Blackbody Radiation: Wien's Law, Stefan's Law and Planck

Why Hot Objects Glow

Heat an iron rod in a blacksmith's fire and it glows dull red, then orange, then yellow-white as it gets hotter. The Sun looks yellow-white, cooler stars look red and hotter stars blue-white. Your own body glows too, but in infrared light that thermal cameras can see and your eyes cannot. All of these are examples of blackbody radiation, the electromagnetic radiation every object emits because of its temperature. Explaining its spectrum forced physicists to invent quantum theory. This tutorial covers what a blackbody is, the shape of its spectrum, Wien's displacement law, the Stefan–Boltzmann law, Planck's quantum hypothesis and the "ultraviolet catastrophe", and applications from stars to the Earth's temperature.

What Is a Blackbody?

A blackbody is an ideal object that absorbs all the electromagnetic radiation falling on it, reflecting none, at every wavelength. A good blackbody is also the best possible emitter: at a given temperature, no surface can radiate more.

A practical blackbody is a small hole in the wall of a hollow box. Radiation entering the hole bounces around inside and is almost all absorbed before it can escape. When the box is heated, the radiation coming out of the hole has the ideal blackbody spectrum, which depends only on the temperature, not on what the box is made of. The Sun and stars, and many everyday hot objects, are close to blackbodies.

Real surfaces emit a fraction of the blackbody amount called the emissivity, ε, between 0 and 1. Matt black surfaces have ε close to 1; shiny metal surfaces have small ε, which is why the inside of a vacuum flask is silvered.

The Blackbody Radiation Spectrum

A graph of the intensity of the radiation against wavelength has a characteristic shape for every temperature:

it starts at zero for very short wavelengths, rises to a peak, and then falls off gradually at long wavelengths;
at a higher temperature, the curve is higher at every wavelength, so the total power emitted is much greater;
at a higher temperature, the peak moves to shorter wavelengths.

This explains the colour sequence of a heated object. At about 1,000 K the peak is in the infrared, and only the short-wavelength tail reaching into the visible red is seen. As the temperature rises the peak moves towards the visible, and more yellow, green and blue light is added, so the glow becomes orange, then yellow-white, then bluish-white.

Wien's Displacement Law

The wavelength of peak emission is inversely proportional to the absolute temperature:

λmax × T = b, where b = 2.898 × 10⁻³ m K (Wien's constant).

Example 1. The Sun's surface temperature is about 5,800 K. Find the wavelength at which it emits most strongly.

λmax = 2.898 × 10⁻³ / 5,800 = 5.0 × 10⁻⁷ m = 500 nm, in the green-blue part of the visible spectrum. (The Sun still looks white or yellowish because it emits strongly across the whole visible range.) It is no coincidence that human eyes are most sensitive near this wavelength.

Example 2. Find the peak wavelength emitted by the human body at 310 K.

λmax = 2.898 × 10⁻³ / 310 = 9.3 × 10⁻⁶ m = 9.3 μm, in the far infrared. Thermal imaging cameras and infrared thermometers are designed to detect wavelengths around 8 to 14 μm for this reason.

Example 3. A red star has its peak at 966 nm. Find its surface temperature.

T = 2.898 × 10⁻³ / 966 × 10⁻⁹ = 3,000 K. Astronomers find star temperatures in exactly this way.

The Stefan–Boltzmann Law

The total power radiated by a blackbody, summed over all wavelengths, rises with the fourth power of its absolute temperature:

P = σAT⁴, where σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ (the Stefan–Boltzmann constant) and A is the surface area.

For a real surface, P = εσAT⁴.

The fourth power makes temperature enormously important: doubling the absolute temperature multiplies the radiated power by 2⁴ = 16.

Example 4. How much power does each square metre of a blackbody at 1,000 K radiate?

P/A = 5.67 × 10⁻⁸ × (1000)⁴ = 56,700 W/m², about 57 kW per square metre.

Example 5. A person with a skin area of 1.8 m² and a skin temperature of 306 K stands in a room at 293 K. The emissivity of skin is about 0.97. Find the net power lost by radiation.

The body also absorbs radiation from the room walls, so the net loss is P = εσA(T⁴ − Tsurroundings⁴) = 0.97 × 5.67 × 10⁻⁸ × 1.8 × (306⁴ − 293⁴) ≈ 138 W. That is comparable to a person's resting metabolic power, which is why radiation is a major way the body loses heat.

Example 6. The Sun has radius 6.96 × 10⁸ m and surface temperature 5,800 K. Find its luminosity (total power output).

L = 4πR²σT⁴ = 4π × (6.96 × 10⁸)² × 5.67 × 10⁻⁸ × 5800⁴ ≈ 3.9 × 10²⁶ W.

The Earth's Temperature

At the Earth's distance (1.5 × 10¹¹ m), the Sun's power is spread over a sphere, giving an intensity of about L/(4πd²) ≈ 1,380 W/m², the "solar constant".

The Earth reflects about 30% of this (its albedo is about 0.3) and absorbs the rest over its cross-section πR². It radiates as a blackbody from its whole surface, 4πR². Balancing absorbed and emitted power:

(1 − 0.3) × 1380 × πR² = σT⁴ × 4πR², so T⁴ = 0.7 × 1380 / (4 × 5.67 × 10⁻⁸), giving T ≈ 255 K (about −18 °C).

The actual average surface temperature is about 288 K (15 °C). The difference of about 33 K is the greenhouse effect: gases such as water vapour and carbon dioxide absorb part of the Earth's outgoing infrared radiation (peaking around 10 μm, by Wien's law) and re-emit some of it back down. Adding more of these gases strengthens the effect.

The Ultraviolet Catastrophe and Planck's Quantum Idea

In the late 1800s, classical physics (the Rayleigh–Jeans law) predicted that the intensity of blackbody radiation should keep rising without limit as the wavelength got shorter. This would mean every hot object emits infinite energy in the ultraviolet and beyond, which is obviously wrong. The failure was called the ultraviolet catastrophe.

In 1900 Max Planck found a formula that matched the measured spectrum perfectly. To derive it, he had to assume that the energy of the oscillators emitting the radiation could only change in small, fixed packets, or quanta:

E = hf, where h = 6.63 × 10⁻³⁴ J s is Planck's constant and f is the frequency.

High-frequency quanta need a lot of energy, and at a given temperature there is rarely enough thermal energy to emit them, so the spectrum falls back to zero at short wavelengths. This was the birth of quantum physics. Einstein extended the idea in 1905 to explain the photoelectric effect, treating light itself as made of photons of energy hf. A photon of 500 nm green light, for example, carries hc/λ ≈ 4.0 × 10⁻¹⁹ J, about 2.5 eV.

Applications

Astronomy: star colours and spectra give their surface temperatures (Wien's law); combined with their luminosity, Stefan's law gives their radii.
Cosmic microwave background: the universe is filled with almost perfect blackbody radiation at 2.7 K, peaking at about 1 mm, the afterglow of the Big Bang.
Thermal imaging and infrared thermometers measure temperature without contact.
Lighting: filament lamps are hot blackbody-like radiators at about 2,700 K that put most of their energy into infrared, which is why they are inefficient compared with LEDs.

Summary

A blackbody absorbs all radiation and is the best possible emitter; blackbody radiation depends only on temperature.
Wien's law: λmax T = 2.898 × 10⁻³ m K; hotter bodies peak at shorter wavelengths.
Stefan–Boltzmann law: P = εσAT⁴; doubling T multiplies the power by 16.
Classical physics failed (the ultraviolet catastrophe); Planck's quanta, E = hf, explained the spectrum and began quantum theory.
A simple energy balance gives the Earth's temperature without an atmosphere as about 255 K.