The total rate of radiation emitted by a black-body radiator is indeed proportional to the fourth power of its temperature, a principle known as the Stefan-Boltzmann law. This relationship is derived from Planck's radiation law, which describes the spectral energy density of electromagnetic radiation emitted by a black body in thermal equilibrium. By integrating Planck's law over all wavelengths, one can derive the total energy emitted per unit area, confirming the fourth power dependency on temperature. The mathematical proof involves integrating the intensity of radiation across all frequencies, leading to the conclusion that total emission increases significantly with temperature. This foundational concept is crucial in thermodynamics and astrophysics, illustrating the behavior of black-body radiation.
