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).