10.3 Modern Physics

The Photoelectric Effect

The photoelectric effect is the emission of electrons from a metal surface when light of sufficiently high frequency falls on it. It was discovered by Heinrich Hertz in 1887 during his experiments with spark-gap transmitters, and explained by Albert Einstein in 1905 using the idea that light delivers energy in discrete packets called photons, each carrying energy E = hf (h = Planck's constant ≈ 6.63 × 10⁻³⁴ J·s).

Einstein's photoelectric equation is:

KE_max = hf − Φ

where Φ (the work function) is the minimum energy needed to eject an electron from the metal, and f is the frequency of the incident light. The threshold frequency is:

f₀ = Φ / h

Below f₀, no electrons are emitted no matter how intense the light — a result classical wave theory could not explain and which established the quantum (particle) nature of light.

Worked example 1. A metal has a work function of 2.0 eV. Find the threshold frequency. (1 eV = 1.6 × 10⁻¹⁹ J, h = 6.63 × 10⁻³⁴ J·s.)

Φ = 2.0 × 1.6 × 10⁻¹⁹ = 3.2 × 10⁻¹⁹ J. f₀ = Φ / h = 3.2 × 10⁻¹⁹ / 6.63 × 10⁻³⁴ ≈ 4.8 × 10¹⁴ Hz (red/orange light).

Dual Nature of Light

Light exhibits wave–particle duality: it propagates as a wave (interference, diffraction, polarization) but interacts with matter in discrete quanta (photons). The photoelectric effect and the Compton effect demonstrate the particle side; Young's double-slit experiment demonstrates the wave side. de Broglie extended duality to matter: λ = h/p for any particle with momentum p.

Atomic Models

The evolution of atomic theory is a common PAF topic:

1. Dalton (1808) — solid, indivisible sphere.
2. J.J. Thomson (1897) — discovered the electron; 'plum-pudding' model with electrons embedded in a positive sphere.
3. Rutherford (1911) — gold-foil experiment; dense, positively charged nucleus with electrons orbiting at a distance.
4. Bohr (1913) — electrons in quantized circular orbits; photons emitted/absorbed on transitions: ΔE = hf.
5. Quantum mechanical model — electrons described by probability clouds (orbitals), not fixed paths.

Bohr's model successfully explained the hydrogen spectrum and introduced the idea of quantized energy levels — a concept that underpins lasers, LEDs, and many cockpit display technologies.

Nuclear Physics: Fission, Fusion, and E = mc²

Einstein's mass–energy equivalence:

E = mc²

states that mass and energy are interchangeable. In any nuclear reaction, the mass defect Δm is released as energy:

• Nuclear fission — a heavy nucleus (e.g., U-235, Pu-239) splits into smaller nuclei when struck by a neutron, releasing ~200 MeV per fission and additional neutrons that sustain a chain reaction. Basis of nuclear power plants and atomic bombs.
• Nuclear fusion — light nuclei (e.g., hydrogen isotopes deuterium + tritium) combine into a heavier nucleus (helium), releasing even more energy per unit mass. Powers the Sun and hydrogen bombs; the goal of controlled fusion research.

Worked example 2. A reaction has a mass defect of 1.0 × 10⁻²⁸ kg. How much energy is released? (c = 3.0 × 10⁸ m/s.)

E = mc² = 1.0 × 10⁻²⁸ × (3.0 × 10⁸)² = 1.0 × 10⁻²⁸ × 9.0 × 10¹⁶ = 9.0 × 10⁻¹² J.

Photoelectric KE_max — Worked Example

Worked example 3. Light of frequency 8.0 × 10¹⁴ Hz falls on a metal whose work function is 2.0 eV. Find the maximum kinetic energy of the emitted electrons (in eV). (h = 6.63 × 10⁻³⁴ J·s, 1 eV = 1.6 × 10⁻¹⁹ J.)

Photon energy E = hf = 6.63 × 10⁻³⁴ × 8.0 × 10¹⁴ = 5.30 × 10⁻¹⁹ J = 5.30 × 10⁻¹⁹ / 1.6 × 10⁻¹⁹ ≈ 3.31 eV. Work function Φ = 2.0 eV. So KE_max = 3.31 − 2.0 = 1.31 eV. The surplus (1.31 eV) appears as the electron's kinetic energy; the threshold frequency for this metal is f₀ = Φ/h ≈ 4.8 × 10¹⁴ Hz, and since 8.0 × 10¹⁴ Hz > f₀, emission occurs.

de Broglie Wavelength — Worked Example

Worked example 4. Find the de Broglie wavelength of an electron moving at 1.0 × 10⁷ m/s. (m_e = 9.11 × 10⁻³¹ kg, h = 6.63 × 10⁻³⁴ J·s.)

p = m v = 9.11 × 10⁻³¹ × 1.0 × 10⁷ = 9.11 × 10⁻²⁴ kg·m/s. λ = h/p = 6.63 × 10⁻³⁴ / 9.11 × 10⁻²⁴ ≈ 7.28 × 10⁻¹¹ m (about 0.073 nm, comparable to atomic spacing — which is why electron microscopes resolve atoms).

Threshold Frequency vs Intensity — The Key Trap

The single most-tested modern-physics trap: below threshold frequency, no emission occurs, no matter how large the intensity. Classical wave theory predicted that increasing intensity should always eventually eject electrons; experiment showed it does not. Only increasing the frequency (or shortening the wavelength) beyond f₀ helps. Once above threshold, increasing intensity ejects more electrons per second (larger photocurrent) but does not change their maximum kinetic energy — only increasing frequency raises KE_max.

Fission vs Fusion — Quick Comparison

Both release energy because the products have higher binding energy per nucleon — the mass defect Δm is converted via E = mc².

Key Discoveries Timeline

Aviation & Defence Links

• Photoelectric sensors and lidar — used in terrain-following and obstacle-avoidance systems.
• Radar relies on photon (electromagnetic wave) reflection — the same wave–particle duality at radio frequencies.
• Nuclear propulsion and weapons — Pakistan's deterrent relies on fission and boosted-fission designs; understanding E = mc² is therefore operationally relevant for PAF officers.