13.1 Modern & Nuclear Physics

13.1 Modern & Nuclear Physics

Quick Answer: Modern physics on this paper is FSc-level: photons carry E = hf; the photoelectric equation is hf = φ + K_max; Bohr levels scale as −1/n²; radioactive decay is exponential with T½ = ln 2 / λ; fission and fusion release energy from mass defect. Learn equations, units, and typical MCQ traps—not reactor engineering detail.

The PAF Aeronautical Engineering Initial Test emphasises FSc Physics. Modern and nuclear topics often appear as short formula MCQs: photoelectric stopping potential, Bohr wavelength, half-life remaining fraction, or which radiation is most penetrating. Treat this section as a compact toolkit for those items.

Photons and the Photoelectric Effect

A photon has energy

$E = hf = \frac{hc}{\lambda}$

where h ≈ 6.63 × 10⁻³⁴ J·s (Planck’s constant), f is frequency, and λ is wavelength. In electron-volts, E (eV) ≈ 1240 / λ(nm) is a useful FSc approximation.

Photoelectric effect: light ejects electrons from a metal surface only if f exceeds a threshold frequency f₀. Einstein’s equation:

$hf = \phi + K_{\max}$

• Work function φ = hf₀ is the minimum energy to free an electron from the metal.
• Maximum kinetic energy of photoelectrons: K_max = hf − φ.
• Stopping potential V₀: eV₀ = K_max, so V₀ = (hf − φ)/e.

Classic trap: intensity does not raise K_max. Only frequency (or wavelength) does, once emission occurs. Intensity controls how many electrons leave per second.

Worked example — photoelectric. A metal has φ = 2.0 eV. Light of wavelength 400 nm strikes it. Photon energy ≈ 1240/400 = 3.1 eV. Then K_max = 3.1 − 2.0 = 1.1 eV, and V₀ = 1.1 V. If intensity doubles at the same λ, V₀ stays 1.1 V; current roughly doubles.

Atomic Models (Exam Focus)

Bohr hydrogen energies:

$E_n = -\frac{13.6\,\text{eV}}{n^2}$

Ground state (n = 1): −13.6 eV. First excited state (n = 2): −3.4 eV. Ionisation from ground state needs 13.6 eV.

A transition from nᵢ to n_f emits (or absorbs) a photon with

$|\Delta E| = hf = 13.6\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\text{ eV}$

(for emission, nᵢ > n_f). Lyman series ends at n = 1 (UV); Balmer at n = 2 (visible).

Worked example — Bohr. Photon for n = 3 → n = 2: |ΔE| = 13.6(1/4 − 1/9) = 13.6(5/36) ≈ 1.89 eV. Wavelength ≈ 1240/1.89 ≈ 656 nm (Balmer H-α)—a standard recognition item.

Radioactivity and Half-Life

Unstable nuclei decay spontaneously. Common emissions:

Decay law:

$N = N_0 e^{-\lambda t}, \quad A = \lambda N = A_0 e^{-\lambda t}$

Half-life T½ is the time for N (or A) to fall to half:

$T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$

After n half-lives, remaining fraction = (1/2)^n. After 3 half-lives, 1/8 remains; after 4, 1/16.

Worked example — half-life. A sample’s activity falls from 800 Bq to 100 Bq. Ratio = 1/8 = (1/2)³, so three half-lives elapsed. If T½ = 2 h, elapsed time = 6 h.

Mean life τ = 1/λ = T½ / ln 2 ≈ 1.44 T½—occasionally asked as a definition MCQ.

Mass Defect, Binding Energy, Fission and Fusion

Nuclear mass is less than the sum of free nucleon masses. Mass defect Δm relates to binding energy:

$BE = \Delta m\, c^2$

Higher binding energy per nucleon means a more stable nucleus. The curve peaks near iron/nickel; that is why:

• Fission: heavy nuclei (e.g. ²³⁵U) split into medium-mass fragments, releasing energy.
• Fusion: light nuclei (e.g. H isotopes) combine toward mid-mass, releasing even more energy per nucleon in stellar and experimental contexts.

FSc papers usually ask qualitative comparisons (which process releases energy, what α/β/γ are) plus E = mc² scale sense—not reactor kinetics. For aeronautical interest, remember that materials, propulsion, and sensors may involve radiation environments later in a career, but the initial test stays at textbook nuclear basics.

Exam Pattern Tips for This Topic

1. Convert eV ↔ joules carefully when asked (1 eV = 1.6 × 10⁻¹⁹ J).
2. Photoelectric: threshold + stopping potential problems dominate; intensity traps are common.
3. Half-life: prefer fraction (1/2)^n over messy continuous exponential unless λ is given.
4. Distinguish ionising power from penetrating power for α, β, γ.
5. Do not invent PAF cut-offs from nuclear topics—these are academic physics marks within the published/advertised pattern; confirm cycle details on joinpaf.gov.pk.

Key Takeaways

• Photoelectric: frequency sets K_max; intensity sets emission rate.
• Bohr: E_n = −13.6 eV / n²; spectral series from ΔE = hf.
• Decay: exponential; T½ = 0.693/λ; remaining = (1/2)^n after n half-lives.
• α / β / γ differ in charge, mass, ionisation, and penetration.
• Fission and fusion release nuclear binding energy via mass defect.