structural marking (causal reasoning) with operational marking (formula/substitution/answer), violating the dimension's requirement that each mark count match the number of distinct, independent award criteria.

FIX: Restructure (c) to have 3 independent operations: (1) state ε_rms ∝ ω from differentiation, (2) apply Ohm's law to show I_rms ∝ ω, (3) calculate or state power scales as ω². For (d), either: (A) split into 3 sequential checks (extrapolation prediction at 40 rad s⁻¹, identify one physical breakdown, state consequence for feasibility), or (B) reframe as a 2-mark part (position + supporting evidence, with limiting considerations as discussion). Current M1/M2/M3 in (d) do not function as independent award criteria but as branches of a single evaluation argument.
topic pass

The ERQ comprehensively tests both topics. B.5 (Current and circuits) is present throughout: Ohm's law, circuit analysis, rms current, power dissipation in resistive elements, and impedance concepts. D.4 (Induction) is genuinely integrated: Faraday's law and the induced emf expression (parts a–c), the ω-scaling of induced emf, and the physical assumptions underlying the induction model (part d). Neither topic is merely nominal; both are essential to solving each part.

Iteration 2 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder strictly. Part (a) is AO1 (State with justification). Parts (b)(i)–(b)(ii) are AO2 (Calculate/Determine with numerical reasoning). Part (c) is AO3 (Deduce via causal reasoning from data scaling). Part (d) is AO3 (Evaluate with position, supporting evidence, and limiting consideration). The progression from recall → calculation → deduction → evaluation is properly ordered.
command_structure fail

Part (c) fails the AO3 'Deduce' command-term structure. Per §4.4.1, 'Deduce' requires an explicit causal chain (X therefore Y), but M1 conflates two separate logical steps without a clear 'therefore' warrant: it states Faraday's law predicts ε_rms ∝ ω, then asserts "data confirm" the proportionality, but does not explicitly deduce why constant I_rms/ω ratio logically proves I_rms ∝ ω from first principles. The causal pathway from the physics principle to the data interpretation is implicit rather than explicit. Additionally, part (d) 'Evaluate' is correctly structured (position M1 + supporting M2 + limiting M2 + judgement M3) but the mark scheme permits alternative neglected effects in M2 without standardised marking criteria—a minor procedural weakness in consistency.

FIX: Part (c) M1 should be rewritten as: 'From Faraday's law, ε = NBAω sin(ωt), therefore ε_rms = NBA ω/√2 ∝ ω. Since I_rms = ε_rms/(R+r) and (R+r) is constant, I_rms ∝ ω. Data confirm this: the ratio I_rms/ω = 0.064 A·s/rad is constant, therefore I_rms scales linearly with ω.' This makes the causal chain explicit: [Faraday predicts ε_rms ∝ ω] → [Ohm's law with constant total resistance] → [therefore I_rms ∝ ω] → [data support this scaling]. For part (d), add a brief bracketed note to M2: '[Award M2 for any one of: self-inductance, back-emf from eddy currents, temperature-dependent resistance change, or magnetic saturation, provided a mechanism is stated.]'
cross_theme pass

Both B.5 (Current and circuits) and D.4 (Induction) are genuinely integrated throughout the question. Faraday's law (D.4) is used to derive the induced emf and explain the scaling of I_rms with ω in parts (b)(i) and (c); circuit analysis (B.5) is used in all parts to calculate currents, power dissipation, and impedance. The induction concept is not merely name-checked—it is central to the physics of the solution (flux linkage, emf induction, and self-inductance as a limiting effect). Part (d) appropriately uses self-inductance from induction theory to evaluate the reliability of the circuit-based extrapolation.
discriminator pass

Part (d) explicitly asks the student to evaluate the reliability of a linear extrapolation by identifying a specific physical effect (self-inductance, eddy currents, temperature dependence, or magnetic saturation) that the simple model Φ_link = NBA cos(ωt) neglects. This is a genuine model-critique task requiring judgement about model validity and limitations, not merely another calculation.
rule_of_one fail

Part (b)(i) is marked as 3 marks but contains 3 award criteria that are reasonably distinct (peak emf from rms, predicted peak emf formula, comparison). However, Part (b)(ii) is marked as 3 marks but the mark scheme lists only 2 truly independent operations: (M1) calculate P_R, and (M2+M3 combined) determine uncertainty. M3 merely applies the fractional uncertainty from M2 to get the absolute value—this is not a separate distinct operation but rather part of the single 'determine uncertainty' task. The mark count (3) exceeds the number of genuinely distinct operations (2), violating the dimension's contract.

FIX: Restructure Part (b)(ii) to clearly separate 3 independent operations: (M1) calculate I_rms² or ε_rms from current data; (M2) calculate P_R = I_rms² × R with substitution; (M3) calculate fractional uncertainty ΔI/I, then absolute uncertainty ΔP. Alternatively, reduce the mark allocation to 2 marks if the intent is only to calculate power and state its uncertainty as a single compound step.
topic pass

The ERQ robustly integrates both topics. B.5 (Current and circuits) is the primary focus throughout: parts (a), (b)(ii), and (c) all require circuit analysis (power dissipation, Ohm's law, impedance scaling). D.4 (Induction HL) is authentically invoked: part (b)(i) derives the induced emf from Faraday's law (ε ∝ NBAω), part (c) uses Faraday to establish I_rms ∝ ω, and part (d) critically invokes self-inductance as a physical effect neglected by the simple flux model. Neither topic is superficial; both are causally necessary to the problem.

Planner output (stage 1)

Archetype: real-world device
Scenario: A student designs a simple electromagnetic braking system for a bicycle wheel using a coil mounted near a rotating permanent magnet. As the wheel rotates, the changing magnetic flux through the coil induces an emf, which drives a current through a resistive load (a small heating element). The induced current creates a magnetic force opposing the rotation, providing braking effect. The system parameters include wheel angular velocity ω, magnetic field strength B, coil area A, number of turns N, and circuit resistance R.
Cross-pollination part: b(i)
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State
b(i)	3	AO2	Calculate
b(ii)	3	AO2	Determine
c	3	AO3	Derive
d	3	AO3	Evaluate

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