Blackbody Radiation & Planck's Law

A blackbody is an idealised object that absorbs all incident radiation and re-emits it purely based on its temperature. The emitted spectrum is a continuous curve whose shape depends only on $T$.

Classical physics (Rayleigh-Jeans law) predicted spectral radiance $\propto \lambda^{-4}$, meaning intensity would rise without limit as wavelength decreases — the ultraviolet catastrophe. This catastrophically disagreed with experiment at short wavelengths.

Max Planck resolved this in 1900 by assuming energy is emitted only in discrete quanta: $E = hf = hc/\lambda$. This single assumption produced the correct spectrum and, more profoundly, launched quantum mechanics. The quantisation means high-frequency modes are exponentially suppressed — few quanta are energetic enough to be emitted — so the curve has a peak and falls to zero at short wavelengths.

Planck's law (spectral radiance):

$B(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{hc/\lambda k_B T} - 1}$

Wien's displacement law (peak wavelength):

$\lambda_{\max} T = 2.898 \times 10^{-3}\,\text{m·K}$

Stefan-Boltzmann law (total power radiated per unit area):

$P/A = \sigma T^4, \quad \sigma = 5.67 \times 10^{-8}\,\text{W m}^{-2}\text{K}^{-4}$

The Sun's surface temperature is $T \approx 5778\,\text{K}$.

Peak wavelength:

$\lambda_{\max} = \frac{2.898 \times 10^{-3}}{5778} \approx 501\,\text{nm}$

This lies in the green part of the visible spectrum — consistent with the Sun appearing white/yellow-white. A star twice as hot ($11{,}556\,\text{K}$) would peak at $\approx 250\,\text{nm}$ (ultraviolet), appearing blue-white. A 3000 K lamp peaks at $\approx 966\,\text{nm}$ (infrared), emitting mostly heat.

Astronomers determine stellar surface temperatures from the colour (peak wavelength) of their spectra — no physical contact needed, just light. The cosmic microwave background (CMB) is a near-perfect 2.725 K blackbody spectrum, the oldest light in the universe. Infrared cameras used in night vision, medical thermography, and satellite weather imaging all rely on blackbody emission from warm objects.

1. A furnace wall is at $T = 1200\,\text{K}$. Using Wien's law, find the peak wavelength of its thermal emission. Is it visible light, infrared, or ultraviolet?

2. If the temperature of a blackbody doubles from $T$ to $2T$, by what factor does the total radiated power change?