Generated ERQ

✓ passed C.4 Standing waves and resonance × D.2 Electric and magnetic fields 12 marks HL 3 passes 122.88s $0.6717

## ERQ · 12 marks · Topics: C.4 Standing waves and resonance + D.2 Electric and magnetic fields **Stem.** A research student investigates a magnetostrictive transducer used in underwater sonar. A nickel rod of length L = 0.250 m is clamped at its midpoint and free at both ends; longitudinal standing waves are excited along it. The speed of longitudinal waves in nickel is 4900 m s⁻¹. The rod lies along the x-axis inside a region of uniform magnetic field B = 0.42 T directed perpendicular to the rod (along the y-axis). A small conducting bead of mass m = 1.8 × 10⁻⁴ kg and charge q = +6.0 × 10⁻⁹ C is suspended on a fine insulating fibre so that it sits at one free end of the rod, in the same plane as the rod's motion. The maximum longitudinal displacement amplitude of each free end of the rod when driven at its fundamental frequency is measured to be a₀ = 3.2 × 10⁻⁶ m. As the end of the rod oscillates along x with velocity v(t), it carries the bead through the magnetic field, producing a magnetic force on the charge. The student wishes to predict the steady-state oscillation amplitude of the bead, treating the bead as a lightly damped oscillator with natural frequency f₀ = 9.8 kHz and quality factor Q = 35. ### Part (a) Draw [3 marks] · AO1/AO2 · Topic: C.4 On a sketch of the rod (clamped at the midpoint, free at both ends), draw the displacement standing-wave pattern for the fundamental mode. Label the positions of all displacement nodes and antinodes, and state the wavelength of this mode in terms of L. ### Part (b)(i) Calculate [3 marks] · AO2/AO3 · Topic: C.4 Determine the fundamental frequency of longitudinal oscillation of the rod. ### Part (b)(ii) Show that [3 marks] · AO2/AO3 · Topic: C.4 + D.2 The end of the rod oscillates as x(t) = a₀ sin(2πft). As the bead is carried along with the end, it moves through the magnetic field B with velocity v(t). Show that the magnitude of the peak magnetic force experienced by the bead is approximately 2.5 × 10⁻¹¹ N. ### Part (c) Determine [3 marks] · AO3 · Topic: C.4 + D.2 The bead behaves as a driven damped oscillator with the magnetic force from (b)(ii) acting as the driving force. Given that the driving frequency from the rod and the bead's natural frequency f₀ are very close, determine the steady-state oscillation amplitude of the bead at resonance, using A_res = Q·F₀ / (m·(2πf₀)²). ### Part (d) Discuss [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The student's model assumes (i) that the magnetic field remains exactly uniform across the bead's trajectory and (ii) that the bead's motion does not perturb the standing wave on the rod. Discuss how each of these idealisations may break down in the real experiment, and state one consequence for the measured bead amplitude compared with the value calculated in (c). --- ## Mark Scheme ### Part (a) [3 marks] - M1: Sketch shows a displacement node at the midpoint clamp AND displacement antinodes at both free ends [ECF: no] - M2: Pattern shown corresponds to a single half-wavelength fitting between the two ends (one node, two antinodes — fundamental) [ECF: no] - M3: States λ = 2L (= 0.500 m) [ECF: no] ### Part (b)(i) [3 marks] - M1: Identifies f = v/λ with λ = 2L for this clamped-centre / free-ends rod [ECF: no] - M2: Correct substitution: f = 4900 / (2 × 0.250) [ECF: yes from a-M3] - M3: f = 9800 Hz = 9.80 kHz (3 s.f., correct unit) [ECF: yes] ### Part (b)(ii) [3 marks] - M1: Identifies peak end-velocity v_max = 2πf·a₀ (differentiation of x(t)) [ECF: no] - M2: Identifies magnetic force on moving charge F = qv_max B and substitutes: F = (6.0 × 10⁻⁹)(2π × 9800 × 3.2 × 10⁻⁶)(0.42) [ECF: yes from b(i)-M3] - M3: Obtains F ≈ 2.48 × 10⁻¹¹ N ≈ 2.5 × 10⁻¹¹ N, with correct unit and 2 s.f. [ECF: yes] ### Part (c) [3 marks] - M1: Identifies F₀ = peak magnetic force from (b)(ii) and (2πf₀)² as the squared angular natural frequency [ECF: no] - M2: Correct substitution: A_res = (35 × 2.5 × 10⁻¹¹) / [(1.8 × 10⁻⁴) × (2π × 9800)²] [ECF: yes from b(ii)-M3 and b(i)-M3] - M3: A_res ≈ 1.3 × 10⁻¹⁵ m (accept 1.2–1.4 × 10⁻¹⁵ m), correct unit and 2 s.f. [ECF: yes] ### Part (d) [2 marks] — Suggest-type: award 1 mark for each of two distinct assumption-critiques, max 2. - M1: Critique of uniform-B assumption — e.g. real magnet has fringing/inhomogeneity over the bead's trajectory, so the effective qvB driving term varies in phase/amplitude across the cycle → measured bead amplitude lower (or distorted) than the resonance prediction in (c). [ECF: no] - M2: Critique of no-back-reaction assumption — e.g. the bead (and the induced currents/forces it exerts on the rod end) loads/damps the standing wave, reducing a₀ below the quoted 3.2 × 10⁻⁶ m, OR the bead's motion changes the effective driving frequency away from exact resonance → measured amplitude smaller than the Q-amplified value in (c). [ECF: no] ### Marker notes - (a): Accept any clear sketch convention (transverse-style envelope is acceptable for longitudinal displacement provided node/antinode labelling is correct). Do NOT award M1 if antinodes placed at the clamp. - (b)(i): Accept 9.8 kHz or 9800 Hz; penalise once only for s.f. across (b) and (c). - (b)(ii): Alternative route using EMF ε = BvL_eff and F = qE is NOT equivalent here (no closed circuit); the correct mechanism is the magnetic Lorentz force F = qv × B on the moving charge — full marks only for this reasoning. - (c): Accept solutions that use ω₀ = 2πf₀ explicitly. If candidate (incorrectly) uses driving frequency from (b)(i) and natural frequency separately in the full Lorentzian denominator and obtains a comparable resonance-peak value, award full ECF. - (d) discriminator: Accept any model-assumption critique provided it (i) names the idealisation, (ii) explains the physical mechanism of breakdown, AND (iii) states a direction of effect on amplitude. Other acceptable critiques: assumption that the bead is a point charge (real bead has finite size sampling non-uniform B); assumption of linear damping (Q drops at high amplitude); assumption that the fibre suspension exerts no restoring torque. Do NOT award marks for generic "experimental error" or "air resistance" without linking to a stated model assumption.

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder: Part (a) is AO1/AO2 with 'Outline' (knowledge/comprehension); parts (b)(i)–(b)(ii) and (c) are AO2/AO3 with 'Show that', 'Determine', and 'Explain' (application/analysis); part (d) is AO3 with 'Suggest' (evaluation). The progression from foundational standing-wave concepts → calculations → physical reasoning → critical analysis of discrepancies is pedagogically sound and satisfies IB cognitive taxonomy requirements.
cross_theme pass

Part (c) genuinely blends both C.4 (standing waves/resonance) and D.2 (magnetic force on current-carrying conductor). The magnetic force F = BIL is the physical mechanism that drives the standing wave into resonance, making D.2 essential to explaining why resonance occurs—not merely a name-check. Both concepts are actively deployed in the solution.
discriminator pass

Part (d) explicitly asks students to 'Suggest two reasons why the calculated value differs from the measured value', requiring critique of the idealized model assumptions (perfect string, constant tension, ideal clamping, absence of damping, etc.). This is a genuine model-critique question that goes beyond calculation.
rule_of_one fail

Part (b)(ii) is marked as 2 marks but lists only 2 award criteria (wavelength and node separation). However, for a 'Determine' question in a 'Show that' context, these are two separate determinations, not three distinct operations. More critically, Part (d) is marked 2 marks but the scheme awards '1 mark for each distinct valid reason (max 2)' — this is a flexible rubric that does not align with the dimension's requirement for a fixed, predictable mark-to-criterion mapping. The scheme lists 5 possible reasons but only 2 are awarded; this violates the principle that each part's mark count must match the number of distinct, pre-weighted award criteria.

FIX: Part (b)(ii): Restructure to 3 marks by adding an intermediate step (e.g., Mark 1: State λ₂ = L; Mark 2: Calculate distance = λ₂/2; Mark 3: Numerical answer 0.240 m). Alternatively, reduce to 1 mark if only the distance is required. Part (d): Revise to fixed criteria: 'Mark 1: [specific reason 1 only]; Mark 2: [specific reason 2 only]' rather than a pool of 5 options. Pre-select two reasons and award them directly, or expand to 3–4 marks to accommodate the variability.
topic pass

The ERQ successfully integrates both topics. Part (c) explicitly requires C.4 (standing wave resonance condition) AND D.2 (Lorentz force F = BIL on current-carrying conductor in magnetic field). The question structurally demands understanding that resonance occurs when the periodic magnetic force frequency matches the wire's natural frequency, forcing students to connect wave physics with electromagnetic force concepts.

Iteration 1 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder: (a) is AO1/AO2 with Draw command; (b)(i)–(b)(ii) are AO2/AO3 with Calculate/Determine commands requiring application of standing-wave theory; (c) is AO3 with Calculate but embedded in a multi-concept context; (d) is AO3 with Suggest requiring critical evaluation of assumptions. Progression from recall through procedural application to reasoning/evaluation is sound.
cross_theme fail

Part (c) uses only D.2 (electric field calculation) as a computational tool to find the force on the particle; it does not require standing-wave physics from C.4 to be solved. Part (d) asks for qualitative assumptions but does not genuinely integrate both concepts—the particle's failure to oscillate is explained entirely by D.2 principles (field geometry and particle inertia), with no reciprocal dependence on C.4 standing-wave resonance. The question fails the blending requirement because the second topic (D.2) is applied in isolation rather than woven into a solution that demands both concepts work together.

FIX: Rewrite part (c) and (d) so that the test particle's motion directly couples to the standing-wave dynamics. For example: (c) Calculate the amplitude of oscillation of the test particle if it were driven by the modulated electric field produced by the rod's end-face displacement at the fundamental frequency, accounting for resonance of the particle itself (coupling C.4 standing-wave frequency to D.2 particle dynamics). (d) Explain why the particle's natural frequency and damping affect its ability to track the rod's oscillations, connecting the standing-wave resonance condition in C.4 to the electromagnetic forcing mechanism in D.2. This ensures both topics are used in the same causal chain.
discriminator fail

Part (d) asks students to 'suggest two distinct physical reasons for this discrepancy' between predicted and measured oscillation amplitude. While this requires critique of the model's predictions, it does NOT ask students to critique the model itself — i.e., to evaluate assumptions, limitations, or idealizations embedded in the theoretical framework. Instead, it asks for physical explanations of why experimental observations deviate from theory. This is explanation of a mismatch, not critique of model validity or assumptions. A true model-critique question would ask 'Discuss the assumption that the electric field remains uniform despite the rod's motion' or 'Explain why treating the test particle as a point charge subject to a uniform field is inadequate.' The current part (d) falls short of the dimension's requirement.

FIX: Rewrite part (d) to include explicit model-critique language. For example: '(d) The student's prediction assumes (i) the electric field remains uniform and (ii) the test particle responds instantaneously to the applied force. Discuss how each of these assumptions breaks down in this experiment, and explain one consequence of each breakdown for the observed oscillation amplitude.' This directly asks students to critique the model's assumptions rather than merely explain experimental discrepancies.
rule_of_one fail

Part (d) is marked as [2 marks] but the mark scheme lists only 2 award bullets that are NOT structured as distinct operations (formula/substitution/answer). Instead, they are two separate conceptual reasons. The dimension requires that a 2-mark part have exactly 2 award criteria following the Calculate/Determine structure (operation-based), but this is a Suggest part asking for two distinct physical reasons—a fundamentally different task type. The mark scheme's structure does not align with the prescribed dimensioning rule for Calculate/Determine parts.

FIX: Clarify the mark scheme for part (d): either (1) restructure it as a Calculate part with explicit formula and substitution steps if only one reason is truly required, or (2) explicitly rewrite the part instruction and marks as 'Suggest [2 marks]: Award 1 mark for each of two distinct physical reasons (spatial-scale mismatch OR frequency/inertia mismatch), up to maximum 2 marks' to signal that the two marks correspond to two independent conceptual bullets rather than sequential operational steps. Alternatively, reduce part (d) to [1 mark] if only one reason is the primary discriminator.
topic fail

Part (c) nominally mentions D.2 by naming the electric force calculation, but the question does NOT actually require understanding of electric/magnetic field concepts. The force is computed using only F = qE = qV/d—a simple substitution that treats the field as a static, uniform entity. There is no meaningful integration of field theory (e.g., field variation, energy storage, field interaction with motion, or any property of fields beyond their magnitude). The question is fundamentally about standing waves (C.4) with a superficial arithmetic application of electrostatics; D.2 is not genuinely tested.

FIX: To authentically test both topics, redesign part (c) and onwards to require genuine field concepts. For example: (c) Ask students to calculate how the rod's oscillating displacement modulates the capacitor's electric field strength at the particle's location, requiring them to understand field inhomogeneity or to relate displacement to field change. (d) Ask them to compare the oscillating electric force amplitude against the electrostatic restoring force or to analyze resonance of the charged particle in an oscillating field, requiring knowledge of how fields interact dynamically with charged motion (D.2). Alternatively, replace the capacitor scenario with a magnetic-field setup: e.g., the vibrating rod oscillates within a uniform magnetic field, and students must calculate the induced EMF and resulting magnetic force on a charge, genuinely integrating C.4 (standing-wave motion) with D.2 (magnetic field effects).

Iteration 2 — passed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder: Part (a) is AO1/AO2 with Draw; Parts (b)(i)–(b)(ii) are AO2/AO3 with Calculate/Show that; Part (c) is AO3 with Determine (applied to resonance formula); Part (d) is AO3 with Discuss (applied to model assumptions). The progression from recall/simple application → procedural calculation → synthesis/evaluation is sound.
cross_theme pass

Both C.4 (standing waves) and D.2 (magnetic/electric fields) are genuinely blended throughout the question. C.4 underpins parts (a)–(b)(i) (fundamental mode, wavelength, frequency), while D.2 is essential in (b)(ii)–(c): the magnetic force F = qvB drives the bead's oscillation, and the resonance calculation requires both the standing-wave velocity amplitude and the Lorentz force law. Part (d) critiques model assumptions that apply to both domains simultaneously (uniform B affects the driving force; bead back-reaction affects both the standing wave and resonance condition).
discriminator pass

Part (d) explicitly asks students to discuss how two stated model idealizations break down in the real experiment and to state one consequence for the measured amplitude. This is a genuine model-critique question that requires physical reasoning about assumptions, not mere calculation.
rule_of_one pass

Each part's mark count aligns well with the number of distinct award criteria. Parts (a)–(c) each have 3 marks and 3 criteria (concept/identification, substitution, final answer). Part (d) has 2 marks and 2 criteria (one critique per assumption). The scheme is internally consistent and fair.
topic pass

The ERQ robustly tests both C.4 (standing waves and resonance) and D.2 (electric and magnetic fields). Part (a)–(b)(i) establish the standing-wave physics; part (b)(ii) requires calculating the magnetic Lorentz force F = qvB on a charge moving through a magnetic field (D.2 core concept), where the velocity comes from the rod's oscillation (C.4); part (c) applies resonance theory to the driven damped oscillator; part (d) critically examines assumptions in both domains. Neither topic is superficial or nominal—both are essential to the physical reasoning.

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