The total power per unit area radiated by a black body at a...

The total power per unit area radiated by a black body at a temperature $T$ is the area under the $I(\lambda, T)$ -versus- $\lambda$ curve as shown in Figure 39.3 . (a) Show that this power per unit area is $$ \int_{0}^{\infty} I(\lambda, T) d \lambda=\sigma T^{-4} $$ where $I(\lambda, T)$ is given by Planck's radiation law and $\sigma$ is a constant independent of $T$. This result is known as Stefan's law. (See Section 19.6.) To carry out the integration, you should make the change of variable $x=h c / \lambda k_{\mathrm{B}} T$ and use $$ \int_{0}^{\infty} \frac{x^{3} d x}{e^{x}-1}=\frac{\pi^{4}}{15} $$ (b) Show that the Stefan-Boltzmann constant $\sigma$ has the value $$ \sigma=\frac{2 \pi^{3} k_{\mathrm{B}}^{4}}{15 c^{2} h^{3}}=5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4} $$

The total power per unit area radiated by a black body at a temperature $T$ is the area under the $I(\lambda, T)$ -versus- $\lambda$ curve as shown in Figure 39.3 . (a) Show that this power per unit area is $$
\int_{0}^{\infty} I(\lambda, T) d \lambda=\sigma T^{-4}
$$ where $I(\lambda, T)$ is given by Planck's radiation law and $\sigma$ is a constant independent of $T$. This result is known as Stefan's law. (See Section 19.6.) To carry out the integration, you should make the change of variable $x=h c / \lambda k_{\mathrm{B}} T$ and use $$
\int_{0}^{\infty} \frac{x^{3} d x}{e^{x}-1}=\frac{\pi^{4}}{15}
$$ (b) Show that the Stefan-Boltzmann constant $\sigma$ has the value $$
\sigma=\frac{2 \pi^{3} k_{\mathrm{B}}^{4}}{15 c^{2} h^{3}}=5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4}
$$

Key Concepts

Black Body Radiation

Black body radiation refers to the electromagnetic radiation emitted by an idealized object that absorbs all incident radiation, regardless of frequency or angle of incidence. This concept is foundational in thermal physics and quantum mechanics because it provides insights into the energy distribution of thermal emissions and is used as a model for studying the interactions between matter and radiation.

Planck's Radiation Law

Planck's radiation law describes the spectral distribution of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. This law was pivotal in the development of quantum theory as it introduced the concept of quantized energy levels, explaining why classical physics could not account for the observed spectrum of black body radiation.

Stefan-Boltzmann Law

The Stefan-Boltzmann law quantifies the total power radiated per unit area by a black body as being proportional to the fourth power of its absolute temperature. This law, derived from integrating Planck's radiation law over all wavelengths, is fundamental in understanding radiative heat transfer and is widely applicable in fields such as astrophysics and climate science.

Mathematical Techniques in Deriving Physical Laws

The derivation of physical laws such as the Stefan-Boltzmann law often involves sophisticated mathematical techniques, including changing variables in integrals and utilizing known integral identities. These methods allow for the transformation and simplification of complex expressions to reveal underlying physical relationships, and they are essential tools in both theoretical and applied physics.

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