The radiance of a blackbody peaks at \(660 \mathrm{nm}\). (a) What's its temperature? (b) How does its radiance at 400 nm compare with that at \(700 \mathrm{nm} ?\)

Blackbody radiation refers to the theoretical concept of an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. That is, a blackbody is a perfect emitter and absorber of radiation. What sets a blackbody apart in the world of thermodynamics and quantum mechanics is its emission of electromagnetic radiation, which is solely determined by its temperature. This emitted radiation forms what we call a blackbody spectrum, and it's unique because no other factors but temperature, such as the body's material or surface structure, affect it.

The blackbody spectrum is continuous and covers all wavelengths, but importantly, it peaks at a certain wavelength that is inversely related to the temperature of the body. Higher temperatures correspond to peaks at shorter wavelengths, which is indicative of more energetic radiation. Conversely, cooler temperatures peak at longer wavelengths, revealing less energetic radiation. This phenomenon is precisely described by Wien's Displacement Law, which provides a crucial bridge between the peak wavelength and the blackbody's temperature.

Planck's law is a quantum-theoretical approach to describe the electromagnetic radiation emitted by a blackbody in thermal equilibrium at a definite temperature. It is named after Max Planck, who proposed this quantum theory in 1900. Planck's law mathematically characterizes how the energy emitted by a blackbody at different wavelengths varies.

According to Planck's law, the radiance of a blackbody—that is, the amount of energy emitted per unit surface area in a unit time and a unit solid angle within a specified wavelength range—has a specific distribution over wavelengths that depends on the body's temperature. The formula for Planck's law involves constants such as the speed of light, Planck's constant, and the Boltzmann constant. The innovative step that Planck took was to introduce the idea of quantization, which asserts that the energy of the oscillators in the blackbody radiation field is quantized, meaning it can only take on discrete values rather than any value along a continuum.

Calculating the temperature of a blackbody from its peak emission wavelength relies heavily on Wien's Displacement Law. This law states that the wavelength at which the radiance of a blackbody reaches its maximum is inversely proportional to the temperature of the blackbody. The mathematical expression for Wien's Displacement Law is: \(T = \frac{b}{\lambda_{max}}\), where \(T\) represents the temperature of the blackbody in Kelvin, \(\lambda_{max}\) is the peak emission wavelength in meters, and \(b\) is Wien's displacement constant (approximately \(2.897 \times 10^{-3} m \cdot K\)).

For instance, in our given exercise, with the peak wavelength of \(660 nm\), we can calculate the temperature by first converting the wavelength into meters (since the constant \(b\) is in meters) and then applying the formula. The temperature calculation helps in understanding the energy output and characteristics of the blackbody radiation for celestial bodies like stars, as well as practical applications like identifying the efficiencies of heating elements or furnaces.The relationship given by Wien's Displacement Law also helps astronomers to estimate the temperature of distant stars by analyzing the peak wavelength of the radiation they emit, assuming that the stars behave like blackbodies.