Generated ERQ

✓ passed C.4 Standing waves and resonance × D.2 Electric and magnetic fields 12 marks HL 3 passes 135.76s $0.6773

## ERQ · 12 marks · Topics: C.4 Standing waves and resonance + D.2 Electric and magnetic fields **Stem.** A physics researcher studies the resonant modes of a thin conducting wire of length L = 0.480 m and linear mass density μ = 1.20 × 10⁻³ kg m⁻¹, stretched between two fixed insulating supports under tension T = 27.0 N. The wire carries a steady current I = 4.50 A and is placed in a uniform horizontal magnetic field of magnitude B = 0.0850 T, directed perpendicular to the wire. The magnetic force per unit length acts as a uniform transverse drive. The researcher slowly varies the frequency of the current (the current is sinusoidal: I(t) = I₀ sin(2πft), with I₀ = 4.50 A) and measures the amplitude of transverse oscillations of the wire's midpoint with a laser displacement sensor. A strong resonance peak is observed at one particular frequency, with peak-to-peak midpoint displacement 6.4 mm. The surrounding air provides light damping. ### Part (a) Draw [2 marks] · AO1 · Topic: C.4 On a sketch of the wire between its two supports, draw the displacement pattern of the lowest-frequency standing-wave mode that can be resonantly excited by the spatially-uniform magnetic driving force described in the stem, and label the position(s) of any antinode(s). ### Part (b)(i) Calculate [3 marks] · AO2 · Topic: C.4 Calculate the frequency of the alternating current that drives the mode you drew in (a). ### Part (b)(ii) Show that [3 marks] · AO2/AO3 · Topic: C.4 + D.2 The amplitude of the driving force per unit length is F₀/L = BI₀. By equating the work done by this spatially-uniform sinusoidal driving force on the wire over one cycle at resonance to the energy dissipated per cycle, show that only standing-wave modes with an odd number of half-wavelengths along L can be resonantly driven by this configuration, and hence justify your choice in (a). ### Part (c) Determine [2 marks] · AO3 · Topic: C.4 + D.2 The magnetic field is now tilted so that it makes an angle θ with the horizontal, while remaining perpendicular to the wire's axis. The researcher finds that the resonance frequency identified in (b)(i) shifts downward by 0.6 %. By considering how the vertical component of the magnetic force per unit length acts together with gravity to modify the effective tension along the wire (treating the wire as nearly horizontal), determine θ. ### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The peak midpoint amplitude (3.2 mm) predicted from a simple lossless standing-wave model using the driving force in (b)(ii) is much larger than the measured 3.2 mm peak. Suggest one physical assumption in this combined wave + electromagnetic model whose failure causes the measured amplitude to be smaller than the idealised prediction, and state explicitly the direction of the resulting discrepancy. --- ## Mark Scheme ### Part (a) [2 marks] - Mark 1: Sketch shows the fundamental mode (n = 1) — a single antinode at the midpoint, nodes only at the two fixed supports [ECF: no] - Mark 2: Antinode at L/2 labelled, and zero displacement clearly indicated at both supports [ECF: no] ### Part (b)(i) [3 marks] - Mark 1: Identifies f₁ = v/(2L) with v = √(T/μ) [ECF: no] - Mark 2: Correct substitution: v = √(27.0/1.20 × 10⁻³) = 150 m s⁻¹; f₁ = 150/(2 × 0.480) [ECF: yes] - Mark 3: f₁ = 156 Hz (accept 156–157 Hz) to 3 s.f. with units [ECF: yes] ### Part (b)(ii) [3 marks] - Mark 1: States that the work per cycle is W ∝ ∫₀ᴸ y_n(x) dx, where y_n(x) = sin(nπx/L) is the mode shape, because the drive force per unit length is spatially uniform [ECF: no] - Mark 2: Evaluates/recognises that ∫₀ᴸ sin(nπx/L) dx = (L/nπ)(1 − cos(nπ)), which vanishes for even n and is non-zero for odd n [ECF: yes] - Mark 3: Concludes that only odd-n modes absorb net energy from the uniform drive, so n = 1 is the lowest excitable mode, consistent with part (a) [ECF: yes] ### Part (c) [2 marks] - Mark 1: Recognises that the vertical component of the magnetic force per unit length, BI₀ sin θ (time-averaged appropriately, or its DC offset if applicable), together with the weight per unit length μg, changes the sag/tension; equivalently, sets up f′/f = √(T′/T) ⇒ T′/T = (0.994)² ≈ 0.988, giving ΔT ≈ −0.012 × 27.0 = −0.32 N over the wire, OR uses force balance giving the required vertical force per unit length. [ECF: no] - Mark 2: Solves for θ using BI₀ sin θ balancing the required change (with μg = 0.0118 N m⁻¹ included), obtaining θ ≈ 2° (accept 1.5°–2.5°) with units [ECF: yes] ### Part (d) [2 marks] - Mark 1: Identifies ONE specific assumption that fails, e.g.: - (i) damping (air drag / internal friction) was neglected — the idealised model has infinite Q - (ii) the drive current is assumed perfectly sinusoidal and in phase with displacement at resonance — phase lag from finite Q reduces stored energy - (iii) the magnetic field is assumed perfectly uniform along L — non-uniformity reduces the spatial overlap integral with the n = 1 mode - (iv) the wire is assumed perfectly flexible (no bending stiffness) — stiffness shifts and broadens the resonance - (v) ohmic heating / temperature rise alters μ or T during measurement [ECF: no] - Mark 2: States that this assumption's failure causes the measured amplitude to be SMALLER than the idealised prediction (i.e. the discrepancy is in the direction of reduced response), with brief mechanism (energy loss / reduced effective drive / off-resonance operation) [ECF: yes] ### Marker notes - (a): accept a clear half-sine sketch; do NOT award mark 2 if antinode is unlabelled or if nodes are drawn anywhere except at the supports. - (b)(ii): accept argument via symmetry — for even n, the mode shape is antisymmetric about L/2 while the drive is symmetric, so the overlap integral vanishes. Full marks if the symmetry argument is stated precisely. - (c): accept alternative routes — e.g., interpreting the shift via a change in effective μ_eff under a static vertical force gradient, provided the algebra yields θ in the 1.5°–2.5° band. Reject answers that ignore gravity or that use the horizontal component. - (d): each candidate assumption must be paired with the correct direction (measured < predicted). Answers giving only "experimental error" or "air resistance" without linking to amplitude reduction earn 1 mark maximum. Reject answers that would predict a LARGER measured amplitude (wrong direction).

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder strictly. Part (a) is AO1/AO2 with Draw (state/sketch nodes and antinodes). Parts (b)(i)–(b)(ii) and (c) are AO2/AO3 with Calculate/Determine (apply standing-wave formula, percentage calculation, force balance). Part (d) is AO3 with Suggest (evaluate sources of systematic error in the experimental model). Command terms and cognitive progression are correctly sequenced.
cross_theme fail

Part (c) uses only D.2 (electric field equilibrium) to find charge; C.4 (standing waves) is not genuinely needed to solve it. The dust grain's position in the standing wave is mentioned in the stem but plays no role in the calculation—the charge is determined purely from force balance in the electric field. Topic C.4 is abandoned after part (b).

FIX: Restructure part (c) to require both concepts: e.g., 'The dust grain at a specific antinode position (5L/10) experiences acoustic radiation pressure in addition to gravity and electric force. Calculate the net charge required for equilibrium, accounting for the acoustic pressure from the standing wave at that antinode.' This would mandate explicit use of standing-wave amplitude/energy (C.4) alongside the electric field (D.2).
discriminator pass

Part (d) explicitly asks students to 'Suggest two reasons why the speed of sound calculated in (b)(i) may differ from the true speed of sound in the air column.' This is a genuine model-critique question that probes assumptions underlying the calculation (end corrections, node localization uncertainty, temperature effects, frequency precision, humidity, tube uniformity). It does not ask for further calculation and aligns well with the IB Physics dimension of critiquing the model used.
rule_of_one fail

Part (a) is assigned 2 marks but lists only 2 award bullets, which is correct. However, part (b)(ii) is assigned 3 marks but contains 3 award bullets that do not cleanly separate into three distinct operations: mark 2 is purely numerical (result of mark 1 calculation), and mark 3 conflates the percentage value with a physical reasoning statement. The 'Calculate/Determine then state' structure should yield two separate marks (calculation + statement), not three. This violates the dimensional requirement that mark counts and award-bullet counts diverge by no more than 1.

FIX: Restructure part (b)(ii) as a 2-mark Determine question: Mark 1 = calculate percentage difference (formula + substitution + numerical result); Mark 2 = state whether consistent with warming and justify briefly. Alternatively, keep 3 marks but split the reasoning into a distinct mark: e.g. Mark 1 = calculation, Mark 2 = percentage result stated, Mark 3 = physical reasoning (√T relationship and plausibility check). As written, the boundary between marks 2 and 3 is ambiguous.
topic pass

The ERQ effectively tests both C.4 and D.2. Parts (a), (b), and (d) substantively probe standing waves, resonance, and node/antinode formation in a closed pipe (C.4). Part (c) requires equilibrium of electric and gravitational forces on a charged dust grain in a uniform electric field (D.2), with the field strength and charge calculation being non-trivial applications of electrostatics. Neither topic is superficially invoked; both are operationally central to the scenario.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly: (a) is AO1/AO2 with Draw command; (b)(i)–(b)(ii) are AO2/AO3 with Show that/Determine; (c) is AO3 with Calculate (integrating both C.4 and D.2); (d) is AO3 with Suggest (discrimination of assumptions). Progression from recall through procedural application to critical evaluation is intact.
cross_theme fail

Part (c) nominally invokes D.2 (electric field) but does not genuinely blend both topics. The electric field is used only to calculate the electric force; the standing-wave physics (C.4) is used only to supply the acoustic restoring force constant k and displacement x. The two concepts operate in parallel force-balance terms rather than being synthetically integrated—the student could solve for charge without understanding standing waves (given k and x as data), and the acoustic physics contributes no new insight into the electric/magnetic phenomena. Part (d) further weakens integration by treating assumptions as a discriminator rather than requiring genuine conceptual blending in the solution.

FIX: Restructure part (c) to require students to derive or justify the restoring-force model using wave properties (e.g., by explaining how pressure antinodes relate to particle displacement nodes and deriving k from the acoustic impedance or wave energy density), or ask students to calculate how the electric field modifies the boundary conditions or resonant frequency of the standing wave. Alternatively, introduce a magnetic field component (e.g., ask how a superimposed magnetic field would affect the trajectory or equilibrium position of the charged bead in the standing-wave potential well), thereby forcing genuine interplay between D.2 and C.4 in the solution logic.
discriminator pass

Part (d) explicitly asks students to 'Suggest one reason why the charge value calculated in (c) is likely to differ from the true charge on the bead.' This is a genuine model-critique question that requires identifying a broken assumption or limitation of the model (linear restoring force, field uniformity, neglected forces, point-charge approximation, etc.) and explaining how it leads to error. The mark scheme confirms this dimension is properly assessed.
rule_of_one fail

Part (d) is marked as [2 marks] but the mark scheme lists only 2 award bullets without clear operational distinction. More critically, the scheme provides 5 separate acceptable answers in the 'discriminator' section without clarifying which specific answer earns which mark. For a 2-mark 'Suggest' part, the operational structure (identify assumption + explain consequence) is sound, but the mark scheme fails to unambiguously assign marks to distinct steps within each candidate response, making consistent application impossible.

FIX: Rewrite part (d) mark scheme as: Mark 1: Identifies one specific assumption in the acoustic levitation model (e.g., linear restoring force, uniform E-field, absence of Gor'kov force, negligible buoyancy, point-charge treatment). Mark 2: Explains how violation of that assumption causes an over- or under-estimate of the calculated charge magnitude. Provide the 5 exemplar assumptions as acceptable targets, each with explicit direction of bias (over/under) to guide marking.
topic pass

The ERQ genuinely integrates both topics. Parts (a)–(b) establish the standing-wave geometry (C.4), part (c) requires simultaneous force balance involving the acoustic restoring force (C.4), gravity, and electric force in a uniform field (D.2), and part (d) probes assumptions in the combined model. Neither topic is superficial; both are essential to solving the problem.

Iteration 2 — failed

ao_ladder fail

The cognitive ladder is broken. Part (d) is labelled 'Suggest [2 marks] · AO3' but the command term 'Suggest' paired with identifying and explaining a physical assumption failure is AO3-appropriate only if the follow-up requires genuine evaluation or synthesis. However, the core problem is that part (c) 'Determine [2 marks] · AO3' asks students to calculate a specific numerical value (θ ≈ 2°) using given data and algebra—this is AO2 (Calculate/Determine with a closed answer), not AO3. The final tier must be clearly AO3 (Explain/Discuss/Evaluate with open-ended reasoning), but part (c) is a straightforward calculation that does not rise to that level.

FIX: Restructure the question so that part (c) remains AO2 (e.g., 'Calculate the angle θ...'), and elevate part (d) to genuine AO3 by requiring students to 'Evaluate the experimental design' or 'Discuss the relative importance of different sources of error in this resonance measurement' with multiple valid answers, rather than asking them to identify a single assumption. Alternatively, swap the content: make (c) the AO3 part (e.g., 'Explain why the frequency shift provides evidence for the vertical component of the magnetic force') and (d) a lower-order AO2 confirmation step. As written, the ladder ascends only to AO2 by part (c) and does not reach true AO3 synthesis.
cross_theme fail

The question fails the dimension requirement because Topic D.2 (Electric and magnetic fields) is not genuinely blended throughout the problem-solving; it appears only as a superficial driver in parts (b)(ii) and (c), while the core physics is entirely Topic C.4 (standing waves). Parts (a) and (b)(i) are pure standing-wave mechanics with no electromagnetic content. Part (b)(ii) uses the magnetic force formula BI₀ only to set up a uniform driving force—the actual solution (evaluating the integral of sin(nπx/L)) is pure wave physics. Part (c) adds a tilted field but reduces to a force-balance adjustment to tension, not genuine electromagnetic-wave interaction. The two topics do not meaningfully integrate; D.2 is merely a vehicle for specifying the drive, not a concept that must be understood and applied alongside C.4.

FIX: Redesign the question so that D.2 concepts are genuinely required in the solution, not just in setup. For example: (1) Ask students to derive how the Lorentz force on a current-carrying wire in a non-uniform magnetic field modifies the effective restoring force and hence the wave equation itself, making D.2 theory essential to finding resonance frequencies. (2) Require analysis of the phase relationship between current oscillation and displacement at resonance using both electromagnetic induction (Faraday's law / motional EMF) and mechanical resonance theory. (3) Pose a scenario where the magnetic field strength varies sinusoidally along the wire length, so that the spatial integral in part (b)(ii) genuinely requires understanding both the magnetic force distribution (D.2) and mode orthogonality (C.4) together. This would make D.2 and C.4 inseparable in the solution path.
discriminator fail

Part (d) asks students to suggest a physical assumption whose failure explains why measured amplitude is smaller than predicted, which is a genuine model-critique task. However, the stem contains a typo/error: it states 'The peak midpoint amplitude (3.2 mm) predicted from a simple lossless standing-wave model... is much larger than the measured 3.2 mm peak.' This is logically contradictory — if both values are 3.2 mm, the predicted value is NOT larger. This internal inconsistency undermines the validity of part (d) as a critique question, since students cannot meaningfully critique a model when the comparison statement is incoherent.

FIX: Correct the stem in part (d) to read: 'The peak midpoint amplitude (e.g. 5.2 mm) predicted from a simple lossless standing-wave model using the driving force in (b)(ii) is much larger than the measured 3.2 mm peak.' Or specify: 'The peak midpoint amplitude predicted from a simple lossless standing-wave model is approximately 5.2 mm, but the measured value is only 3.2 mm.' This removes the contradiction and allows part (d) to function as a valid model-critique question.
rule_of_one fail

Part (d) has 2 marks but lists 5 distinct award criteria (damping, phase lag, field non-uniformity, bending stiffness, ohmic heating), violating the strict 1:1 mapping between marks and operations. Additionally, Part (c) conflates two separate operations (setting up the frequency ratio and solving for θ) into a single mark 1, leaving mark 2 to do both substitution and final answer—asymmetric structure.

FIX: Part (d): Restructure to exactly 2 marks by (1) Award mark 1 for identifying ONE specific physical assumption (e.g., 'damping was neglected') without listing alternatives; Award mark 2 for stating the direction (measured < predicted) and brief mechanism. Part (c): Split mark 1 into two sub-bullets: (1a) Set up f′/f = √(T′/T) and calculate T′/T = 0.988; (1b) Recognize ΔT ≈ −0.32 N arises from vertical magnetic + gravitational force; Award mark 2 for solving BI₀ sin θ = μg × L (or equivalent) to yield θ ≈ 2°. This ensures 3 operations per 3-mark part and 2 per 2-mark part.
topic pass

The ERQ robustly tests both C.4 (standing waves and resonance) and D.2 (electric and magnetic fields). Part (a)–(b) establish standing-wave modes on the wire. Parts (b)(ii) and (c) genuinely integrate the magnetic field concepts: part (b)(ii) uses the magnetic force F = BI₀ as the spatially-uniform driving mechanism to derive which modes can resonate (odd-n selection rule), and part (c) applies the magnetic force's vertical component to modify effective tension. The physics is tightly coupled—neither topic is superficial or nominally mentioned.

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