I(ν) = (2hν^3/c^2)(1/(e^(hν/kt)-1))

Definition

This equation represents the spectral radiance of a black body as described by Planck's law, which quantifies the amount of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. The term connects temperature, frequency, and energy in the context of black-body radiation, showcasing how the energy distribution among different frequencies leads to the observed spectrum.

5 Must Know Facts For Your Next Test

1. The equation shows that spectral radiance increases with frequency ($$ u$$$), reflecting the quantum nature of energy distribution in black-body radiation.
2. At higher temperatures, the peak frequency shifts towards higher values according to Wien's displacement law, which can be derived from this equation.
3. The denominator $$e^{(h u/kt)} - 1$$ accounts for the quantum statistical effects that dictate how energy is distributed among photons at thermal equilibrium.
4. As temperature approaches absolute zero, the spectral radiance approaches zero, indicating that a black body emits very little radiation.
5. This equation successfully resolved the ultraviolet catastrophe predicted by classical physics, demonstrating that energy quantization must be considered for accurate predictions.

Review Questions

• How does Planck's law explain the relationship between temperature and spectral radiance for a black body?
  • Planck's law, represented by the equation i(ν) = (2hν^3/c^2)(1/(e^(hν/kt)-1)), illustrates that as the temperature of a black body increases, its spectral radiance also increases across all frequencies. This relationship shows how energy is redistributed among frequencies and highlights that at higher temperatures, more high-frequency photons are emitted. The peak frequency also shifts towards higher values due to Wien's displacement law, further illustrating how temperature affects emitted radiation.
• In what ways does the equation i(ν) = (2hν^3/c^2)(1/(e^(hν/kt)-1)) differ from classical predictions of black-body radiation?
  • The equation differs significantly from classical predictions by accounting for quantum mechanics. Classical physics suggested that spectral radiance would increase indefinitely with frequency, leading to the ultraviolet catastrophe. However, Planck's law introduces quantized energy levels through $$h$$ (Planck's constant), which results in a finite emission spectrum that decreases at higher frequencies instead. This shift towards understanding energy distribution as quantized solved discrepancies between theory and observed experimental data.
• Evaluate the implications of Planck's law on modern physics and technology based on the understanding of i(ν).
  • Planck's law has had profound implications on modern physics by laying the groundwork for quantum theory. The realization that energy is quantized changed our understanding of atomic and subatomic processes and led to developments in various fields such as thermodynamics, statistical mechanics, and quantum mechanics. Technologies like lasers and semiconductor devices rely on these principles. Furthermore, understanding black-body radiation has applications in astrophysics, helping us analyze celestial bodies' temperatures and compositions based on their emitted spectra.

Related terms

Black Body: An idealized physical object that absorbs all incoming radiation and re-emits energy uniformly across all frequencies, serving as a perfect emitter and absorber of radiation.

Planck's Constant (h): A fundamental constant that relates the energy of a photon to its frequency, playing a crucial role in quantum mechanics and black-body radiation.

Stefan-Boltzmann Law: A law that states the total energy radiated per unit surface area of a black body is proportional to the fourth power of its absolute temperature.