(a) Write the Planck distribution law in terms of the...

(a) Write the Planck distribution law in terms of the frequency $f,$ rather than the wavelength $\lambda,$ to obtain $I(f) .$ (b) Show that $$ \int_{0}^{\infty} I(\lambda) d \lambda=\frac{2 \pi^{5} k^{4}}{15 c^{2} h^{3}} T^{4} $$ where $I(\lambda)$ is the Planck distribution formula of Eq. (39.24). Hint: Change the integration variable from $\lambda$ to $f$. You'll need to use the following tabulated integral: $$ \int_{0}^{\infty} \frac{x^{3}}{e^{\alpha x}-1} d x=\frac{1}{240}\left(\frac{2 \pi}{\alpha}\right)^{4} $$ (c) The result of part (b) is $I$ and has the form of the Stefan-Boltzmann law, $I=\sigma T^{4}$ (Eq. 39.19). Evaluate the constants in part (b) to show that $\sigma$ has the value given in Section 39.5 .

(a) Write the Planck distribution law in terms of the frequency $f,$ rather than the wavelength $\lambda,$ to obtain $I(f) .$ (b) Show that
$$
\int_{0}^{\infty} I(\lambda) d \lambda=\frac{2 \pi^{5} k^{4}}{15 c^{2} h^{3}} T^{4}
$$
where $I(\lambda)$ is the Planck distribution formula of Eq. (39.24). Hint:
Change the integration variable from $\lambda$ to $f$. You'll need to use the following tabulated integral:
$$
\int_{0}^{\infty} \frac{x^{3}}{e^{\alpha x}-1} d x=\frac{1}{240}\left(\frac{2 \pi}{\alpha}\right)^{4}
$$
(c) The result of part (b) is $I$ and has the form of the Stefan-Boltzmann law, $I=\sigma T^{4}$ (Eq. 39.19). Evaluate the constants in part (b) to show that $\sigma$ has the value given in Section 39.5 .

Key Concepts

Planck Radiation Law

The Planck radiation law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. It was pivotal in establishing quantum theory, as it introduced the concept of energy quanta. The law gives the intensity of the radiation as a function of either frequency or wavelength, showing how energy is distributed over the spectrum and explaining phenomena such as the ultraviolet catastrophe that earlier classical theories could not resolve.

Frequency and Wavelength Conversion

Converting the Planck distribution from wavelength to frequency form involves using the fundamental relationship c = f?, where c is the speed of light, and appropriately adjusting the differential elements. This conversion requires careful treatment of the Jacobian factor to ensure that the spectral radiance remains correctly normalized under change of variables. It represents a general technique in physics for converting between different representations of functions that depend on interrelated variables.

Stefan-Boltzmann Law

The Stefan-Boltzmann law relates the total energy radiated per unit surface area of a black body to the fourth power of its absolute temperature. By integrating the Planck distribution over all wavelengths or frequencies, one recovers this T^4 dependence with a proportionality constant known as the Stefan-Boltzmann constant. This law is a cornerstone result in thermal physics and astrophysics, linking microscopic quantum behavior to macroscopic thermal emission.

Integral Evaluation in Quantum Statistical Mechanics

Evaluating integrals in the context of Planck’s law is crucial for deriving macroscopic thermodynamic quantities, such as the total energy density or radiated power. This often involves integrals of functions with exponential denominators, which can be evaluated using known tabulated integrals or special functions. Such techniques illustrate the interplay between mathematical methods and physical theory in obtaining exact results from statistical mechanics.

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