Stefan's Law — Stefan-Boltzmann Law of Radiation

Stefan's Law Statement: The total energy radiated per unit surface area of a black body per unit time (power radiated per unit area) is directly proportional to the fourth power of the black body's absolute temperature.

Mathematical Expression: P = σAT⁴

Alternatively, power per unit area (intensity): E = σT⁴

Where: • P = total power radiated (W = Watts) • E = energy radiated per unit area per second (W/m²) • σ = Stefan-Boltzmann constant = 5.67 × 10⁻⁸ W·m⁻²·K⁻⁴ • A = surface area of the black body (m²) • T = absolute temperature of the body (K, Kelvin)

Proportionality: E ∝ T⁴

If temperature doubles (T → 2T): E → σ(2T)⁴ = 16σT⁴ Power increases 16 times when temperature doubles!

Historical note: Stefan experimentally established this law in 1879. Boltzmann derived it theoretically from thermodynamic principles in 1884. This is why it's called the Stefan-Boltzmann Law.

Value of Stefan-Boltzmann Constant: σ = 5.67 × 10⁻⁸ W·m⁻²·K⁻⁴ = 5.67 × 10⁻⁸ J·s⁻¹·m⁻²·K⁻⁴

Dimensional Formula of σ: From P = σAT⁴: σ = P / (A × T⁴) [σ] = [W] / ([m²][K⁴]) = [ML²T⁻³] / [L²K⁴] = [MT⁻³K⁻⁴]

Relation to fundamental constants: σ = (2π⁵k_B⁴) / (15h³c²) Where: • k_B = Boltzmann constant = 1.38 × 10⁻²³ J/K • h = Planck's constant = 6.626 × 10⁻³⁴ J·s • c = speed of light = 3 × 10⁸ m/s

This derivation comes from integrating Planck's blackbody radiation spectrum over all frequencies.

Black Body: A perfect black body absorbs all incident radiation (absorptivity = 1) and emits the maximum possible radiation at a given temperature.

For real bodies (non-black): P = εσAT⁴

Where ε = emissivity (0 < ε ≤ 1) • ε = 1: perfect black body • ε < 1: real (grey) body • Example: Human skin ε ≈ 0.98; polished silver ε ≈ 0.02

Net radiation for a body at temperature T in surroundings at T₀: P_net = εσA(T⁴ - T₀⁴)

Example: A body at 600 K in 300 K surroundings: P_net = σA(600⁴ - 300⁴) = σA(1.296×10¹¹ - 8.1×10⁹) = σA × 1.215 × 10¹¹

Newton's Law of Cooling is an approximation of Stefan's Law valid when (T - T₀) is small.

Example 1: Power radiated by the Sun Sun's surface temperature: T = 5778 K Sun's radius: R = 6.96 × 10⁸ m Surface area: A = 4πR² = 4π × (6.96×10⁸)² = 6.09 × 10¹⁸ m² P = σAT⁴ = 5.67×10⁻⁸ × 6.09×10¹⁸ × (5778)⁴ = 5.67×10⁻⁸ × 6.09×10¹⁸ × 1.114×10¹⁵ ≈ 3.85 × 10²⁶ W (This matches the known solar luminosity ✓)

Example 2: Temperature of a black body If energy radiated = 5.67 W/m²: E = σT⁴ 5.67 = 5.67×10⁻⁸ × T⁴ T⁴ = 10⁸ T = (10⁸)^(1/4) = 10² = 100 K

Example 3: Ratio of energies at two temperatures T₁ = 1000 K, T₂ = 2000 K E₁/E₂ = T₁⁴/T₂⁴ = (1000/2000)⁴ = (1/2)⁴ = 1/16 So doubling temperature increases radiation 16 times.

1. Astrophysics and Stellar Physics: • Calculating the luminosity (total power) of stars • Estimating surface temperatures of stars from their luminosity • Solar constant ≈ 1361 W/m² (solar radiation at Earth's distance)

2. Climate Science and Global Warming: • Earth's energy balance: solar input vs. Earth's radiation out • Greenhouse effect analysis • Stefan's law explains why Earth's temperature is self-regulating

3. Industrial Furnaces and High-Temperature Measurement: • Pyrometers use Stefan's law to measure furnace temperatures without contact • Optical pyrometers measure temperatures above 700°C

4. Incandescent Lamps: • Tungsten filament at ~2700 K radiates visible light • Efficiency calculated using Stefan's law

5. Infrared (Thermal) Cameras: • Detect heat radiation from objects • Medical thermography — detecting tumours (slightly higher temperature) • Night vision devices

6. Cryogenics: • Radiation heat loss from liquid nitrogen/helium containers • Vacuum flask (Dewar) design minimises radiated heat exchange