Integrate Eq. 3.41 to obtain Eq. 3.26 . Use the definite integral ∫0^∞ x^3 d x /(e^x-1)=π^4 / 15

step 1 So we need to show that we can relate this Stefan Boltzmann constant to Plunk's constant. So let

Integrate Eq. 3.41 to obtain Eq. 3.26 . Use the definite...

Integrate Eq. 3.41 to obtain Eq. $3.26 .$ Use the definite integral $\int_{0}^{\infty} x^{3} d x /\left(e^{x}-1\right)=\pi^{4} / 15$ to obtain Eq. 3.42 relating the Stefan-Boltzmann constant to Planck's constant.

Key Concepts

Use of Standard Definite Integrals

Standard definite integrals, such as the integral of x³/(e^x - 1) from zero to infinity, are used in the derivation of macroscopic laws from microscopic principles. Recognizing and employing these known integrals simplify computations and facilitate the derivation of important constants, for instance, connecting the Stefan-Boltzmann constant to Planck's constant by analytically integrating the spectral density expression.

Definite Integration in Thermal Physics

Definite integration plays a crucial role in thermal physics by allowing the summing over all contributions of a continuous variable—in this case, frequency or wavelength—to yield total physical quantities like energy or power. In the context of blackbody radiation, techniques such as a change of variable are employed to simplify the integration of Planck's law, often leading to standard integrals whose values, like the calculated integral of x³/(e^x - 1), are known.

Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a blackbody is proportional to the fourth power of its absolute temperature. It is derived by integrating Planck's radiation law over all wavelengths. The derived constant in this law, known as the Stefan-Boltzmann constant, encapsulates fundamental physical constants, including Planck’s constant, and connects thermodynamic temperature with radiative power.

Planck's Radiation Law

This law describes the spectral energy distribution of electromagnetic radiation emitted by a blackbody in thermal equilibrium. It is fundamental to modern quantum physics and furnishes the basis for calculating the energy output over various frequencies by integrating the spectral radiance. The law introduces Planck’s constant as a scaling factor, linking the microscopic quantum behavior to macroscopic thermal radiation phenomena.

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