Generated ERQ

✓ passed C.4 Standing waves and resonance × D.4 Induction (HL) 14 marks HL 3 passes 122.25s $0.7829

``` ## ERQ · 14 marks · Topics: C.4 Standing waves and resonance + D.4 Induction (HL) · Archetype: theory_application **Integration:** primary=C.4 Standing waves and resonance, secondary=D.4 Induction (HL) (strength: supporting) **Stem.** A student investigates standing waves in an air column using a vertical glass tube, open at the top and connected at the bottom to a water reservoir whose height can be lowered to extend the air column. A tuning fork of frequency f = 512 Hz is held just above the open end. To detect the instant at which the water surface passes a given height, a small cylindrical permanent magnet of mass 4.20 g floats on the water surface, and a flat circular coil of N = 240 turns and mean radius r = 1.50 cm is fixed around the tube at a known height. As the reservoir is lowered, the magnet descends with the water and the coil registers an induced EMF on a data-logger. The student records the two successive air-column lengths at which loud acoustic resonance is heard: L₁ = 0.162 m and L₂ = 0.494 m. The speed of sound in air at the laboratory temperature is to be determined. Assume the magnet behaves as a point dipole of moment p = 0.082 A m². ### Part (a) Define [2 marks] · AO1 · Topic: C.4 State the boundary conditions, in terms of displacement nodes and antinodes, that apply at (i) the open end of the tube and (ii) the water surface, for a standing sound wave inside the air column. ### Part (b)(i) Calculate [3 marks] · AO2 · Topic: C.4 Using L₁ and L₂, calculate the wavelength of the sound in the air column and hence determine the speed of sound v. ### Part (b)(ii) Determine [3 marks] · AO2 · Topic: C.4 The position of the open end of the tube is not exactly the position of the displacement antinode; there is an "end correction" e such that the effective length of the air column is L + e. Determine the value of e from the student's data. ### Part (c) Show that [4 marks] · AO2/AO3 · Topic: C.4 + D.4 As the floating magnet descends through the coil at near-constant speed u, the axial magnetic flux linkage through the coil is approximately Φ(z) = (μ₀ N p)/(2(r² + z²)^{3/2}) · r², where z is the distance from the magnet to the plane of the coil. Show that the induced EMF reaches its maximum magnitude when the magnet is at z = r/2, and not when the magnet is in the plane of the coil (z = 0). ### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR · Topic: C.4 + D.4 Using the result of part (c), suggest how the student's value of the speed of sound from (b)(i) is affected if the data-logger is used to mark each resonance position from the EMF peak rather than from the audible resonance, and justify whether the student's calculated v is altered. --- ## Mark Scheme ### Part (a) [2 marks] — Define - M1: At the open end (top) — displacement antinode (pressure node) [no ECF] - M2: At the water surface (closed end) — displacement node (pressure antinode) [no ECF] ### Part (b)(i) [3 marks] — Calculate - M1: Recognises L₂ − L₁ = λ/2, so λ = 2(0.494 − 0.162) = 0.664 m [no ECF] - M2: Applies v = fλ with f = 512 Hz [substitution] [ECF from M1] - M3: v = 512 × 0.664 = 340 m s⁻¹ (accept 339–340 m s⁻¹) [ECF from M1] ### Part (b)(ii) [3 marks] — Determine - M1: Recognises L₁ + e = λ/4 (fundamental antinode-to-node distance) [no ECF] - M2: e = λ/4 − L₁ = 0.166 − 0.162 [substitution, ECF from (b)(i) λ] - M3: e = 0.004 m = 4 × 10⁻³ m (accept 3–5 mm) [ECF from (b)(i)] ### Part (c) [4 marks] — Show that (derivation, ~1 step per mark) - M1: EMF ε = −dΦ/dt = −(dΦ/dz)(dz/dt) = −u · dΦ/dz, so |ε| is maximised when |dΦ/dz| is maximised [identifies Faraday's law applied to moving magnet] - M2: Differentiates: dΦ/dz = (μ₀ N p r²/2) · d/dz[(r² + z²)^{−3/2}] = −(3μ₀ N p r² z)/[2(r² + z²)^{5/2}] - M3: Sets d²Φ/dz² = 0 (extremum of dΦ/dz); obtains (r² + z²)^{5/2} − 5z²(r² + z²)^{3/2} = 0, i.e. r² + z² = 5z², giving z² = r²/4 - M4: Therefore z = r/2; since this is non-zero, the EMF peak occurs before the magnet reaches the coil plane (z = 0 gives dΦ/dz = 0, i.e. zero EMF, not maximum) [conclusion] ### Part (d) [2 marks] — Suggest (proposal + physics-based warrant per §4.4.1) - M1 [2 marks combined]: Proposes that each EMF-marked position is offset from the true water-surface (acoustic-node) position by a fixed amount Δ = r/2 = 0.75 cm above the coil plane, **with** the physics warrant that, by part (c), the induction signal peaks when the magnet is at z = r/2 above the coil — this same geometric offset is added to both L₁ and L₂ readings, so the difference L₂ − L₁ (and hence λ and v) is unchanged; only the end correction e in (b)(ii) absorbs the shift. Award full 2 marks for a proposal that explicitly states (i) a constant offset and (ii) cites the z = r/2 induction result to justify why v is unaffected. - Partial credit (1 mark): correct conclusion that v is unchanged but without explicit reference to the z = r/2 offset from part (c); OR identifies the r/2 offset but does not conclude that v survives. ### Marker notes - Alternative method accepted for (c): students may set dε/dz = 0 directly (equivalent to d²Φ/dz² = 0) and reach z² = r²/4 in one step — award M2+M3 together; M1 and M4 still required. - Part (b)(ii) ECF: accept e in range 3–5 mm using student's λ from (b)(i). - Part (d): accept proposals that note the offset Δ may differ between L₁ and L₂ only if the descent speed u changes — credit M1 fully if student notes that for constant u the offset is identical at both resonances. Do NOT credit answers that invoke buoyancy or response-time without linking to the z = r/2 induction result. - Show-that target in (c): z = r/2 given exactly; any student-derived expression algebraically equivalent to z² = r²/4 acceptable. ```

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder: (a) is AO1 (Define); (b)(i)–(ii) are AO2 (Calculate/Determine); (c) is AO2/AO3 (Show that, with physics reasoning); (d) is AO3 (Suggest with justification). Command terms and cognitive demand increase monotonically. Total 14 marks distributed appropriately across tiers.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 'Suggest' structure. The command requires 'a proposal + a brief physics-based justification' as two separate marking points. M1 correctly states the proposal, but M2 provides justification that applies to the *mechanism* of why the offset causes error, not a physics-based warrant *linking the proposal to a physical principle*. The scheme fails to award an explicit mark for the causal link (e.g., 'because the EMF peak occurs at magnet-coil-centre, not at the displacement antinode, the measured water level is systematically offset from the true resonance condition').

FIX: Restructure Part (d) marking as: M1 = proposal (e.g. 'EMF peak does not coincide with acoustic resonance position; the magnet reaches maximum flux rate when at coil-centre, not when the water surface is at the resonance antinode'); M2 = explicit physics-based justification (e.g. 'The true resonance occurs when the water surface (boundary condition) aligns with the displacement antinode; the induced-EMF peak instead occurs at a fixed geometric offset (coil radius offset from antinode); therefore every measured L contains a constant systematic shift'). Alternatively, award M1 for any valid proposal from the marker notes and M2 specifically for linking the proposal to the physical principle that resonance position is defined by boundary conditions relative to the antinode, whereas EMF peak is defined by dΦ/dt geometry.
cross_theme pass

The question genuinely integrates C.4 (standing waves) and D.4 (induction). Part (c) requires both: calculating the induced EMF using Faraday's law (D.4) applied to a magnet descending through a coil, while the physical setup serves the standing-wave investigation. Part (d) deepens integration by requiring students to explain why the EMF peak (induction geometry) does not align with the acoustic resonance position (standing-wave boundary condition)—neither topic is merely name-checked; both are structurally essential.
discriminator pass

Part (d) requires the student to 'Suggest one physical reason why this synchronisation may introduce a systematic error.' This explicitly asks for model critique: identifying a mismatch between the theoretical resonance condition (antinode position) and the experimental marker (EMF peak), and explaining why this creates systematic error. This is genuine model-critique flavor, not just calculation.
rule_of_one fail

Part (d) is marked as [2 marks] — Suggest, but the mark scheme lists only M1 (proposal) and M2 (physics justification) as separate criteria without subdividing M1 into distinct operations. For a 2-mark Suggest part in IB HL, the structure is acceptable if M1 covers the proposal statement and M2 covers the reasoned justification. However, the divergence arises because 'Suggest' parts typically require only the proposal + one clear justification, and the mark scheme correctly identifies these two elements. The real issue is internal consistency: M1 lists four alternative proposals (float position, coil offset, flux-peak timing, antinode location) as a single award criterion, which is acceptable for a Suggest part. The scheme is internally sound. However, upon re-examination, part (b)(ii) shows a minor tension: it has 3 marks but the logic (state formula → substitute → calculate) maps cleanly. The critical check: does each part's mark count match the number of distinct operations? Part (d) [2 marks] has M1 (proposal) and M2 (justification), which is appropriate for a Suggest part and represents two distinct cognitive steps. This passes. All other parts (a), (b)(i), (b)(ii), (c) show 2, 3, 3, 4 marks respectively with matching award criteria counts. Upon strict inspection, all parts pass the mark-count-to-operation alignment. No divergence exceeds 1.
topic pass

The ERQ effectively tests both topics. Part (c) explicitly requires Faraday's law of electromagnetic induction (D.4) to calculate the induced EMF in the coil as the magnet descends, while simultaneously requiring understanding of standing wave resonance and flux change geometry (C.4). Part (d) then demands integration: explaining how the EMF-peak position differs from the acoustic resonance position—a genuine physical coupling between the two topics. Neither topic is superficial; both are essential to the scenario.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly: (a) is AO1 with Define; (b)(i)–(b)(ii) are AO2 with Calculate/Determine; (c) is AO2/AO3 with Show that (bridging tier); (d) is AO3 with Suggest. The progression from recall → procedure → derivation → critical analysis is intact and well-sequenced.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 'Suggest' structure. M1 provides only a proposal (the offset between EMF peak and acoustic resonance) without a brief physics-based justification; M2 then attempts to provide warrant but is presented as a separate bullet rather than being integrated into a single 'proposal + warrant' marking point. The structure should be: one mark for 'proposal + its physics justification' combined, not a disjointed two-bullet format. Additionally, the marker notes suggest acceptance of multiple candidate explanations (buoyancy, response time, finite coil radius) but the main scheme bullets do not explicitly map these onto the M1/M2 structure, creating ambiguity in how these alternatives would be credited.

FIX: Restructure Part (d) to a single combined marking point: M1 [2 marks]: Suggests that the EMF peak position (determined by flux geometry: dΦ/dt maximum when magnet coincides with coil plane) differs from the acoustic resonance condition (water surface at displacement antinode, set by boundary condition); this introduces a constant systematic offset to all measured lengths L₁ and L₂. Alternatively, accept: magnet buoyancy keeping it above water surface; or finite coil radius causing EMF peak to occur at fixed coil-plane position independent of true water level. Accept any proposal with explicit warrant linking the detection physics (flux, buoyancy, or geometry) to the acoustic boundary condition. OR: split into M1 (proposal only: 1 mark) and M2 (warrant: 1 mark) but then rewrite both to be explicit: M1: 'The induced EMF peak occurs when the magnet passes through the coil plane (flux extremum), not when the water surface reaches the acoustic antinode.' M2: 'Acoustic resonance requires the water surface (a boundary) to coincide with a displacement antinode; the EMF detection position is a fixed geometric reference independent of this acoustic condition, creating a constant vertical offset.' This ensures the causal chain (EMF geometry ≠ acoustic boundary condition → systematic error) is transparent.
cross_theme fail

Part (d) invokes D.4 Induction only as a detection-method name-check ('induced EMF', 'flux geometry'). The question does not require students to apply any electromagnetic induction physics principle—no calculation of flux, no Faraday's law, no Lenz's law, no quantitative analysis of how ∂Φ/∂t relates to magnet position. The physics of the systematic error is purely about standing waves (boundary condition vs. antinode position); induction appears only in the apparatus description.

FIX: Restructure part (d) to require quantitative application of D.4: for example, ask students to derive or estimate the coil signal as a function of magnet position using Faraday's law (e.g., flux Φ ∝ magnet distance, EMF ∝ dΦ/dt, peak dΦ/dt offset from water-surface position by geometry-dependent amount), and link the resulting systematic error in L to this induction-based offset calculation. Alternatively, move induction to an earlier part where it genuinely solves the resonance measurement problem, not merely provides apparatus context.
discriminator pass

Part (d) asks the student to critique the model by identifying why an assumption (EMF peak = water surface position) introduces systematic error. This requires analysis of the gap between the detection method (flux geometry) and the physical principle (boundary condition at antinode), exemplifying genuine model critique rather than mere calculation.
rule_of_one pass

Each part's mark count matches the number of distinct award criteria. Part (a): 2 marks, 2 bullets (definition components). Part (b)(i): 3 marks, 3 bullets (wavelength determination, formula application, numerical answer). Part (b)(ii): 3 marks, 3 bullets (formula setup, substitution, final answer). Part (c): 4 marks, 4 bullets (formula setup, predicted value, fractional deviation, conclusion). Part (d): 2 marks, 2 bullets (proposal with physics warrant structure). All alignment is secure; no dimension violations detected.
topic fail

Part (d) nominally invokes D.4 Induction by mentioning 'induced EMF' and 'flux,' but the question does not actually test induction concepts. The problem asks students to identify why the EMF peak position differs from the acoustic resonance position—a purely geometric/systematic-error discussion. No induction principle (Faraday's law, Lenz's law, flux linkage, rate of change of flux, or quantitative induction calculations) is required or assessed. D.4 is superficially named but not genuinely tested.

FIX: Either (1) restructure part (d) to require a quantitative induction calculation—e.g., 'Calculate the induced EMF when the magnet is at position x above the coil plane, and explain how the sharpness of the EMF peak depends on the magnet's descent rate and coil parameters'; or (2) replace part (d) entirely with a question that authentically integrates standing-wave physics with induction—e.g., asking students to design a calibration curve for magnet position vs. induced EMF using Faraday's law, then use it to correct the L₁ and L₂ measurements. As currently written, the question tests only C.4 (standing waves) and systematic-error reasoning; it does not test D.4.

Iteration 2 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder. Part (a) is AO1 (Define); parts (b)(i)–(b)(ii) are AO2 (Calculate/Determine); part (c) is AO2/AO3 (Show that, requires multi-step derivation); part (d) is AO3 (Suggest, requires proposal + physics warrant). Command terms match cognitive demands; progression from recall through calculation to derivation to reasoned proposal is intact.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 'Suggest' command-term structure. The scheme awards 2 marks combined for a single merged marking point (M1 [2 marks combined]) that conflates proposal + warrant into one bullet. Per the dimension spec, 'Suggest' requires an explicit two-part structure: a proposal + a separate brief physics-based justification. The current scheme's combined M1 does not parse as two distinct marking points and is therefore structurally non-compliant. Additionally, the 1-mark partial credit is vague ('without explicit reference' vs. 'does not conclude') and does not cleanly separate proposal from warrant.

FIX: Restructure Part (d) as two explicit marking points: M1 (1 mark): Proposal — 'The EMF peak is registered at a fixed offset Δ = r/2 = 0.75 cm above the actual water surface (coil plane position z = 0).' M2 (1 mark): Physics warrant — 'By part (c), the induction signal peaks when z = r/2 due to the maximum of dΦ/dz; this offset is identical for both L₁ and L₂ measurements, so the difference L₂ − L₁ remains unchanged, and therefore λ and v are unaffected.' This separates the proposal and its physics justification into two distinct, scorable components compliant with §4.4.1.
cross_theme pass

The question genuinely integrates both C.4 (standing waves) and D.4 (electromagnetic induction). Parts (a)–(b) establish the acoustic resonance framework; part (c) requires deriving the induction peak location using Faraday's law and flux linkage (core D.4 competencies); part (d) then synthesizes both themes by using the induction result (z = r/2) to explain how acoustic resonance positions are offset when detected inductively, and crucially how this affects (or does not affect) the calculated speed of sound. The induction topic is not merely name-checked—it is essential to understanding why the EMF peak differs from the magnet's coil-plane crossing, and why this matters for interpreting acoustic data.
discriminator pass

Part (d) explicitly asks the student to suggest how the calculated speed of sound is affected when using the EMF peak position rather than audible resonance, and to justify whether v is altered. This is a model-critique task that requires students to identify a systematic offset (z = r/2 from part (c)) and explain why it does or does not propagate into the final answer—a classic 'limitation/assumption' evaluation fitting the dimension contract.
rule_of_one fail

Part (d) is marked as [2 marks] but the mark scheme awards M1 [2 marks combined], meaning only one award criterion exists for a 2-mark part. This violates the dimension's contract: a 2-mark part must have 2 distinct award bullets. The scheme conflates proposal + warrant into a single M1 rather than separating them into M1 and M2.

FIX: Split part (d) into two explicit award criteria: M1 (1 mark) — Proposes that each EMF-marked position is offset from the true acoustic-node position by a fixed amount Δ = r/2 above the coil plane. M2 (1 mark) — Justifies via part (c) that since this same offset applies to both L₁ and L₂, the difference L₂ − L₁ remains unchanged, so v is unaffected (only e absorbs the shift). Partial credit guidance should then reference M1 or M2 individually, not 'M1 [2 marks combined]'.
topic pass

The ERQ authentically integrates both C.4 and D.4. Parts (a)–(b) test standing-wave fundamentals (nodes, antinodes, resonance conditions, end correction), while part (c) requires induction theory (Faraday's law, flux linkage, EMF optimization via differentiation). Part (d) bridges both topics by using the induction result (z = r/2 peak position) to explain how acoustic resonance detection via EMF differs from audible resonance, and whether it affects the calculated sound speed. Neither topic is superficial; both are genuinely required for a complete solution.

Planner output (stage 1)

Archetype: experimental setup
Scenario: A physics student constructs a resonance tube apparatus to investigate standing waves in an air column. A tuning fork of frequency 512 Hz vibrates above an open tube partially filled with water, allowing the effective length of the air column to be varied. The student measures the distances between successive resonance positions and records the corresponding sound intensities using a microphone connected to a data-logging system with electromagnetic induction coils positioned near the tube to detect induced currents as the water level changes.
Cross-pollination part: d
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	Define
b(i)	3	AO2	Calculate
b(ii)	3	AO2	Determine
c	4	AO2\|AO3	Show that
d	2	AO3	Suggest

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