What is blackbody radiation?
Why hot objects glow, and why Planck had to quantize energy.
A blackbody is an ideal object that absorbs all electromagnetic radiation that hits it (no reflection, no transmission). When it is at temperature \(T\), it also emits radiation with a characteristic spectrum that depends only on \(T\). This emitted light is called blackbody radiation.
If you plot the emitted intensity versus wavelength, you find two key experimental trends:
- As \(T\) increases, the total emitted power increases dramatically.
- The peak of the spectrum shifts to shorter wavelengths (toward blue) as \(T\) increases.
The “ultraviolet catastrophe” (classical failure)
Classical physics predicted that a blackbody should emit more and more energy at very short wavelengths (high frequency), leading to an infinite total emitted energy. This incorrect prediction is known as the ultraviolet catastrophe.
The experimental spectrum does not diverge in the ultraviolet; instead it rises, reaches a maximum, and then falls off.
Planck’s solution: quantized energy
Planck resolved the problem by proposing that energy exchange between matter and radiation occurs in discrete packets (quanta). The energy of a quantum of radiation of frequency \( \nu \) is
\[ E = h\nu, \]
where \(h\) is Planck’s constant. This idea leads to the correct blackbody spectrum, known as Planck’s law (written here as a function of wavelength):
\[ B_\lambda(T) = \frac{2hc^2}{\lambda^5}\, \frac{1}{e^{hc/(\lambda kT)}-1}. \]
Here \(k\) is Boltzmann’s constant, \(c\) is the speed of light, and \(B_\lambda(T)\) describes how the intensity is distributed over wavelength. The exponential term in the denominator suppresses high-frequency (short-wavelength) emission, preventing the ultraviolet catastrophe.
Two extremely useful results
Wien’s displacement law tells you where the spectrum peaks:
\[ \lambda_{\max}T = b, \]
where \(b\) is Wien’s constant. As \(T\) increases, \( \lambda_{\max} \) decreases (the peak shifts to shorter wavelengths).
Stefan–Boltzmann law gives the total power emitted per unit area (all wavelengths combined):
\[ \frac{P}{A} = \sigma T^4, \]
where \( \sigma \) is the Stefan–Boltzmann constant. This explains why the emitted power rises very rapidly with temperature.
Big idea: blackbody radiation forced the introduction of energy quantization (\(E=h\nu\)), which became a foundation of quantum mechanics.
Your turn
Key points (one glance)
- A blackbody absorbs all incident radiation and emits a spectrum depending only on \(T\).
- Classical physics predicted the ultraviolet catastrophe; experiments show emission drops at short wavelengths.
- Planck’s hypothesis: energy is exchanged in quanta \(E=h\nu\).
- Planck’s law (wavelength form): \(B_\lambda(T)=\frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/(\lambda kT)}-1}\).
- Wien’s law: \( \lambda_{\max}T=b \) (hotter → peak shifts to shorter wavelengths).
- Stefan–Boltzmann law: \(P/A=\sigma T^4\) (total emitted power grows as \(T^4\)).