Blackbody radiation is the thermal electromagnetic radiation emitted by an ideal object that absorbs all incident
radiation and re-emits energy according only to its temperature. The spectrum is not arbitrary: it is determined by
Planck’s law. This law was historically important because it solved one of the major failures of classical physics,
the ultraviolet catastrophe.
In wavelength form, Planck’s law for spectral radiance is
Planck’s law.
\[
B(\lambda,T)=\frac{2hc^2}{\lambda^5}\,\frac{1}{e^{hc/(\lambda kT)}-1}
\]
Here, \(B(\lambda,T)\) is the spectral radiance, \(\lambda\) is the wavelength, and \(T\) is the absolute
temperature. The constants are Planck’s constant \(h\), the speed of light \(c\), and Boltzmann’s constant \(k\).
In SI form, the result is commonly expressed in \(\mathrm{W\,sr^{-1}\,m^{-3}}\), meaning power per unit projected
area, per unit solid angle, and per unit wavelength interval.
The key quantum feature of the formula is the exponential denominator. At short wavelengths, the factor
\(e^{hc/(\lambda kT)}\) becomes very large, which suppresses the spectrum strongly. That suppression is exactly what
classical theory failed to predict.
Dimensionless exponential argument.
\[
x=\frac{hc}{\lambda kT}
\]
The size of \(x\) tells you which region of the spectrum you are in. If \(x\) is small, the long-wavelength limit is
approached and the Rayleigh–Jeans law becomes a reasonable approximation. If \(x\) is large, quantum effects dominate
and the spectrum drops rapidly.
Wien’s displacement law
One of the most useful results associated with blackbody radiation is Wien’s displacement law. It tells us where the
spectrum reaches its maximum:
Wien’s law.
\[
\lambda_{\max}T=b,
\qquad
b\approx 2.897771955\times10^{-3}\ \mathrm{m\cdot K}
\]
This can be rearranged as
\[
\lambda_{\max}=\frac{b}{T}.
\]
The interpretation is simple and important: hotter objects peak at shorter wavelengths. This is why cooler stars tend
to look redder and hotter stars shift toward blue-white light.
Sample calculation: Sun-like blackbody
For a Sun-like surface temperature,
\[
T=5800\ \mathrm{K},
\]
Wien’s law gives
Compute the peak wavelength.
\[
\begin{aligned}
\lambda_{\max}
&= \frac{2.897771955\times10^{-3}}{5800} \\
&\approx 5.00\times10^{-7}\ \mathrm{m}
\end{aligned}
\]
Converting to nanometers:
\[
\lambda_{\max}\approx 500\ \mathrm{nm}.
\]
This lies in the visible range. The Sun does not emit only at this wavelength, but its thermal spectrum peaks near
it, which is one reason sunlight is intense in the visible part of the spectrum.
Classical Rayleigh–Jeans law
Before Planck’s result, classical reasoning led to the Rayleigh–Jeans law:
Rayleigh–Jeans law.
\[
B_{\mathrm{RJ}}(\lambda,T)=\frac{2ckT}{\lambda^4}
\]
This approximation works reasonably well at sufficiently long wavelengths, but it predicts that radiance grows
without limit at short wavelengths. That unphysical divergence is called the ultraviolet catastrophe. Planck’s law
fixes this by introducing the exponential suppression factor that comes from quantized energy exchange.
Wien approximation
In the short-wavelength regime, Planck’s law approaches the Wien approximation:
Wien approximation.
\[
B_{\mathrm{W}}(\lambda,T)=\frac{2hc^2}{\lambda^5}\,e^{-hc/(\lambda kT)}
\]
This form is useful when the exponential factor is large. It highlights the rapid falloff of the spectrum on the
short-wavelength side of the peak.
Stefan–Boltzmann law
If Planck’s law is integrated over all wavelengths and over the outward hemisphere, one obtains the total radiant
exitance:
Stefan–Boltzmann law.
\[
M=\sigma T^4
\]
Here, \(\sigma\) is the Stefan–Boltzmann constant. This shows that total emitted power per unit area rises with the
fourth power of temperature. As a result, even modest temperature increases can greatly increase the total thermal
emission.
Physical meaning and applications
Planck’s law is fundamental in astronomy, thermal imaging, climate science, materials science, and optics. It helps
estimate stellar temperatures, interpret infrared camera readings, model furnace radiation, and understand how hot
objects glow. At low temperatures, the peak lies in the infrared, so objects may emit strongly without appearing
bright to the human eye. As temperature rises, the peak shifts toward shorter wavelengths and visible glow becomes
more prominent.
The law was also historically revolutionary. To explain the observed spectrum, Planck introduced the idea that energy
is exchanged in discrete quanta. That step became one of the foundations of quantum theory.
| Concept |
Main relation |
Meaning |
| Planck spectral radiance |
\(B(\lambda,T)=\dfrac{2hc^2}{\lambda^5}\dfrac{1}{e^{hc/(\lambda kT)}-1}\) |
Exact blackbody spectrum per unit wavelength |
| Wien peak law |
\(\lambda_{\max}T=b\) |
Hotter objects peak at shorter wavelengths |
| Rayleigh–Jeans law |
\(B_{\mathrm{RJ}}=\dfrac{2ckT}{\lambda^4}\) |
Classical long-wavelength approximation |
| Wien approximation |
\(B_{\mathrm{W}}=\dfrac{2hc^2}{\lambda^5}e^{-hc/(\lambda kT)}\) |
Short-wavelength approximation |
| Stefan–Boltzmann law |
\(M=\sigma T^4\) |
Total emitted power per unit area |