Stefan-Boltzmann & Wien's Law Reference Cheat Sheet

Key Facts

• The Stefan-Boltzmann law for total emitted power is $P = e\sigma AT^4$, where $P$ is power, $e$ is emissivity, $A$ is surface area, and $T$ is absolute temperature.
• The Stefan-Boltzmann constant is $\sigma = 5.67 \times 10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}$.
• The emitted power per unit area is $j = \frac{P}{A} = e\sigma T^4$.
• The net radiated power between an object and its surroundings is $P_{\text{net}} = e\sigma A\left(T^4 - T_{\text{env}}^4\right)$.
• Wien's displacement law is $\lambda_{\max}T = b$, where $b = 2.90 \times 10^{-3}\ \mathrm{m\,K}$.
• The peak wavelength from Wien's law is $\lambda_{\max} = \frac{b}{T}$, so increasing $T$ shifts the peak toward shorter wavelengths.
• A perfect blackbody has emissivity $e = 1$, while real surfaces have $0 < e < 1$.
• Temperature must be converted to kelvins using $T_{K} = T_{^{\circ}\mathrm{C}} + 273.15$ before using radiation laws.

Vocabulary

Blackbody

A blackbody is an ideal object that absorbs all incoming radiation and emits the maximum possible thermal radiation at each temperature.

Emissivity

Emissivity, written as $e$, is a number from $0$ to $1$ that compares a real surface's radiation to that of a perfect blackbody.

Stefan-Boltzmann Law

The Stefan-Boltzmann law states that radiated power is proportional to surface area and the fourth power of absolute temperature, $P = e\sigma AT^4$.

Wien's Displacement Law

Wien's displacement law states that the peak wavelength of thermal radiation satisfies $\lambda_{\max}T = b$.

Peak Wavelength

The peak wavelength $\lambda_{\max}$ is the wavelength at which a hot object emits the greatest intensity of radiation.

Absolute Temperature

Absolute temperature is temperature measured in kelvins, where $0\ \mathrm{K}$ represents absolute zero.