Show that Stefan's law results from Planck's radiation law. Hint: To compute the total power of black body radiation emitted across the entire spectrum of wavelengths at a given temperature, integrate Planck's law over the entire spectrum \(P(T)=\int_{0}^{\infty} I(\lambda, T) d \lambda .\) Use the substitution \(x=h c / \lambda k T\) and the tabulated value of the integral \(\int_{0}^{\infty} d x x^{3} /\left(e^{x}-1\right)=\pi^{4} / 15\).

Understanding the connection between temperature and thermal radiation is crucial in thermal physics, and the Stefan-Boltzmann Law makes this relationship clear. Essentially, the law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time is proportional to the fourth power of the black body's temperature, mathematically expressed as
\[P(T) = \text{\(\sigma\)} T^4\] where P(T) is the power radiated per unit area, T is the absolute temperature, and \(\sigma\) is the Stefan-Boltzmann constant. This constant relates to other fundamental constants, signifying the deep connection between electromagnetic radiation and thermodynamics.For a student to grasp this concept, it's important to visualize this law in action. For instance, as an object gets hotter, it does not just radiate more energy, but the rate at which its energy output increases is exponential due to the power of four relationship. A practical example is the incandescent light bulb, where a slight increase in temperature causes a significant increase in brightness.

Black body radiation represents an idealized physical body that perfectly absorbs all incident electromagnetic radiation, irrespective of frequency or angle of incidence. A black body also radiates energy at all frequencies, with its emission solely determined by its temperature, characterized by Planck's radiation law. This concept is a cornerstone of quantum mechanics and has been instrumental in the development of technologies such as thermal cameras and understanding stellar physics.In the context of the textbook problem, the black body stands as a reference model that helps physicists and students alike make predictions about the behavior of real-world materials at various temperatures. The term 'black body' is somewhat of a misnomer since such bodies can emit visible light and do not necessarily appear black. For example, a heated iron object at lower temperatures may first emit just infrared light (beyond human vision) but will glow red and eventually white as its temperature increases, clearly demonstrating black body radiation in action.

Integrating spectral radiance over the entire spectrum allows us to calculate the total power output of a black body. Spectral radiance itself is the amount of radiant energy with respect to a particular wavelength. By considering every possible wavelength (integrating from zero to infinity), we can find the overall radiated energy, which leads us to understand how objects radiate energy across the electromagnetic spectrum as a function of temperature.In performing the integration, as demonstrated in the exercise solution, the introduction of a variable substitution using the dimensionless variable \(x\) is a common mathematical technique. It simplifies the integral into a form where a known value can be applied, drastically reducing the complexity of the calculation. Integrating spectral radiance provides a bridge between the microscopic interactions within a material and the macroscopic observable phenomena, such as the heat and light we experience in daily life from various sources.

Thermal radiation physics explores how objects emit radiation due to their temperatures. Different from conductive and convective heat transfer, thermal radiation does not need a medium; it can occur in a vacuum, such as the heat from the Sun reaching Earth. It is a fundamental mode of heat transfer and critical to many areas of research and practical applications, ranging from climatology to engineering.The laws of thermal radiation, including the Stefan-Boltzmann and Planck's laws, are rooted in quantum mechanics and statistical thermodynamics. As an object gets hotter, it not only radiates more intensely but also shifts its peak emission to shorter wavelengths. This shift is why the heating element on a stove changes from red to white as it gets hotter. Understanding this concept is essential for industries focused on energy efficiency, as it defines the optimal working temperatures for machines, engines, and insulation materials.