law

Stefan–Boltzmann law

The Stefan-Boltzmann law states that the total energy radiated by a blackbody per unit area is proportional to the fourth power of its temperature.
18 related articles · 6 formulas · 1 who worked here

Description

Imagine a stove: the stronger you heat it, the more heat it radiates — that's intuitive. But the Stefan–Boltzmann law says that the radiated power increases not just a lot, but as the fourth power of temperature! That is, if you heat an iron from 300°C to 600°C (twice as hot on the absolute scale), it will radiate 16 times more heat. Analogy: it's like having a garden hose — you turn the tap slightly, and the water pressure increases dramatically. So it is with light and heat from a heated body.

History

Irish physicist John Tyndall conducted measurements of thermal radiation in the 1860s, which led Stefan to analyze the data of Dulong and Petit. Josef Stefan, an Austrian physicist of Slovenian origin, published a note in 1879 in which he claimed that the radiation from a platinum wire is proportional to the fourth power of its absolute temperature. Ludwig Boltzmann, impressed by the insight, undertook a rigorous proof: in 1884, he considered radiation as a working substance in a Carnot cycle and derived the law from the second law of thermodynamics, using the light pressure discovered by James Clerk Maxwell. Later, in 1900, Max Planck, in deriving his famous formula, confirmed and generalized this law, showing that σ can be expressed through Planck's constant, the speed of light, and Boltzmann's constant.

How it works

The Stefan–Boltzmann law is widely applied in astrophysics to determine the luminosities of stars, as well as to calculate the energy balance of planets, including Earth: knowing the solar constant and albedo, one can estimate the planet's equilibrium temperature. In engineering, the law is used in designing furnaces, steam boilers, as well as in infrared sensors and thermal imagers. Limits of applicability: the law holds strictly only for black bodies or gray bodies with a constant emissivity; for real bodies, a correction factor (emissivity) is introduced. At very high temperatures, when effects of general relativity become noticeable (e.g., near black holes), the calculation of energy release requires modifications.

Fun fact

Thanks to this law, we understand why the Sun, while not so large by cosmic standards, shines more than 400 times brighter than its entire surface would if it had the temperature of a human body! If the Sun cooled to 4000°C instead of the current 5500°C, we would receive half as much light — life on Earth would be different. And an incandescent light bulb shines dimly precisely because its filament only heats up to 2500°C — if it were like the Sun, it would instantly evaporate.

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Formulas

$$\langle h\nu \rangle = c \cdot k_B T$$
The average energy of photons in blackbody radiation is proportional to temperature.
What is in the formula
⟨hν⟩variableAverage photon energy — energy joule
TvariableTemperature of the blackbody — temperature kelvin
cconstantA numerical factor that depends on the frequency integration limits and the specific definition of 'average' (e.g., over the full spectrum). — proportionality constant = 2.701
k_BconstantFundamental constant converting temperature to energy scale — boltzmann constant = 1.380649e-23 joule per kelvin
⟨ ⟩operatorStatistical average over the photon distribution — ensemble average
\langlehν\rangle=\frac{π^4}{30\zeta(3)}k_bt\approx2.701k_bt · 2601.22247
$$L = 4\pi R^2 \sigma T^4$$
Luminosity relates to radius and temperature.
What is in the formula
LvariableLuminosity (total power radiated) — luminosity watt
RvariableRadius of the radiating body — radius metre
TvariableEffective temperature of the surface — temperature kelvin
\sigmaconstantProportionality constant relating blackbody radiated power to temperature — stefan boltzmann constant = 5.670374419e-8 watt per metre squared per kelvin to the fourth
r=\sqrt{\frac{l}{4πσt^4}} · 2606.06575v1
l=4πr^2σt^4 · 2606.28993
l=4πr^2σt^4 · 2606.29435
l=4πr^2σt^4 · 2607.06684
$$T \propto \left(\frac{1}{\tau}\right)^{1/4}$$
Temperature decreases as time to the -1/4 power.
What is in the formula
Tvariabletemperature — temperature kelvin
τvariabletime since start of cooling — time second
t(τ)\approx(\frac{\etagm_p^2}{2τ4πr_p^3εσ})^{1/4} · 2607.02653v1
$$F_{\text{total}} = \sigma T^4$$
Total flux equals Stefan-Boltzmann radiation.
What is in the formula
F_{\text{total}}variabletotal radiative flux emitted per unit area — radiative flux watt per square metre
Tvariableabsolute temperature of the radiating body — temperature kelvin
\sigmaconstantStefan-Boltzmann constant relating temperature to radiated power — stefan boltzmann constant = 5.670374419e-8 watt per square metre per kelvin to the fourth
f_{\rmrad}+f_{\rmconv}=σ_{\rmsb}t_{\rmeff}^4 · 2607.24141v1
$$\lambda_{\rm max} = \frac{b}{T}$$
The peak wavelength of blackbody radiation is inversely proportional to temperature.
What is in the formula
lambda_maxvariableWavelength at which blackbody emission is maximum — wavelength metre
TvariableAbsolute temperature of the blackbody — temperature kelvin
bconstantWien's displacement constant, relating peak wavelength to temperature — wien displacement constant = 2.897771955e-3 metre kelvin
λ_{\rmmax}=\frac{b}{t} · 2606.30750
$$T_{\text{eff}} = \left( \frac{1}{\sigma} \int F_\lambda \, d\lambda \right)^{1/4}$$
Temperature of blackbody with same total flux.
What is in the formula
T_effvariableEffective temperature — temperature kelvin
F_lambdavariableSpectral flux density per unit wavelength — spectral flux density watt per metre cubed
lambdavariableWavelength — wavelength metre
sigmaconstantRelates total blackbody flux to temperature — stefan boltzmann constant = 5.670374419e-8 watt per metre squared per kelvin to the fourth
∫operatorIntegration over all wavelengths — integral
t_{\rmeff}=(\frac{1}{σ_{\rmsb}}\intf_{\rmp,surf,λ}dλ)^{1/4} · 2607.01316v1

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