Modern Physics

Modern Physics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Modern Physics covers atomic-scale phenomena and high-velocity regimes that Newtonian mechanics cannot explain. The photoelectric effect establishes the photon model: light delivers energy in quanta of E = hf, and electrons escape a metal only when frequency exceeds the threshold frequency f₀, satisfying KE_max = hf − W = h(f − f₀), where W = hf₀ is the work function. Wave–particle duality extends to matter via the de Broglie wavelength λ = h/mv. The Bohr model quantizes electron orbits with angular momentum mvr = nh/2π, giving radii rₙ = 0.529 n² Å and energies Eₙ = −13.6/n² eV. X-rays from a Coolidge tube obey the Duane–Hunt limit λ_min = hc/eV. Radioactive decay follows N = N₀e^(−λt) with half-life T₁/₂ = 0.693/λ; α-emission drops A by 4, Z by 2; β-emission raises Z by 1. Mass–energy equivalence E = Δmc² governs fission and fusion energy release.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Photoelectric Effect

Einstein explained Hertz’s observation by treating light as photons. A surface emits electrons only when f > f₀; below f₀, no electron escapes regardless of intensity. Maximum kinetic energy: KE_max = hf − W, where W is the work function characteristic of the metal. Stopping potential V_s depends on frequency alone, while saturation current scales with intensity. The slope of a KE_max vs f graph equals Planck’s constant h ≈ 6.626 × 10⁻³⁴ J·s, while its x-intercept gives f₀.

Wave–Particle Duality

de Broglie (1924) assigned wavelength λ = h/p = h/mv to any particle with momentum. Electrons accelerated through 150 V have λ ≈ 0.1 nm, suitable for crystal diffraction. Davisson–Germer confirmed this experimentally. Light, in turn, shows particle behaviour via the photoelectric and Compton effects (Δλ = h/(mₑc)(1 − cos θ)).

Bohr Model of Hydrogen

Electrons occupy stationary orbits of quantized angular momentum L = mvr = nh/2π. Allowed radii scale as rₙ = n²a₀ (a₀ = 0.529 Å), and energies as Eₙ = −13.6/n² eV. A photon of energy hf = E_i − E_f is emitted on transition. The model successfully predicts hydrogen and hydrogen-like ion spectra but fails for multi-electron atoms.

X-Rays

Produced when high-speed electrons strike a tungsten target. Bremsstrahlung yields a continuous spectrum with a sharp cutoff at λ_min = hc/eV. Characteristic K-series lines appear when inner-shell vacancies are filled. MDCAT rarely demands numerical X-ray problems but frequently tests the inverse relationship between λ_min and accelerating voltage.

Radioactivity

Unstable nuclei decay spontaneously. α-decay (⁴₂He nucleus): A → A−4, Z → Z−2. β⁻-decay (electron + antineutrino): A unchanged, Z → Z+1. γ-decay: neither A nor Z changes, only nuclear energy drops. Activity A = λN follows first-order kinetics with T₁/₂ = 0.693/λ. Average life τ = 1/λ.

Nuclear Reactions

Fission splits heavy nuclei (²³⁵U, ²³⁹Pu) into fragments plus neutrons, releasing ~200 MeV per event. Fusion combines light nuclei (D + T → ⁴He + n), releasing 17.6 MeV, but requires ~10⁸ K plasmas. Both obey E = Δmc².

Lasers

Three requirements: population inversion, metastable state, and resonant cavity. Output is coherent, monochromatic, and collimated — distinguishing laser light from ordinary sources.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Worked Example

Light of wavelength 400 nm falls on a metal with W = 2.0 eV. Photon energy E = hc/λ = (1240 eV·nm)/400 nm = 3.10 eV. Maximum KE = 3.10 − 2.0 = 1.10 eV. Stopping potential V_s = 1.10 V. Threshold wavelength λ₀ = hc/W = 1240/2.0 = 620 nm — below 620 nm the surface emits electrons, above it the photoelectric current is zero. This problem appears in roughly 1 of every 6 MDCAT Physics MCQs from this unit.

Edge Cases & Common Mistakes

• Current vs. energy confusion: Increasing intensity raises the number of emitted electrons (higher current) but does not raise their maximum kinetic energy. Examiners pair “double the intensity” with the wrong answer choice.
• Bohr model limits: It violates the Heisenberg uncertainty principle because it assigns definite r and v simultaneously. Use it only for hydrogen and H-like ions (He⁺, Li²⁺), where the nucleus is a point charge and screening is absent.
• Decay constant units: λ has units s⁻¹, not seconds. Mixing up λ with T₁/₂ is a frequent error; remember T₁/₂ = 0.693/λ.
• α vs. β vs. γ penetrance: α stopped by paper, β by ~5 mm aluminium, γ needs thick lead — a common matching-column item.
• Compton shift sign: Δλ is positive (wavelength increases) after scattering, an asymmetry Compton’s classical wave theory could not explain.
• Stopping potential independence from intensity is the single most-tested fact in MDCAT’s photoelectric block.

Connections

Modern Physics unifies with Waves & Optics through interference patterns of electrons, with Electromagnetism via the photoelectric work-function concept, and with Kinetic Theory through mass-defect energy release in fission reactors and stellar cores.

Practice Prompts

1. A radioactive sample has N₀ = 10⁶ nuclei and λ = 0.02 s⁻¹. Find the activity at t = 50 s and the time for 90% decay.
2. An electron is accelerated from rest through 100 V. Calculate its de Broglie wavelength and the smallest orbit it could occupy if captured by a proton.

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