Short Answer, Model Answers
Q1 (a): Interference (Young's double slit) demonstrates superposition, constructive and destructive combination requires waves. Diffraction demonstrates bending around obstacles, particles travel in straight lines and cannot diffract. Polarisation proves light is a transverse wave, only transverse waves can be restricted to one plane of oscillation. Refraction (Foucault's experiment) shows light slows in denser media, consistent with the wave model.
Q1 (b): Newton's model predicted that light would travel faster in water (surface attraction accelerates corpuscles). In 1850, Foucault directly measured the speed of light in water and found it slower than in air by a factor of $n$, directly contradicting Newton and confirming the wave model.
Q1 (c): Huygens' principle: every point on a wavefront acts as a source of secondary spherical wavelets; the new wavefront is the tangent (envelope) to all these wavelets. At a narrow aperture, wavelets at the edge of the aperture spread out into the geometric shadow, there is no edge to constrain them. This spreading is diffraction. Particles would simply travel through the aperture in straight lines with no spreading.
Q2 (a): Wave model contradictions: (i) Classical waves have continuous energy that builds up at the surface, all frequencies should eventually eject electrons. The existence of a threshold frequency below which no ejection ever occurs contradicts this. (ii) Wave intensity determines amplitude and hence energy density, $K_{\max}$ should increase with intensity. Experiment shows $K_{\max}$ depends only on frequency. (iii) Classical waves take time to build up enough energy, emission should be delayed. Experiment shows it is instantaneous. Einstein's resolution: each photon has energy $E = hf$. One photon gives all its energy to one electron. For ejection, $hf$ must exceed the work function $\phi$. Excess energy becomes kinetic energy: $K_{\max} = hf - \phi$. Intensity determines the number of photons, not their energy.
Q2 (b)(i): $f_0 = \phi/h = 2.10/(4.14\times10^{-15}) = 5.07\times10^{14}$ Hz.
Q2 (b)(ii): $\lambda_0 = c/f_0 = (3.00\times10^8)/(5.07\times10^{14}) = 5.92\times10^{-7}$ m $= 592$ nm.
Q2 (b)(iii): $E = hc/\lambda = (4.14\times10^{-15})(3.00\times10^8)/(380\times10^{-9}) = 3.27$ eV. $K_{\max} = 3.27 - 2.10 = 1.17$ eV $= 1.17 \times 1.6\times10^{-19} = 1.87\times10^{-19}$ J.
Q2 (c): Tripling intensity means three times as many photons arrive per second → three times as many electrons ejected per second → photocurrent triples. However, each photon has the same energy ($hf$), so $K_{\max} = hf - \phi$ is unchanged. This is a distinctive prediction of the photon model: intensity affects number but not energy of ejected electrons.
Q3 (a): The Compton effect: X-rays directed at a target are scattered in various directions. The scattered X-rays have a longer wavelength than the incident X-rays (the wavelength increase depends on scattering angle). Classical wave theory cannot explain this: when a wave scatters off an object, its frequency should not change, the object oscillates and re-radiates at the same frequency. The observed wavelength shift is impossible within the wave model.
Q3 (b): The photon model treats X-ray photons as particles with momentum $p = h/\lambda$ and energy $E = hf$. When a photon collides with a (nearly) free electron, momentum and energy are conserved, just like a billiard-ball collision. The photon transfers some momentum and energy to the electron, so the scattered photon has lower energy (lower $f$, longer $\lambda$). The wavelength shift calculated from relativistic collision equations matches experiment exactly.
Q3 (c): The student's claim is too narrow. Young's double-slit experiment does demonstrate wave behaviour, interference fringes cannot be produced by classical particles. However, this is not the full story. The photoelectric effect and the Compton effect both demonstrate particle-like behaviour: quantised energy transfer, particle collisions with momentum conservation. Neither the pure wave model nor the pure particle model can explain all observations. The modern synthesis, wave-particle duality, holds that light exhibits whichever aspect the experimental setup probes. Both aspects are equally real and fundamental. A complete description of light requires quantum electrodynamics.
Q4 (a): Rømer timed the eclipses of Jupiter's moon Io. As Earth moved farther from Jupiter in its orbit, eclipses were observed later than predicted; as Earth moved closer, they were observed earlier. This happens because light takes longer to cross the larger Earth-Jupiter distance, so the timing shift directly reveals a finite, measurable speed of light rather than an error in Io's orbital period.
Q4 (b): Fizeau sent light through a gap in a rapidly rotating toothed wheel to a mirror 8 km away and back through the wheel; at certain rotation rates a tooth blocked the returning beam (extinction), and the known distance, tooth count and rotation rate at extinction gave $c$. Foucault replaced the wheel with a rotating mirror: the light travels to a fixed distant mirror and back while the rotating mirror turns through a small, precisely measurable angle, deflecting the returned beam; this angle, the rotation rate and the distance give $c$ directly.
Q4 (c): $c = \text{distance}/\text{time} = 3.0\times10^{11} / 1000 = 3.0\times10^8$ m/s, matching the accepted value of $c$ to two significant figures.
Q4 (d): In Rømer's era, physicists used a known distance (Earth's orbital diameter) and a measured time delay to calculate an unknown speed of light. Since 1983 the metre is defined as the distance light travels in $1/299\,792\,458$ s, fixing $c$ by definition. This reverses the relationship: distance is now calculated from a measured travel time multiplied by the fixed value of $c$ (e.g. lunar laser ranging, GPS), rather than speed being calculated from distance and time.
Q5 (a): By Wien's Law, $\lambda_{\max} = b/T$, so shorter peak wavelength means higher temperature. Ranking hottest to coolest: Star P (290 nm, hottest), Star Q (500 nm), Star R (950 nm, coolest).
Q5 (b): Star P's curve does not just peak further left, it lies above Star R's curve at essentially every wavelength (hotter objects radiate more strongly across the whole spectrum, not only near their own peak), while both curves still rise from near zero, reach a single peak, and fall back toward zero at long wavelength.
Q5 (c): Classical (Rayleigh-Jeans) theory predicts that intensity keeps increasing without limit as wavelength decreases toward the ultraviolet, an unphysical divergence called the ultraviolet catastrophe. None of the three real curves does this, each rises, peaks, and falls back toward zero at short wavelength, because at short wavelengths (high frequencies) the energy of a single quantum becomes too large for thermal agitation to supply, so emission at those wavelengths is strongly suppressed rather than unbounded.
Q5 (d): Planck assumed that oscillators exchange energy with radiation only in discrete quanta, $E = nhf$, rather than continuously. This quantisation makes high-frequency (short-wavelength) emission exponentially improbable, correctly reproducing the observed peak-and-fall shape of the black-body curve.
