Synthesis — The Wave Model of Light

Q1 — Sample Band 6 response (8 marks), annotated

Graph structure and interpretation: From eV_s = hf − φ, rearranging gives V_s = (h/e)f − φ/e. A graph of V_s (y-axis) vs f (x-axis) should be a straight line with gradient = h/e and y-intercept = −φ/e [1]. The x-intercept gives the threshold frequency f₀, from which φ = hf₀.

Work function: The first row gives f₀ = 5.5 × 10¹⁴ Hz (V_s = 0.00 V — just at threshold). φ = hf₀ = (4.14 × 10⁻¹⁵)(5.5 × 10¹⁴) = 2.28 eV. This agrees with the accepted value for sodium (2.28 eV) [1].

Estimating h: Using (6.2, 0.29) and (10.3, 1.99): gradient = (1.99 − 0.29) / [(10.3 − 6.2) × 10¹⁴] = 1.70 / (4.1 × 10¹⁴) = 4.15 × 10⁻¹⁵ V·s [1 mark for method]. Since gradient = h/e, h = gradient × e = 4.15 × 10⁻¹⁵ × 1.60 × 10⁻¹⁹ / 1.60 × 10⁻¹⁹ = 4.15 × 10⁻¹⁵ eV·s [1 mark for value — accept 4.10–4.20]. Comparison: accepted h = 4.14 × 10⁻¹⁵ eV·s; excellent agreement [1].

Evaluation of photon model: Millikan’s data support Einstein’s photon model in three ways: (1) the linear V_s–f relationship with a sharp threshold matches K_max = hf − φ exactly; (2) no electrons are detected at f = 5.5 × 10¹⁴ Hz regardless of intensity; (3) the gradient gives a value of h consistent with Planck’s constant [1]. Specific contradiction of wave model: the wave model predicts that V_s should increase with intensity (bigger waves → more energy → faster electrons). Millikan found that at fixed frequency, changing intensity changes only the photocurrent magnitude, not the stopping voltage — this is impossible if energy is delivered continuously by a wave [1].

Limitation and improvement: A limitation is surface contamination of the sodium: sodium oxidises rapidly in air, changing the work function and making threshold measurements unreliable [1]. Improvement: conduct the experiment in an ultra-high vacuum chamber to prevent oxidation; repeat measurements at each frequency five times and average to reduce random error [1].

Marking criteria summary (8 marks): 1 = correctly states graph shape (linear) and identifies gradient = h/e and x-intercept = f₀; 1 = correct work function calculation (2.28 eV) from threshold row; 1 = correct method for gradient using two data points; 1 = correct value of h (4.10–4.20 × 10⁻¹⁵ eV·s) with comparison to accepted value; 1 = photon model supported with two or more specific data-linked reasons; 1 = names a specific result that contradicts the wave model (stopping voltage independent of intensity); 1 = one valid limitation; 1 = one valid improvement.

Q2 — Sample Band 6 response (7 marks), annotated

Hypothesis: If the photon model is correct, then the photoelectric current will begin within the instrumental time resolution (effectively instantaneously, <1 μs) after above-threshold light is turned on. If the wave model is correct, there will be a measurable time lag (potentially seconds to minutes at low intensity) before the first electron is detected. Independent variable: time at which light is switched on. Dependent variable: time delay (lag) before photocurrent is detected. [1 — testable hypothesis with IV and DV]

Procedure: (1) Set up the photocell in a dark enclosure. Connect the ammeter in series and the oscilloscope to measure photocurrent vs time with 1 μs resolution. (2) Select a frequency clearly above the threshold for sodium (f > 5.5 × 10¹⁴ Hz; use UV at λ = 300 nm, f = 10¹⁵ Hz) and reduce the intensity to the minimum detectable by the ammeter. (3) Use a shutter mechanism to switch the light on at a precisely known time (t = 0) recorded by the oscilloscope’s trigger channel. (4) Record the oscilloscope trace showing photocurrent vs time and measure any time lag between t = 0 and the first current spike. Repeat 10 times and average [1 — four clear steps including lag-detection method]. (5) Repeat at reduced intensity (1% of original) to test whether a larger lag appears at lower intensity.

Falsification: If no current is detected (even at very low intensity, above threshold frequency) for several minutes, this would suggest the wave model is correct — the photon model would be falsified. Alternatively, if the time lag increased systematically as intensity decreased, this would support the wave model’s energy-accumulation prediction [1].

Limitations: (1) The oscilloscope and ammeter have minimum detection limits; extremely low-intensity currents may be below the noise floor, making it impossible to determine whether any lag shorter than the noise timescale actually occurred [1]. (2) Sodium is highly reactive and the surface must be kept clean; oxidation can shift the threshold and create a leakage current that mimics a photoelectric signal even before light is turned on [1].

Improvement: Use a photomultiplier tube (PMT) capable of detecting single electrons to push the sensitivity limit to single-photon events; this allows the test to be conducted at the absolute minimum intensity (single photon at a time) [1].

Expected result (confirms photon model): The oscilloscope will show current onset within the noise floor of the instrument (effectively instantaneous) at all intensities above threshold. No time lag proportional to 1/intensity will be observed, directly falsifying the wave model [1].

Marking criteria summary (7 marks): 1 = testable hypothesis naming IV and DV; 1 = four steps including precise lag-detection method (oscilloscope trigger); 1 = states what result would falsify the wave model (measurable lag increasing with lower intensity); 1 = one valid limitation; 1 = second valid limitation; 1 = one specific improvement (PMT or ultra-high vacuum); 1 = precise physics terminology throughout (threshold frequency, photon, work function, intensity, photocurrent, photon model vs wave model).