Planck’s Law: Blackbody Radiation and Quantum Physics

Posted on Dec 31, 2025 in Physics

Planck’s Radiation Law: Detailed Analysis

Introduction:

Planck’s radiation law is a fundamental principle in quantum physics that explains the spectral distribution of electromagnetic radiation emitted by a blackbody in thermal equilibrium. This law resolved the limitations of classical physics, especially the “ultraviolet catastrophe” predicted by the Rayleigh-Jeans law at short wavelengths.

Background: Resolving Spectral Failures

Before Planck, scientists used Wien’s law and Rayleigh-Jeans law to describe blackbody radiation, but both failed to match experimental results across the full spectrum:

  • Rayleigh-Jeans law worked well for long wavelengths but diverged at short wavelengths (predicting infinite energy — the ultraviolet catastrophe).
  • Wien’s law fit well at short wavelengths but failed at long wavelengths.

To resolve this, Max Planck proposed a new theoretical model in 1900 by introducing the concept of quantized energy levels.

Planck’s Postulates

  1. Energy Quantization: Energy is not continuous but emitted or absorbed in discrete packets called quanta.
  2. Energy of a Quantum: $E=h\nu$, where:
    • $E$ = energy of radiation
    • $h$ = Planck’s constant ($6.626 \times 10^{-34} \text{ Js}$)
    • $\nu$ = frequency of radiation

This revolutionary idea led to the correct formulation of the blackbody radiation spectrum.

Planck’s Radiation Law (Mathematical Expression)

The spectral energy density per unit wavelength $u(\lambda,T)$ is given by:

$$u(\lambda, T) = \frac{8\pi hc}{\lambda^5} \cdot \frac{1}{e^{\frac{hc}{\lambda kT}} – 1}$$

Where:

  • $u(\lambda,T)$ = energy per unit volume per unit wavelength at temperature $T$
  • $\lambda$ = wavelength
  • $T$ = absolute temperature (in Kelvin)
  • $h$ = Planck’s constant
  • $c$ = speed of light
  • $k$ = Boltzmann constant ($1.38 \times 10^{-23} \text{ J/K}$)

Explanation of Terms

  • The term $\frac{8\pi hc}{\lambda^5}$ shows that energy increases as the wavelength decreases.
  • The denominator $e^{\frac{hc}{\lambda kT}} – 1$ ensures that energy approaches zero at very small wavelengths, preventing the ultraviolet catastrophe.

Graphical Representation

  • The radiation curve for different temperatures shows a peak (maximum energy emitted) that shifts to shorter wavelengths as temperature increases.
  • This behavior aligns with Wien’s Displacement Law, which states that $\lambda_{\text{max}} T = \text{constant}$.

Significance of Planck’s Law

  1. Explains Blackbody Radiation Completely: Matches experimental data at all wavelengths and temperatures.
  2. Foundation of Quantum Theory: Introduced the concept of quantization, leading to the development of quantum mechanics.
  3. Eliminates Ultraviolet Catastrophe: Corrects the divergence issue in classical theories.

Applications

  • Design and analysis of thermal radiation sensors
  • Understanding the cosmic microwave background radiation
  • Used in astrophysics to determine the temperature of stars
  • Foundation for quantum mechanics and modern physics

Conclusion

Planck’s radiation law marked a turning point in physics. By assuming that electromagnetic energy could only be emitted in quantized packets, Planck not only solved the blackbody radiation problem but also laid the groundwork for quantum mechanics. This law continues to influence numerous fields, from astrophysics to thermal imaging and quantum computing.