The Stefan-Boltzmann law is derived from:

Explanation:

The Stefan-Boltzmann Law

• The Stefan-Boltzmann law is a fundamental principle in thermal radiation, stating that the total energy radiated per unit surface area of a black body is directly proportional to the fourth power of its absolute temperature. Mathematically, the Stefan-Boltzmann law is expressed as:

E = σ × T⁴

Where:

• E: Total energy radiated per unit surface area (W/m²)
• σ: Stefan-Boltzmann constant (5.67 × 10^-8 W/m²K⁴)
• T: Absolute temperature of the black body (K)

The Stefan-Boltzmann law is derived from Planck's Law, which describes the spectral distribution of electromagnetic radiation emitted by a black body in thermal equilibrium. By integrating Planck's law over all wavelengths, the Stefan-Boltzmann law can be obtained. This integration process effectively sums up the contributions of radiation from all wavelengths, yielding the total emissive power of the black body.

Planck's law:

• Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T.
• Planck’s law for the energy Eλ radiated per unit volume by a cavity of a blackbody in the wavelength interval λ to λ + Δλ can be written in terms of Planck’s constant (h), the speed of light (c = λ × v), the Boltzmann constant (k), and the absolute temperature (T):

Energy per unit volume per unit wavelength:

\({E_\lambda } = \frac{{8\pi hc}}{{{\lambda ^5}}} \times \frac{1}{{{e^{\frac{{hc}}{{kT\lambda }} - 1}}}}\)

Energy per unit volume per unit frequency:

\({E_\nu } = \frac{{8\pi h}}{{{c^3}}} \times \frac{{{\nu ^3}}}{{{e^{\frac{{hv}}{{kT}} - 1}}}}\)

So Planck’s distribution function:

\(E\left( {\omega ,T} \right) = \frac{1}{{{e^{\frac{{h\omega }}{\tau }}} - 1}}\)

Using planck’s law, when we plot Ebλ with λ, we get the curve as shown below.

As temperature increases, the peak of the curve shift towards a lower wavelength.