7.5 Thermal Radiation, Stefan-Boltzmann, and Wien's Law

Treat radiation as electromagnetic thermal transfer, then quantify total power and spectral peak shifts with absolute-temperature laws.

Estimated time: 27 minutes

Radiation Without a Medium

Radiation differs from conduction and convection because no material medium is required. Any object above absolute zero emits electromagnetic radiation due to charged-particle motion in matter. This is why energy from the Sun can cross interplanetary vacuum and still heat Earth.

Emission and absorption occur simultaneously. Net radiative transfer depends on both the object and surroundings. A warm object in colder surroundings has net outward transfer; a cooler object near a hotter source can have net inward transfer even while still emitting radiation itself.

Stefan-Boltzmann Law and Net Exchange

$P = \epsilon\sigma A T^4,\quad P_{net}=\epsilon\sigma A\left(T_{obj}^4-T_{env}^4\right)$

Total radiative power rises with fourth power of absolute temperature, and net flow depends on both object and environment temperatures.

The fourth-power dependence is the key operational feature. Small increases in high Kelvin temperatures can produce large power changes. This is why radiative cooling and heating dominate in many high-temperature systems and why thermal-control design in space engineering focuses heavily on emissivity, area, and view factors.

A perfect black body has emissivity $\epsilon = 1$ and acts as an ideal absorber and emitter benchmark. Real surfaces have $\epsilon < 1$ , often wavelength-dependent. In introductory modeling, treating $\epsilon$ as an effective constant already captures most first-order design behavior.

Wien's Law and Spectral Peak

$\lambda_{max}T = 2.90\times10^{-3}\,\text{m K}$

Higher temperature shifts peak emission toward shorter wavelengths.

Wien's law links observed spectral peak to surface temperature. Cooler bodies peak in infrared, while hotter bodies peak in visible or ultraviolet ranges. This provides a practical temperature-estimation method in astrophysics and remote sensing when direct contact measurements are impossible.

Stellar and Planetary Applications

For stars approximated as radiating spheres, luminosity combines Stefan-Boltzmann with geometric area. Comparing luminosity and surface temperature lets you infer relative radius scales. This is a core bridge between thermal radiation laws and observational astronomy in IB contexts.

Planetary temperature modeling adds absorbed solar flux, albedo, and emitted infrared balance. Even simplified equilibrium models expose why greenhouse effects and emissivity differences materially alter surface conditions compared with naive distance-only expectations.

Simulation: Black-Body Spectra and Net Radiation

Adjust emissivity, object temperature, and surroundings temperature to compare $T^4$ power scaling with Wien peak shifts.

Heat Transfer + Radiation Lab

Emissivity epsilon: 0.94Emitter temperature (K): 580.0Surroundings temperature (K): 295.0Radiating area (m^2): 1.60

Black-body style spectra

Emitter power

9650.3 W

Absorbed background

645.8 W

Net radiative transfer

9004.5 W

Wien peak (emitter)

5.000 um

Move between conduction and radiation views to compare Fourier temperature-gradient transport with fourth-power thermal radiation behavior.

Note

Always convert to Kelvin before applying Stefan-Boltzmann or Wien laws. Using deg C directly breaks the physical scaling.

Test Yourself

Estimate the wavelength of peak emission for a star with surface temperature $5800\,\text{K}$ using Wien's law. Enter your answer in micrometers ( $\mu\text{m}$ ).

Hint: Compute $\lambda_{\max} = 2.90\times10^{-3}/T$ , then convert meters to micrometers.

Your numerical answer (um)

7.4 Conduction and Convection

Chapter 7 Wrap-Up