Planck's radiation law, expressed in terms of energy per...

Planck's radiation law, expressed in terms of energy per unit range of wavelength instead of frequency, becomes $$ E_{\lambda}=\frac{8 \pi c h}{\lambda^{5}\left(e^{c h / \lambda k T}-1\right)} $$ Use the variable $x=c h / \lambda k T$ to show that the total energy per unit volume at temperature $T^{\circ}$ absolute is given by $$ \int_{0}^{\infty} E_{\lambda} \mathrm{d} \lambda=a T^{4} \mathrm{~J} \mathrm{~m}^{-3} $$ where $$ a=\frac{8 \pi^{5} k^{4}}{15 c^{3} h^{3}} $$ (The constant $c a / 4=\sigma$, Stefan's Constant in the Stefan-Boltzmann Law.) Note that $$ \int_{0}^{\infty} \frac{x^{3} \mathrm{~d} x}{e^{x}-1}=\frac{\pi^{4}}{15} $$

Planck's radiation law, expressed in terms of energy per unit range of wavelength instead of frequency, becomes
$$
E_{\lambda}=\frac{8 \pi c h}{\lambda^{5}\left(e^{c h / \lambda k T}-1\right)}
$$
Use the variable $x=c h / \lambda k T$ to show that the total energy per unit volume at temperature $T^{\circ}$ absolute is given by
$$
\int_{0}^{\infty} E_{\lambda} \mathrm{d} \lambda=a T^{4} \mathrm{~J} \mathrm{~m}^{-3}
$$
where
$$
a=\frac{8 \pi^{5} k^{4}}{15 c^{3} h^{3}}
$$
(The constant $c a / 4=\sigma$, Stefan's Constant in the Stefan-Boltzmann Law.) Note that
$$
\int_{0}^{\infty} \frac{x^{3} \mathrm{~d} x}{e^{x}-1}=\frac{\pi^{4}}{15}
$$

Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths scales with the fourth power of its temperature. This law is directly related to the integrated form of Planck's law, where the constant derived from the integration is connected to Stefan’s constant, ultimately bridging quantum theory and classical thermodynamics.

Blackbody Radiation Energy Density

The energy density of blackbody radiation is obtained by integrating the spectral energy distribution over all wavelengths. This total energy per unit volume depends on temperature as T?, linking the microscopic quantum description of radiation with macroscopic thermodynamic properties.

Integration of Bose-Einstein Integrals

Integrals of the form ?x³/(e? - 1) dx frequently arise in the context of quantum statistical mechanics when calculating the total energy density of bosonic systems, such as photons in blackbody radiation. Recognizing and evaluating these integrals is crucial for deriving the T? dependence in the energy density of a blackbody.

Planck's Radiation Law

This law describes the spectral energy distribution of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. It was pivotal in the development of quantum theory, as it correctly models how energy is radiated at different wavelengths, demonstrating that energy emission is quantized.

Variable Substitution in Integration

Changing variables, such as using x = c h / (? k T), is a common mathematical technique used to simplify integrals by recasting them into a form that is easier to evaluate. In the context of Planck's law, this substitution transforms the wavelength-dependent expression into a dimensionless integral that can be solved analytically.

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