Based on the figure, at a temperature of 300 \(\mathrm{K}\) and a wavelength of \(30 \times 10^{-6} \mathrm{m}\) , the brightness of a black- body will most likely be: A. less than \(5 \times 10^{6}\) watts per \(\mathrm{m}^{3}\) and \(40 \times 10^{6}\) watts B. between \(5 \times 10^{6}\) watts per \(m^{3}\) and \(40 \times 10^{6}\) watts per \(m^{3}\) . C. between \(41 \times 10^{6}\) watts per \(m^{3}\) and \(130 \times 10^{6}\) watts per \(m^{3}\) . D. greater than \(130 \times 10^{6}\) watts per \(m^{3}\) .

Black-body radiation is a fundamental concept in physics that describes the type of electromagnetic radiation emitted by an object that absorbs all the radiation that falls on it. A perfect black body is an idealized object that does not reflect or transmit any radiation, meaning it is a perfect absorber and equally a perfect emitter of radiation.

When an object is heated, it emits radiation over a wide range of wavelengths, but the intensity of radiation peaks at a particular wavelength dependent on the temperature of the object. This phenomenon is described by Planck's law, which provides the spectral distribution of a black body's emission enabling us to calculate the brightness for a given temperature and wavelength.

Understanding this concept helps in several practical applications, including thermal imaging, astrophysics to determine the temperature of stars and planets, and even climatology for understanding the Earth's radiation budget.

Planck's constant, denoted as \(h\), is a fundamental constant in physics that has the value \(6.626 \times 10^{-34} m^2kg/s\). It plays a crucial role in the field of quantum mechanics, representing the smallest action—effectively a 'quantum'—that can be observed in nature.

This constant is not only part of the Planck's law formula that describes the spectral brightness of black-body radiation, but it's also a key component of other quantum phenomena, including the energy of a photon, \(E = hu\), where \(u\) is the frequency of the radiation. Understanding Planck's constant is therefore essential for studying the interaction between matter and electromagnetic radiation at the smallest scales.

The Boltzmann constant \(k\), which is approximately equal to \(1.381 \times 10^{-23} J/K\), is a constant linking the average kinetic energy of particles in a gas with its temperature. This constant plays a fundamental role in statistical mechanics and thermodynamics, encapsulating how individual particle energies are distributed within a system.

In Planck's law formula, the Boltzmann constant is utilized as part of the exponential factor that determines the distribution of energy at different wavelengths for a given temperature. It's also a critical component of the ideal gas law and makes appearances in the entropy and Boltzmann distribution equations, reflecting its broad importance for understanding thermal properties of materials at the atomic and molecular levels.