Derive Stefan's Law From Planck's Radiation Law

Deriving Stefan-Boltzmann Law from Planck's Radiation Law

The Stefan-Boltzmann law, a cornerstone of thermal physics, states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time is directly proportional to the fourth power of the black body's thermodynamic temperature. This seemingly simple relationship is a powerful consequence of the more fundamental Planck's radiation law, which describes the spectral radiance of a black body as a function of wavelength and temperature. This article will detail the mathematical derivation of the Stefan-Boltzmann law from Planck's law, providing a deeper understanding of the connection between these two crucial laws of physics.

Understanding the Starting Points: Planck's Law and the Stefan-Boltzmann Constant

Before embarking on the derivation, let's briefly review the fundamental equations:

Planck's Radiation Law: This law describes the spectral radiance (energy emitted per unit time per unit area per unit solid angle per unit wavelength) of a black body at a given temperature and wavelength. It's expressed as:

B(λ, T) = (2hc²/λ⁵) * [1/(e^(hc/λkT) - 1)]

Where:

• B(λ, T) is the spectral radiance
• h is Planck's constant (6.626 x 10⁻³⁴ Js)
• c is the speed of light (3 x 10⁸ m/s)
• k is Boltzmann's constant (1.38 x 10⁻²³ J/K)
• λ is the wavelength
• T is the absolute temperature

Stefan-Boltzmann Law: This law states that the total power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature:

P/A = σT⁴

Where:

• P/A is the power radiated per unit area
• σ is the Stefan-Boltzmann constant (5.67 x 10⁻⁸ Wm⁻²K⁻⁴)
• T is the absolute temperature

Our goal is to derive the Stefan-Boltzmann law (and the value of σ) from Planck's law. This involves integrating Planck's law over all wavelengths to find the total power radiated.

The Mathematical Derivation: Integrating Planck's Law

The total power radiated per unit area, P/A, is obtained by integrating the spectral radiance, B(λ, T), over all wavelengths from 0 to ∞:

P/A = ∫₀^∞ B(λ, T) dλ  = ∫₀^∞ (2hc²/λ⁵) * [1/(e^(hc/λkT) - 1)] dλ

This integral is not trivial; it requires a substitution to simplify the expression. Let's use the substitution:

x = hc/λkT

This implies:

λ = hc/xkT  and  dλ = -(hc/xk²T²)dx

As λ goes from 0 to ∞, x goes from ∞ to 0. Therefore, the integral becomes:

P/A = ∫∞^₀ (2hc²/((hc/xkT)⁵)) * [1/(eˣ - 1)] * (-(hc/xk²T²)dx)

Simplifying this expression:

P/A = (2k⁴T⁴/h³c²) ∫₀^∞ (x³/(eˣ - 1)) dx

The integral ∫₀^∞ (x³/(eˣ - 1)) dx is a well-known definite integral with a value of π⁴/15. Substituting this value, we get:

P/A = (2k⁴T⁴/h³c²) * (π⁴/15)

This expression now beautifully reveals the Stefan-Boltzmann law. We can rearrange the equation to explicitly show the Stefan-Boltzmann constant:

P/A = [(2π⁴k⁴)/(15h³c²)] * T⁴

The term [(2π⁴k⁴)/(15h³c²)] is precisely the Stefan-Boltzmann constant, σ. By plugging in the values of the fundamental constants (h, k, and c), we can calculate the value of σ:

σ = (2π⁴k⁴)/(15h³c²) ≈ 5.67 x 10⁻⁸ Wm⁻²K⁻⁴

This calculated value matches the experimentally determined value of the Stefan-Boltzmann constant, thereby validating the derivation.

Implications and Extensions of the Derivation

This derivation demonstrates the profound connection between Planck's radiation law and the Stefan-Boltzmann law. Planck's law, a quantum description of blackbody radiation, underpins the classical Stefan-Boltzmann law, which is remarkably simple in its form yet powerful in its applications.

The derivation highlights several crucial points:

• Quantum nature of radiation: The derivation implicitly incorporates the quantization of energy, a cornerstone of quantum mechanics, through Planck's constant (h) in Planck's law. The classical Rayleigh-Jeans law, which attempts to explain blackbody radiation classically, fails to predict the correct behavior at high frequencies (or short wavelengths), leading to the ultraviolet catastrophe. Planck's law, with its quantization of energy, resolves this issue.

• Universality of the Stefan-Boltzmann law: The derivation shows that the Stefan-Boltzmann law is a universal law applicable to all black bodies, irrespective of their specific material properties. This universality arises because the derivation relies solely on fundamental constants and the nature of blackbody radiation, not on material-specific parameters.

• Applications in astrophysics and other fields: The Stefan-Boltzmann law finds extensive applications in astrophysics, allowing astronomers to estimate the surface temperatures of stars based on their observed luminosity. It also plays a significant role in thermal engineering, climate modeling, and other fields where radiative heat transfer is important.

• Beyond ideal black bodies: While the derivation strictly applies to ideal black bodies, it can be extended to real-world objects by introducing an emissivity factor (ε), which accounts for the deviation of real bodies from ideal blackbody behavior. The modified equation becomes:

P/A = εσT⁴

Where ε is the emissivity (0 ≤ ε ≤ 1), with ε = 1 for a perfect black body.

Conclusion

The derivation of the Stefan-Boltzmann law from Planck's radiation law provides a compelling demonstration of the power and elegance of physics. It beautifully connects the quantum world of Planck's law with the macroscopic observable described by the Stefan-Boltzmann law. This derivation not only solidifies our understanding of blackbody radiation but also underpins numerous applications across diverse scientific and engineering fields, making it a fundamental concept in modern physics. The successful derivation further highlights the crucial role of fundamental constants and the beauty of mathematical integration in revealing fundamental physical laws from seemingly complex expressions. The simplicity of the final result, the Stefan-Boltzmann law, belies the depth of the underlying quantum physics encapsulated in Planck's groundbreaking work. The journey from Planck's law to the Stefan-Boltzmann law is a testament to the interconnectedness of seemingly disparate concepts in physics and the predictive power of mathematical modeling.