Generated ERQ

✓ passed D.3 Motion in electromagnetic fields × A.1 Kinematics 12 marks HL 3 passes 112.61s $0.6405

## ERQ · 12 marks · Topics: D.3 Motion in electromagnetic fields + A.1 Kinematics **Stem.** A mass spectrometer's velocity selector ejects protons (charge +1.6 × 10⁻¹⁹ C, mass 1.67 × 10⁻²⁷ kg) into an analyser chamber containing a uniform magnetic field of magnitude **B = 0.50 T** directed along the +x-axis. A proton enters the field at the origin with speed **v = 2.0 × 10⁶ m s⁻¹**, with its velocity vector lying in the x–y plane at an angle of **30° to the +x-axis**. The resulting trajectory is helical: the velocity component parallel to **B** is unchanged, while the perpendicular component executes uniform circular motion in the y–z plane. A detector plate is mounted perpendicular to the x-axis at **x = 1.0 m** from the entry point. Edge effects and gravity are negligible inside the analyser. ### Part (a) State [2 marks] · AO1 · Topic: D.3 State the direction of the magnetic force on the proton at the instant it enters the field, and explain why this force does no work on the proton. ### Part (b)(i) Determine [4 marks] · AO2/AO3 · Topic: D.3 + A.1 By equating the magnetic force on the proton to the centripetal force required for circular motion in the y–z plane (a kinematic condition for uniform circular motion at radius r with tangential speed v_⊥), determine the radius r of the helix. ### Part (b)(ii) Calculate [2 marks] · AO2 · Topic: D.3 Calculate the period T of one revolution in the y–z plane. ### Part (c) Show that [2 marks] · AO2/AO3 · Topic: D.3 + A.1 Using the kinematic equation for uniform motion along the field direction, show that the time for the proton to reach the detector plate at x = 1.0 m is approximately 5.8 × 10⁻⁷ s, and hence determine how many complete revolutions the helix executes between entry and detection. ### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The pitch predicted by this model assumes the axial motion is at constant velocity (a A.1 kinematic idealisation). In a real solenoid analyser, the magnetic field weakens near the ends ("fringing"). Suggest how this fringing would cause the measured number of revolutions before impact to differ from your answer in (c). --- ## Mark Scheme ### Part (a) [2 marks] - M1: Force is in the +z direction (or "out of the x–y plane" / perpendicular to **v** and **B**, consistent with F = qv × B for a positive charge with v in x–y plane and B along +x). [ECF: no] - M2: Force is always perpendicular to velocity, therefore F · v = 0 and no work is done (accept: kinetic energy / speed is unchanged). [ECF: no] ### Part (b)(i) [4 marks] - M1: Resolves perpendicular component v_⊥ = v sin 30° = 1.0 × 10⁶ m s⁻¹ (accept v sin θ). [ECF: no] - M2: States force balance qv_⊥B = mv_⊥²/r (Lorentz force = centripetal kinematic requirement). [ECF: no] - M3: Rearranges to r = mv_⊥/(qB) and substitutes: r = (1.67 × 10⁻²⁷ × 1.0 × 10⁶) / (1.6 × 10⁻¹⁹ × 0.50). [ECF: yes — accepts wrong v_⊥ from M1] - M4: r = 2.1 × 10⁻² m (accept 0.021 m, 2.09 × 10⁻² m). [ECF: yes] ### Part (b)(ii) [2 marks] - M1: Uses T = 2πm/(qB) or T = 2πr/v_⊥ with correct substitution. [ECF: yes from (b)(i)] - M2: T = 1.3 × 10⁻⁷ s (accept 1.31 × 10⁻⁷ s). [ECF: yes] ### Part (c) [2 marks] - M1: Identifies v_∥ = v cos 30° = 1.73 × 10⁶ m s⁻¹ and applies kinematic equation t = x/v_∥ = 1.0 / 1.73 × 10⁶ → t ≈ 5.78 × 10⁻⁷ s ≈ 5.8 × 10⁻⁷ s. [ECF: no — "show that"] - M2: Number of revolutions N = t/T = 5.78 × 10⁻⁷ / 1.31 × 10⁻⁷ ≈ 4.4 revolutions (accept 4 complete revolutions, or 4.4). [ECF: yes from (b)(ii)] ### Part (d) [2 marks] - M1: Identifies that in fringing regions B is smaller, so the cyclotron period T = 2πm/(qB) increases (equivalently, angular speed ω = qB/m decreases) near the ends. [ECF: no] - M2: Consequence: fewer revolutions are completed in the same axial transit time than predicted, OR the assumption of constant v_∥ also fails because the field has off-axis components that exert a force with an x-component, accelerating/decelerating the axial motion and invalidating the uniform-kinematics model used in (c). [ECF: no] ### Marker notes - (a) M1: accept clear diagram with labelled +z arrow; reject "upward" without axes. - (b)(i): full credit for an algebraic derivation r = mv sin θ/(qB) followed by correct numerical answer. If candidate uses v (not v_⊥) throughout, award M2 only; cap at 2/4. - (b)(ii): accept derivation from T = 2πr/v_⊥ using the candidate's r and v_⊥ (ECF route). - (c): "Show that" — working must be visible; bare answer scores 0 for M1. Accept N stated as "4 full + partial" or 4.4. - (d) discriminator: accept any of (i) reduced B → longer period near ends; (ii) radial B_r component in fringe field → axial Lorentz force, violating constant-v_∥ kinematic assumption; (iii) non-uniform B → r and pitch vary along trajectory, so the helix is not geometrically regular. Do NOT accept generic "experimental error" or "air resistance".

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (State/Define), parts (b) and (c) progress through AO2/AO3 (Calculate/Determine/Show that), and part (d) is AO3 (Suggest/Explain), with increasing cognitive demand matching the mark allocation. The structure respects the IB progression from recall through application to evaluation.
cross_theme pass

The ERQ genuinely blends D.3 (Motion in electromagnetic fields) and A.1 (Kinematics). Part (c) is the integration point: it explicitly requires decomposing velocity into parallel and perpendicular components using trigonometry (A.1 concept—uniform motion along x-axis), then applies the kinematic equation t = x/v∥ to find transit time, and combines this with the period from D.3 to calculate revolutions. Both physics concepts are essential to solving the problem, not merely name-checked.
discriminator pass

Part (d) explicitly asks students to 'Suggest one reason why the number of revolutions counted in a real solenoid-based apparatus would differ from the value calculated in part (c).' This is a clear model-critique question that requires identifying a physical limitation (non-uniform field, fringe effects, magnetic mirror, collisions, etc.) and explaining how it breaks the idealized assumptions (constant parallel velocity, uniform field). The mark scheme confirms this is the intended dimension.
rule_of_one fail

Part (d) is allocated 2 marks but the mark scheme lists only 1 distinct award criterion (identification of a physical limitation). The second bullet describes explanation/consequence but is not structured as a separate countable operation with its own award point. This violates the dimension's contract: a 2-mark part must have exactly 2 award criteria.

FIX: Restructure part (d)'s mark scheme to explicitly separate the 2 marks: (1) Identifies a physical limitation [1 mark]; (2) Explains how this affects the transit time/number of revolutions [1 mark]. Currently both are bundled under a single hyphen. Alternatively, reduce part (d) to 1 mark if only identification is required.
topic pass

The ERQ rigorously tests both topics. D.3 (Motion in electromagnetic fields) dominates parts (a)–(b) via the Lorentz force, cyclotron radius, and cyclotron period; A.1 (Kinematics) is genuinely required in part (c), where uniform motion kinematics (v∥ = v cos θ; t = x/v∥) must be applied to the parallel velocity component to find transit time and revolution count. Neither topic is merely nominal—both are structurally essential to solving the problem.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (State/Define); parts (b)(i)–(c) are AO2/AO3 (Calculate/Determine/Show that); part (d) is AO3 (Suggest/Explain). The progression from recall through procedural to higher-order reasoning is sound and matches HL expectations.
cross_theme fail

Part (c) nominally invokes A.1 kinematics (t = x/v∥), but this is merely a plug-in formula application after the electromagnetic pitch calculation. The kinematics concept is not genuinely blended; D.3 dominates throughout (helical motion, Lorentz force, cyclotron period, pitch). A.1 is reduced to name-checking a single kinematic equation rather than being an integral part of solving the core physics problem.

FIX: Redesign part (c) to require students to derive or analyze the axial motion using kinematic concepts more substantially—for example, ask them to integrate the equation of motion under a time-varying magnetic field, or to use kinematic equations to predict deflection patterns caused by field inhomogeneity, or to compare constant-velocity vs. accelerated axial models. Alternatively, restructure so that analyzing the perpendicular circular motion explicitly applies both centripetal kinematics (v²/r) and D.3 electromagnetic force balance simultaneously in a single coherent sub-question.
discriminator pass

Part (d) explicitly asks students to 'Suggest one reason why the measured transit time would differ' and to 'explain how this affects the validity of treating the axial motion as uniform.' This directly critiques the model assumption (constant v_∥ in uniform field) by requiring identification of a real-world limitation (fringing/non-uniformity) and its physical consequence, meeting the dimension's requirement for model critique.
rule_of_one fail

Part (b)(ii) is marked as 2 marks but lists only 1 distinct operation (choice of formula); the substitution and numerical answer are bundled into M2. This violates the dimension rule requiring 2–3 discrete award criteria for a 2-mark Calculate/Determine part. Part (c) is marked as 3 marks but M1 conflates the pitch calculation and the final numerical confirmation into a single bullet, making the structure unclear; however, three distinct operations are nominally present (identify formula, substitute, compute time). Part (d) is marked as 2 marks with 2 bullets (identify limitation + explain consequence), which is acceptable. The primary failure is part (b)(ii).

FIX: In part (b)(ii), separate M1 and M2 as follows: M1: 'Uses T = 2πm/(qB) [or T = 2πr/v_⊥].' M2: 'Substitution: T = 2π × 1.67 × 10⁻²⁷ / (1.6 × 10⁻¹⁹ × 0.50) [or equivalent using r and v_⊥].' M3: 'T = 1.3 × 10⁻⁷ s (accept 1.31 × 10⁻⁷ s).' This ensures 3 discrete operations for the 2-mark part, or reduce the mark allocation to 1 mark if only formula + answer are required.
topic pass

The ERQ genuinely tests both D.3 (Motion in electromagnetic fields) and A.1 (Kinematics). Parts (a)–(b) focus on the Lorentz force, radius of curvature, and period (core D.3), while part (c) critically requires A.1 kinematics—specifically constant-velocity kinematics (t = x/v)—to determine transit time from the pitch and axial speed. Part (d) extends this by questioning the validity of the constant-velocity assumption under real field conditions. The two topics are genuinely integrated and codependent, not merely mentioned.

Iteration 2 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder across all four parts: (a) is AO1/AO2 with State command term requiring direction and reasoning (knowledge + understanding); (b)(i)–(ii) are AO2/AO3 with Determine/Calculate requiring algebraic derivation and substitution (application + analysis); (c) is AO2/AO3 with Show that/determine requiring kinematic proof and numerical synthesis (application + analysis); (d) is AO3 with Suggest requiring evaluation of assumptions and physical reasoning about real-world deviations (synthesis + evaluation). Progression is logical and marks are appropriately distributed across difficulty tiers.
cross_theme pass

The ERQ genuinely blends both D.3 (motion in electromagnetic fields) and A.1 (kinematics). Part (b)(i) explicitly equates magnetic force to centripetal force and uses the kinematic condition for uniform circular motion (v²/r). Part (c) applies the kinematic equation for uniform motion (t = x/v) to the axial component. Both concepts are essential to solving the question, not mere name-checks; the helical trajectory inherently requires both electromagnetic force analysis and kinematic decomposition of motion.
discriminator pass

Part (d) explicitly asks students to critique the model by identifying how fringing (a real-world deviation) causes the measured outcome to differ from the theoretical prediction. It directly challenges the A.1 kinematic idealisation (constant axial velocity) and requires reasoning about physical limitations, exemplifying genuine model critique.
rule_of_one fail

Part (b)(i) is allocated 4 marks but contains 5 distinct award bullets (M1–M5 decomposed as: resolve v_⊥, state force balance, rearrange formula, substitute numerically, state final answer). This violates the strict 1-mark-per-operation contract. Additionally, Part (c) is allocated 2 marks but lists 2 bullets that represent substantively different tasks (deriving time, then calculating revolutions), which should ideally map to 1 mark each—acceptable at the boundary but the M1 'show that' criterion is vague about whether the time derivation alone earns 1 mark or if both steps are required for credit.

FIX: Restructure Part (b)(i) to 3 marks: M1 (resolve v_⊥ and state force balance together), M2 (rearrange and substitute to r = mv_⊥/(qB)), M3 (numerical answer with units). Alternatively, expand to 5 marks if the five operations are intended to be separately awarded. For Part (c), clarify whether M1 covers the kinematic derivation *only* (earning 1 mark for 'show that') and M2 covers the revolution count, or merge them into a single 1-mark 'show that' criterion followed by a separate 1-mark 'number of revolutions' criterion to achieve clearer alignment.
topic pass

The ERQ rigorously integrates both D.3 (Motion in electromagnetic fields) and A.1 (Kinematics) throughout. D.3 concepts (Lorentz force, magnetic force perpendicularity, cyclotron motion, period formula) are central to parts (a)–(c). A.1 concepts (velocity decomposition, kinematic equations for uniform motion, acceleration, time-distance relations) are explicitly required in (b)(i)–(c) to resolve velocity components and calculate transit time. Part (d) deepens the integration by examining where the constant-velocity kinematic idealisation fails. Neither topic is nominal or decorative.

Planner output (stage 1)

Archetype: experimental setup
Scenario: A charged particle enters a uniform magnetic field region at an angle to the field direction, experiencing a Lorentz force that curves its trajectory. Students analyze the helical motion resulting from the superposition of circular motion perpendicular to the field and constant-velocity motion parallel to it. The scenario requires linking electromagnetic force concepts (D.3) with kinematic decomposition (A.1) to explain why the particle does not simply follow a circular path.
Cross-pollination part: c
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State the direction of the Lorentz force on a charged particle moving in a magnetic field, and define the term 'radius of curvature' for circular motion in the perpendicular plane.
b(i)	3	AO2	A proton (charge +1.6 × 10⁻¹⁹ C, mass 1.67 × 10⁻²⁷ kg) enters a uniform magnetic field of magnitude 0.50 T at speed 2.0 × 10⁶ m s⁻¹, at 30° to the field direction. Calculate the radius of the circular component of motion.
b(ii)	2	AO2	Determine the period of one complete revolution in the plane perpendicular to the magnetic field.
c	3	AO2\|AO3	Show that the pitch (axial distance traveled per revolution) of the helical path is 2π m (to 2 s.f.), and use kinematic equations from A.1 to derive the time taken for the particle to travel 1.0 m along the field direction.
d	2	AO3	Suggest why a real apparatus using a solenoid to create the magnetic field would deviate from the ideal helical trajectory predicted by this model, and explain how this limitation affects the validity of assumptions made in part (c).

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