23.1 Blackbody Radiation

What is a Black Body?

A black body is an idealised object that:

• Absorbs all electromagnetic radiation incident upon it, regardless of frequency or angle — its absorptivity is 1 (unity).
• Emits radiation at the maximum possible rate for any object at the same temperature.

Real objects approximate black bodies; for example, a small hole in a hollow cavity behaves like a black body because radiation entering the hole undergoes multiple reflections and is almost entirely absorbed.

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Blackbody Radiation Spectrum

When a black body is heated, it emits a continuous spectrum of electromagnetic radiation. The intensity–wavelength graph has a characteristic shape:

• There is a peak wavelength ${\lambda }_{\max }$ at which maximum energy is emitted.
• At wavelengths shorter or longer than ${\lambda }_{\max }$, the intensity falls off.
• As temperature increases, the peak shifts to shorter wavelengths and the total area under the curve increases (more total energy emitted).

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Wien's Displacement Law (SLO P-12-F-46)

The peak wavelength ${\lambda }_{\max }$ of the emitted spectrum is inversely proportional to the absolute temperature $T$ of the black body:

${\lambda }_{\max }T=b$

where the Wien's displacement constant $b\approx 2.9\times 10^{-3}\text{m K}$.

Implication: A hotter object emits peak radiation at a shorter (bluer) wavelength.

Example

The Sun's surface temperature is approximately $5800\text{K}$. Find the peak wavelength of solar radiation.

${\lambda }_{\max }=\frac{b}{T}=\frac{2.9\times 10^{-3}}{5800}\approx 5.0\times 10^{-7}\text{m}=500\text{nm}$

This lies in the visible green region, consistent with the Sun appearing white/yellow.

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The Ultraviolet Catastrophe

Classical physics (the Rayleigh–Jeans law) predicted that the intensity of blackbody radiation should increase without limit as wavelength decreases — implying infinite energy emitted at ultraviolet and shorter wavelengths. This catastrophic failure of classical theory is called the ultraviolet catastrophe.

Experimental data showed the intensity actually falls at short wavelengths, completely contradicting the classical prediction.

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Planck's Quantum Hypothesis

In 1900, Max Planck resolved the ultraviolet catastrophe by proposing that energy is not emitted or absorbed continuously, but in discrete packets called quanta.

The energy of a single quantum is:

$E=hf$

where:

• $h=6.63\times 10^{-34}\text{J s}$ is Planck's constant
• $f$ is the frequency of the radiation

This assumption correctly predicted the observed blackbody spectrum and marked the birth of quantum physics.

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Stefan-Boltzmann Law (SLO P-12-F-47)

The total power radiated per unit surface area $E$ (also called radiant emittance) by a perfect black body is proportional to the fourth power of its absolute temperature:

$E=\sigma T^4$

where:

• $E$ = power radiated per unit area $[{\text{W m}}^{-2}]$
• $\sigma =5.67\times 10^{-8}{\text{W m}}^{-2}{\text{K}}^{-4}$ is the Stefan-Boltzmann constant
• $T$ = absolute temperature in Kelvin $[\text{K}]$

For a star (or any spherical black body) of radius $R$, the total luminosity is:

$L=\sigma A{T}^4=4\pi R^2\sigma T^4$

Example

A black body at $T=300\text{K}$ radiates how much power per unit area?

$E=\sigma T^4=5.67\times 10^{-8}\times (300{)}^4=5.67\times 10^{-8}\times 8.1\times 10^9\approx 459{\text{W m}}^{-2}$

Key consequence

If the temperature of a black body doubles, the total radiated power increases by a factor of $2^4=16$.

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Summary Table