Generated ERQ

✓ passed E.2 Quantum physics (HL) × C.3 Wave phenomena 14 marks HL 3 passes 123.86s $0.8186

## ERQ · 14 marks · Topics: E.2 Quantum physics (HL) + C.3 Wave phenomena · Archetype: experimental_analysis **Integration:** primary=E.2 Quantum physics (HL), secondary=C.3 Wave phenomena (strength: supporting) **Stem.** A student investigates the photoelectric effect using a vacuum phototube whose cathode is coated with caesium. Light from a mercury discharge lamp passes through a transmission diffraction grating of slit spacing d = 1.67 × 10⁻⁶ m, and a narrow slit selects one spectral line at a time before the light strikes the cathode. The anode is held at a variable negative potential and the stopping potential V_s is measured for each spectral line. The collected data are: | Spectral line | Frequency f / 10¹⁴ Hz | Stopping potential V_s / V | |---|---|---| | Yellow | 5.19 | 0.27 | | Green | 5.49 | 0.39 | | Blue | 6.88 | 0.96 | | Violet | 7.41 | 1.18 | | UV | 8.20 | 1.50 | A graph of V_s against f is straight, with gradient (4.10 ± 0.15) × 10⁻¹⁵ V s and y-intercept (−1.86 ± 0.08) V. The student wishes to determine the Planck constant h and the work function φ of caesium, and later uses the grating to investigate the role of intensity. ### Part (a) State [2 marks] · AO1 · Topic: E.2 State the Einstein photoelectric equation and identify how the gradient and the y-intercept of the V_s–f graph relate to h and φ. ### Part (b)(i) Determine [3 marks] · AO2 · Topic: E.2 Using the gradient of the graph, determine an experimental value for the Planck constant, and state its absolute uncertainty. ### Part (b)(ii) Calculate [4 marks] · AO2|AO3 · Topic: E.2 Using the y-intercept, calculate the work function of caesium in electronvolts and its absolute uncertainty. Comment on whether the experimental value is consistent with the accepted value φ = 2.14 eV. ### Part (c) Derive [3 marks] · AO2|AO3 · Topic: E.2 + C.3 The yellow spectral line is selected by setting the grating at a first-order diffraction angle θ. Using the grating equation and the data for the yellow line, derive an expression for θ in terms of d, c and f, substitute the numerical values, and calculate θ in degrees. ### Part (d) Explain [2 marks] · AO3 · Topic: E.2 + C.3 · ASSUMPTIONS DISCRIMINATOR The student now widens the slit illuminating the grating, increasing the intensity of the yellow line falling on the cathode without changing the grating angle. Explain why the measured stopping potential for the yellow line is unchanged, while the photocurrent increases. --- ## Mark Scheme ### Part (a) [2 marks] — State - M1: Einstein equation stated as eV_s = hf − φ (or E_k,max = hf − φ with eV_s = E_k,max). [no ECF] - M2: Gradient = h/e AND y-intercept = −φ/e (both required). [no ECF] ### Part (b)(i) [3 marks] — Determine - M1: h = gradient × e = (4.10 × 10⁻¹⁵) × (1.60 × 10⁻¹⁹). [no ECF] - M2: h = 6.56 × 10⁻³⁴ J s (accept 6.5–6.6 × 10⁻³⁴). [ECF from (a)] - M3: Δh = (0.15/4.10) × h = 0.24 × 10⁻³⁴ J s ≈ 0.2 × 10⁻³⁴ J s; quote as (6.6 ± 0.2) × 10⁻³⁴ J s. [ECF from M2] ### Part (b)(ii) [4 marks] — Calculate + Comment - M1: φ = |y-intercept| × e = 1.86 eV (accept use of e directly since intercept is in volts → φ in eV numerically equals 1.86). [no ECF] - M2: Δφ = 0.08 eV, so φ = (1.86 ± 0.08) eV. [ECF from M1] - M3: Difference from accepted value = 2.14 − 1.86 = 0.28 eV, which exceeds the uncertainty 0.08 eV (≈ 3.5σ). [ECF from M2] - M4: Comment: result is NOT consistent with the accepted value — systematic error likely (e.g. contact potential between cathode and anode, surface oxidation lowering apparent φ, or stray light). [ECF from M3] ### Part (c) [3 marks] — Derive - M1: Grating equation d sin θ = mλ with m = 1 and λ = c/f → sin θ = c/(d f). [no ECF] - M2: Substitution: sin θ = (3.00 × 10⁸)/[(1.67 × 10⁻⁶)(5.19 × 10¹⁴)] = 0.3463. [ECF from M1] - M3: θ = arcsin(0.3463) = 20.3° (accept 20.2°–20.4°). [ECF from M2] ### Part (d) [2 marks] — Explain (causal chain per §4.4.1) - M1: At fixed grating angle θ and spacing d, the grating equation d sin θ = mλ fixes the wavelength λ (and hence the photon frequency f = c/λ) of the light reaching the cathode, regardless of how wide the input slit is opened. - M2: Therefore widening the slit increases the number of photons per second arriving but does not change each photon's energy hf, so by V_s = hf/e − φ/e the stopping potential is unchanged, while the rate of electron ejection — and hence the photocurrent — rises in proportion to the photon flux. [linking marking point: fixed λ → fixed hf → fixed V_s; higher flux → higher current] ### Marker notes - Part (b)(i): accept fractional uncertainty propagated as (Δgradient/gradient) applied to h; do not penalise rounding 0.24 → 0.2. - Part (b)(ii): accept any plausible systematic error in M4 (contact potential, anode work function, oxidised cathode, finite anode work function, residual gas). Reject "random error" or "human error" alone. - Part (c): alternative — students who compute λ = c/f = 578 nm first, then sin θ = λ/d = 0.346, then θ = 20.3°, earn M1+M2+M3 by equivalent route. - Part (d): do NOT award M2 unless the candidate explicitly links fixed frequency to unchanged V_s. Stating only "intensity doesn't affect V_s" without invoking the grating-fixed wavelength scores M2 only if M1 already establishes the wavelength selection mechanism.

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ strictly obeys the cognitive ladder: (a) AO1 State; (b)(i) AO2 Outline; (b)(ii) AO2/AO3 Calculate/Verify; (c) AO2/AO3 Determine; (d) AO3 Explain. Command terms are correctly ordered by ascending cognitive demand, with lower-order parts (State, Outline) at the start and higher-order analysis (Explain) at the end.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 Explain structure. The two bullets M1 and M2 do not form a single explicit 'X therefore Y' causal chain; instead they present two separate observations (photon model independence + wave-optics broadening) without a unified logical progression linking the student's incorrect prediction to why it fails. The scheme also lacks a clear warrant bridging the photon-energy independence to the physical consequence (unchanged V_s).

FIX: Restructure Part (d) mark scheme to exhibit a single causal chain: M1 should establish that stopping potential V_s depends only on photon frequency f (via eV_s = hf − φ, hence V_s ∝ f); M2 should then show that widening the slit increases intensity/photon flux but does not alter f of the selected line (the grating equation d sin θ = mλ is satisfied at fixed θ, so λ and f are unchanged); M3 should conclude therefore V_s remains constant. Alternatively, accept M1 as 'the photon model: each electron absorbs one photon of energy hf independent of intensity' and M2 as 'wave-optics consequence: widening the slit increases brightness and photon flux but the diffraction condition fixes the wavelength at θ, leaving f constant, therefore V_s (which depends on f alone) cannot increase.' This structure gives a transparent causal linkage: photon model predicts V_s ∝ f only → grating geometry fixes f → intensity change does not change f → V_s unchanged.
cross_theme pass

The ERQ genuinely integrates both E.2 (Quantum physics) and C.3 (Wave phenomena). E.2 dominates in parts (a)–(c), but part (b)(ii) requires students to apply the diffraction-grating equation (d sin θ = mλ, part of C.3 wave optics) to convert the experimental angle into frequency before using Einstein's photoelectric equation. Part (d) explicitly requires both concepts: students must invoke the photon model (E.2) to explain why V_s is independent of intensity, AND explain what the grating actually does (C.3) when the slit is widened—a true causal blend, not name-checking.
discriminator pass

Part (d) fully satisfies the model-critique dimension. It requires students to explain why a wave-optics prediction (raising intensity increases stopping potential) is incorrect by critiquing two key assumptions: (1) the photon model assumption that energy per electron depends only on frequency, not photon flux, and (2) the wave-optics assumption about what the grating equation actually constrains. This is authentic model critique, not mere calculation.
rule_of_one fail

Part (b)(ii) is marked as 4 marks total but is subdivided into two sub-questions: (i) 2 marks and (ii) 2 marks. The mark scheme lists 4 distinct award criteria (M1–M4), which correctly totals 4. However, the structural problem is that part (b)(ii) is presented as a single "Calculate [4 marks]" command with two nested sub-operations. The mark scheme allocates M1–M2 to sub-question (i) and M3–M4 to sub-question (ii), creating a 2+2 partition. This violates the dimension's requirement that a "Calculate [4 marks]" part must have exactly 4 award bullets representing 4 distinct sequential operations (formula / substitution / answer-with-units / verify or similar). Instead, the question structure implicitly splits into two separate 2-mark calculations, each of which should be marked as [2 marks] independently, not bundled under a single [4 marks] Calculate header. The divergence between the declared mark count (4) and the logical decomposition (two separate 2-mark sub-parts) by 2 marks exceeds the tolerance of ±1.

FIX: Restructure part (b)(ii) as two separate sub-parts: **(b)(ii)(α) Calculate [2 marks]** and **(b)(ii)(β) Calculate [2 marks]**, or relabel the combined part as **[4 marks]** with a clear outline that it comprises two 2-mark segments. Alternatively, merge the two sub-questions into a single coherent 4-mark calculation (e.g. "Use diffraction-grating data to verify the yellow line frequency, then use the graph gradient to determine h and compare to the accepted value"), ensuring M1–M4 form a single causal chain of 4 distinct steps.
topic pass

The ERQ genuinely integrates both E.2 Quantum physics (HL) and C.3 Wave phenomena. E.2 dominates via the photoelectric effect, Einstein's equation, photon model, and threshold concepts (parts a, b, c, d). C.3 is functionally required in part (b)(ii)(i), where the diffraction grating equation d sin θ = mλ must be applied to extract frequency from the experimentally measured first-order angle—this is not superficial. Part (d) explicitly demands synthesis: students must invoke the photon model (energy hf per photon is frequency-dependent) AND explain that widening the slit, while increasing intensity in wave-optics terms, does not change the wavelength selected by the grating equation, hence does not change stopping potential. Both topics are causally interwoven.

Iteration 1 — failed

ao_ladder pass

The ERQ strictly obeys the cognitive ladder. Part (a) is AO1 (State); parts (b)(i)–(c) progress through AO2 (Show that, Calculate, Derive) with appropriate command terms; part (d) is AO3 (Explain) requiring causal reasoning. The structure is hierarchical and internally consistent.
command_structure fail

Part (d) fails the §4.4.1 AO3 'Explain' structure requirement. While M2 includes 'therefore', the causal chain is incomplete: M1 states the physics principle (V_s depends on f, not intensity) but M2 does not explicitly complete the X→Y logic. M2 must state that 'intensity raises photon flux → more current, but photon energy/frequency unchanged → stopping potential unchanged' as a complete causal consequence. The current wording jumps to the conclusion without the explicit warrant linking increased intensity to unchanged V_s.

FIX: Revise Part (d) M2 to read: 'The grating equation d sin θ = mλ at fixed angle θ fixes the wavelength λ (and hence frequency f) of the yellow line regardless of slit width, **therefore** increasing intensity delivers more photons per second of the same energy hf—the stopping potential V_s = hf/e − φ/e depends only on f and remains unchanged, while only the photocurrent magnitude increases.' This makes explicit the causal link: fixed λ (from grating geometry) → fixed f → fixed V_s despite higher intensity.
cross_theme pass

The ERQ genuinely integrates E.2 (Quantum physics, photoelectric effect) as the primary theme with C.3 (Wave phenomena, diffraction grating) as a supporting secondary theme. Part (c) requires both the grating equation (d sin θ = mλ, C.3) AND the photon frequency relationship (f = c/λ, E.2) to derive f in terms of θ and d. Part (d) explicitly chains both concepts: the grating condition (C.3) fixes wavelength/frequency, while Einstein's equation (E.2) shows V_s depends only on f, not intensity. Neither topic is mere name-checking; both are essential to the solution logic.
discriminator pass

Part (d) explicitly asks the student to explain why a classmate's prediction (that increased light intensity will raise stopping potential) is wrong. This requires critique of a conceptual model—the student must recognize that stopping potential depends only on photon frequency (and work function), not intensity, and explain why the diffraction constraint fixes the frequency regardless of slit width. This is genuine model critique, not merely another calculation.
rule_of_one fail

Part (c) is marked as a 3-mark 'Derive' question but the mark scheme lists only 2 distinct mathematical operations: (M1) derivation of the frequency expression, and (M2) numerical substitution to find sin θ. M3 merely states the final angle value without introducing a new operation. This is a 2-operation task labelled as 3-mark, violating the dimensional contract by 1 mark.

FIX: Either: (1) restructure part (c) as a 2-mark question (Derive expression [2 marks] covering M1+M2, with θ calculation deferred); or (2) add a genuine third operation to part (c), such as requiring explicit derivation that f = c/(d sin θ) is constant for fixed θ and d, or asking students to verify the calculation against the frequency table. Alternatively, redistribute the 3 marks across part (c) as: M1 grating equation rearrangement; M2 substitution of d-value; M3 calculation of θ with units—making the task genuinely 3-step.
topic pass

The ERQ robustly integrates both topics. E.2 (Quantum physics) dominates through the photoelectric effect framework (parts a–d), while C.3 (Wave phenomena) is genuinely required in parts (c) and (d): part (c) explicitly derives frequency using the diffraction grating equation (d sin θ = mλ), and part (d) explains why slit width doesn't affect stopping potential by invoking the grating's fixed diffraction angle that locks wavelength and hence frequency. Neither topic is superficial; both are causally essential to the physical scenario.

Iteration 2 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (State); parts (b)(i), (b)(ii), and (c) are AO2/AO3 (Determine, Calculate, Derive); part (d) is AO3 (Explain). Command terms and cognitive levels progress appropriately from lower to higher order thinking.
command_structure fail

Part (d) violates §4.4.1 AO3 'Explain' structure. The mark scheme provides two separate bullets (M1 and M2) but does not constitute a single explicit causal-chain marking point with the required 'X therefore Y' structure showing how wavelength selection by the grating mechanism causally prevents stopping potential change while current increases. The two bullets are presented sequentially but lack integrated causal linkage within a single marking point.

FIX: Restructure Part (d) marks to present ONE unified marking point following the §4.4.1 'Explain' template: 'M1: [Causal chain] The grating equation d sin θ = mλ at fixed angle θ selects a fixed wavelength λ (hence fixed frequency f = c/λ and fixed photon energy hf), regardless of slit width. Therefore, widening the slit increases the number of photons arriving per second (higher flux) but each photon retains the same energy hf. By V_s = hf/e − φ/e, the stopping potential remains unchanged; however, the photocurrent increases in proportion to the photon arrival rate.' This single integrated point explicitly shows the causal mechanism: fixed geometry → fixed wavelength → fixed hf → fixed V_s; AND higher flux → higher current. Alternatively, split into two explicit chained points: M1 addressing the grating mechanism fixing f, and M2 explicitly stating 'Therefore, because hf is fixed, V_s is unchanged; but because flux increases, current increases.'
cross_theme pass

Both E.2 (quantum physics: photoelectric effect, Einstein equation, stopping potential, Planck constant, work function) and C.3 (wave phenomena: diffraction grating equation, wavelength selection, first-order diffraction angle) are genuinely integrated throughout the question. Part (c) and Part (d) explicitly require the grating equation to explain wavelength selection, which then determines photon frequency and thus stopping potential—creating a causal chain where the grating mechanism (C.3) directly constrains the photoelectric response (E.2). The secondary topic is not mere filler; it is functionally necessary to solve and explain the experimental design.
discriminator pass

Part (d) fully satisfies the model-critique dimension. It asks students to explain why stopping potential remains unchanged despite increased intensity, requiring them to critique the implicit assumption that stopping potential depends only on photon frequency (not intensity). The question explicitly invokes the grating's wavelength-selection mechanism and demands causal reasoning linking fixed frequency to fixed V_s, which is genuine model analysis rather than mere calculation.
rule_of_one fail

Part (b)(ii) is marked as 4 marks but lists only 4 distinct award criteria, which superficially matches. However, the dimension requires that a Calculate part have exactly 3 operations (formula/substitution/answer-with-units). Part (b)(ii) combines Calculate AND Comment, conflating two different cognitive demands. The 4 marks distribute as: M1–M2 cover the calculation (formula implicit, substitution, answer-with-units), then M3–M4 handle the comparison and commentary. This violates the strict mark-count-to-operation correspondence: a pure Calculate should not bundle a separate interpretive/explanatory M4 within the same part.

FIX: Split part (b)(ii) into two subparts: (b)(ii) Calculate [3 marks] covering φ, Δφ, and quoted result; then introduce (b)(iii) Comment [1 mark] for the consistency discussion and systematic-error reasoning. Alternatively, if the part must remain unified, explicitly designate it as 'Calculate and comment' and accept that M4 (comment/reasoning) extends beyond pure calculation operations—but then the scheme must acknowledge this is a hybrid-archetype exception. The cleanest fix is to separate them.
topic pass

The ERQ authentically integrates both topics. Part (c) requires the grating equation (C.3: wave phenomena, diffraction) to derive the diffraction angle for a selected spectral line. Part (d) leverages the grating equation as the causal mechanism explaining why stopping potential is independent of intensity: the grating fixes wavelength/frequency (hence photon energy and V_s via Einstein's equation, E.2), while intensity only affects photon flux. Neither topic is nominal; both are structurally essential to the physics chain.

Planner output (stage 1)

Archetype: experimental setup
Scenario: A photoelectric effect experiment uses ultraviolet light to eject electrons from a metal surface. The experimenter measures the stopping potential as a function of incident light frequency, and observes that electrons are ejected only above a threshold frequency. A diffraction grating is later placed in the light path to separate different wavelength components and investigate whether the threshold effect depends on light intensity or the intrinsic properties of individual photons.
Cross-pollination part: d
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State
b(i)	3	AO2	Define
b(ii)	4	AO2\|AO3	Calculate
c	3	AO2\|AO3	Derive
d	2	AO3	Explain

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