Tru Physics

Planck’s Law

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Introduction

Planck’s law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. The law is named after Max Planck, who was the first to propose it in 1900. It corrected the known limitations of the Rayleigh-Jeans law and Wien’s displacement law and marked the birth of quantum mechanics.

Basic Formulation

The spectral radiance of a black body, B_{\nu}(T), as a function of frequency \nu at absolute temperature T is given by Planck’s law:

B_{\nu}(T) = \dfrac{2h\nu^3}{c^2}\dfrac{1}{e^{\left(\frac{h\nu}{k_BT}\right)}-1}

where:

  • h is Planck’s constant,
  • k_B is Boltzmann’s constant,
  • c is the speed of light.

Planck’s law can also be expressed in terms of wavelength \lambda:

B_{\lambda}(T) = \dfrac{2hc^2}{\lambda^5}\dfrac{1}{e^{\left(\frac{hc}{\lambda k_BT}\right)}-1}

Importance and Implications

Planck’s law is a cornerstone in the development of quantum mechanics. It was the first physical law to incorporate Planck’s constant and to imply the quantization of energy. This led to the concept of photons, the quantum particles of light, and to the realization that physical systems at the microscopic scale behave differently than macroscopic systems.

Ultraviolet Catastrophe

The formulation of Planck’s law resolved the so-called “ultraviolet catastrophe” predicted by the Rayleigh-Jeans law. According to the Rayleigh-Jeans law, the spectral radiance of a black body should increase without limit as the frequency of the radiation increases. This unrealistic prediction, clearly not observed in nature, was dubbed the “ultraviolet catastrophe”. Planck’s law correctly predicts that the spectral radiance decreases at high frequencies, in agreement with observed physical behavior.

Wien’s Displacement Law and Stefan-Boltzmann Law

Planck’s law includes two important limiting cases. For high frequencies (short wavelengths), Planck’s law reduces to Wien’s displacement law, which describes the shift of the peak frequency of black-body radiation as a function of temperature. For low frequencies (long wavelengths), Planck’s law reduces to the Rayleigh-Jeans law.

By integrating Planck’s law over all frequencies, we obtain the Stefan-Boltzmann law, which states that the total energy radiated per unit surface area of a black body is directly proportional to the fourth power of its absolute temperature:

P = \sigma T^4

where P is the total power radiated per unit area, T is the absolute temperature, and \sigma is the Stefan-Boltzmann constant.

Conclusion

Planck’s law has wide-ranging implications and applications across many fields of physics, including quantum mechanics, statistical mechanics, and astrophysics. It has been experimentally verified over a wide range of frequencies and temperatures, and is one of the key principles underlying our understanding of light and heat radiation.

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