Planck's law
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. This fundamental principle in quantum mechanics and thermodynamics was formulated by Max Planck in 1900. It marked a pivotal moment in the development of quantum theory, providing a solution to the ultraviolet catastrophe predicted by classical physics.
Historical Context
The late 19th century saw significant advancements in understanding black body radiation, a theoretical object that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. Classical physics, particularly the Rayleigh-Jeans law, failed to accurately describe the observed spectrum of black body radiation, especially at shorter wavelengths. This discrepancy, known as the ultraviolet catastrophe, led to the development of Planck's law.
Max Planck introduced the concept of quantized energy levels, proposing that electromagnetic energy could be emitted or absorbed only in discrete quantities, or quanta. This hypothesis was revolutionary, laying the groundwork for the development of quantum mechanics.
Mathematical Formulation
Planck's law can be expressed in terms of the spectral radiance \( B(\nu, T) \), which is the power emitted per unit area of the black body, per unit solid angle, per unit frequency. The formula is given by:
\[ B(\nu, T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{\frac{h\nu}{k_BT}} - 1} \]
where: - \( \nu \) is the frequency of the radiation, - \( T \) is the absolute temperature of the black body, - \( h \) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)), - \( c \) is the speed of light in a vacuum (\(3 \times 10^8 \, \text{m/s}\)), - \( k_B \) is the Boltzmann constant (\(1.381 \times 10^{-23} \, \text{J/K}\)).
Alternatively, Planck's law can be written in terms of wavelength \( \lambda \):
\[ B(\lambda, T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda k_BT}} - 1} \]
Implications and Applications
Planck's law has profound implications in various fields of physics and astronomy. It provides the foundation for understanding black body radiation, which is crucial for studying the thermal properties of stars and other celestial bodies. The law also underpins the Stefan-Boltzmann law and Wien's displacement law, which describe the total energy radiated by a black body and the wavelength at which this radiation is maximized, respectively.
In quantum mechanics, Planck's law was instrumental in the development of the photon concept, leading to the photoelectric effect explanation by Albert Einstein. This effect, in turn, provided strong evidence for the quantization of light and contributed to the broader acceptance of quantum theory.
Derivation of Planck's Law
The derivation of Planck's law involves several key steps, starting with the assumption of quantized energy levels. Planck postulated that the energy of oscillators in a black body is quantized and can be expressed as \( E = nh\nu \), where \( n \) is an integer. By applying statistical mechanics, specifically the Bose-Einstein distribution, Planck derived the expression for the spectral radiance.
The detailed derivation involves calculating the average energy of the oscillators and integrating over all possible frequencies. This process leads to the final form of Planck's law, which accurately describes the observed black body spectrum.
Experimental Verification
Planck's law has been extensively verified through experiments. Early 20th-century experiments, such as those conducted by Robert Williams Wood and Otto Lummer, provided empirical data that matched Planck's theoretical predictions. Modern techniques, including spectroscopy and cosmic microwave background observations, continue to confirm the accuracy of Planck's law.
Extensions and Generalizations
While Planck's law was originally formulated for black bodies, it has been extended to other systems. For instance, grey body radiation accounts for objects that do not perfectly absorb all incident radiation. Additionally, Planck's law has been adapted to describe the emission spectra of plasmas and other non-thermal systems.