What is Stefan-boltzmann: Definition and 31 Discussions

The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time


{\displaystyle j^{\star }}
(also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T:






{\displaystyle j^{\star }=\sigma T^{4}.}
The constant of proportionality σ, called the Stefan–Boltzmann constant, is derived from other known physical constants. Since 2019, the value of the constant is






















{\displaystyle \sigma ={\frac {2\pi ^{5}k^{4}}{15c^{2}h^{3}}}=5.670374\ldots \times 10^{-8}\,\mathrm {W\,m^{-2}\,K^{-4}} ,}
where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. The radiance from a specified angle of view (watts per square metre per steradian) is given by









{\displaystyle L={\frac {j^{\star }}{\pi }}={\frac {\sigma }{\pi }}T^{4}.}
A body that does not absorb all incident radiation (sometimes known as a grey body) emits less total energy than a black body and is characterized by an emissivity,


{\displaystyle \varepsilon <1}






{\displaystyle j^{\star }=\varepsilon \sigma T^{4}.}
The radiant emittance


{\displaystyle j^{\star }}
has dimensions of energy flux (energy per unit time per unit area), and the SI units of measure are joules per second per square metre, or equivalently, watts per square metre. The SI unit for absolute temperature T is the kelvin.


{\displaystyle \varepsilon }
is the emissivity of the grey body; if it is a perfect blackbody,


{\displaystyle \varepsilon =1}
. In the still more general (and realistic) case, the emissivity depends on the wavelength,


{\displaystyle \varepsilon =\varepsilon (\lambda )}
To find the total power radiated from an object, multiply by its surface area,


{\displaystyle A}







{\displaystyle P=Aj^{\star }=A\varepsilon \sigma T^{4}.}
Wavelength- and subwavelength-scale particles, metamaterials, and other nanostructures are not subject to ray-optical limits and may be designed to exceed the Stefan–Boltzmann law.

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  1. jackdale

    The Second Law of Thermodynamics and the Stefan-Boltzmann Law

    I have a question about the Second Law of Thermodynamics and the Stefan-Boltzmann law. These quotes are from http://www.ces.fau.edu/nasa/module-2/correlation-between-temperature-and-radiation.php “The Stefan-Boltzmann law, a fundamental law of physics, explains the relationship between an...
  2. J

    B Understanding the Stefan-Boltzmann Law (when the surroundings are hotter)

    1.If so what would the law mean if ##T_{surroundings}>T##? 2. Stefan-Boltzmann Law is formulated as ##H = A\sigma T^4## where ##H## is the energy emitted per unit time, ##A## is the area of the object, ##T## is the absolute temperature of the object and (3.) I am unclear about whether...
  3. T

    A Stefan-Boltzmann Equation question (qualitative)

    Hi, In the Stefan-Boltzmann equation for radiation heat transfer, there exists expected parameters of any model (area, constants, etc.). However, the temperature is raised to the fourth power. Can someone explain why? I get that it could just be because it has been "experimentally observed."...
  4. D

    B What would happen to Earth's temperatures without an atmosphere?

    Hi everyone I've read that if Earth had no atmosphere it'd be completely frozen over with temperatures around 255K. Why is this the case when the moon has daytime temperatures of 373K (and night time temperatures of 100K)? Why wouldn't the oceans thaw during the day? Are they just too big...
  5. K. Doc Holiday

    B The Stefan-Boltzmann Law and Sunspots

    The Stefan-Boltzmann Law Question: Is this law proven to be true? If so, then what about sunspots? The surface of the sun is roughly double the temperature of an umbra which means it should be 16 times brighter according to this law. It isn't. The luminosity of the photosphere is 10,000...
  6. A

    I Stefan-Boltzmann Law: IvdvcosΘdw Explained

    I am having a doubt of why is the radiative flux IvdvcosΘdw in equilibrium integrated for a spherical black body only from 0 to pi/2 not pi(e.g. For the entire surface for which it is zero), v=frequency dw= solid angle
  7. V

    How much energy does the Earth radiate to space?

    Homework Statement How much energy is radiated from the Earth to space? AE = Area of Earth εE = Emissivity of Earth TE = Temperature of Earth Tsp = Temperature of Space Tsp = 2.7 K (off of NASA web page) Homework Equations P = AE εE σ (TE4 - Tsp4) The Attempt at a Solution Isn't the...
  8. PytrTchaikovsky

    Thermal Radiation: Calc Total Emitted Joules in Certain Temp Range

    Dear forum I am working with thermal radiation. This is the specific formula: P = σ ⋅ A ⋅ T4 P = emitted effect (W, J/s) σ = Stefan-Boltzmann constant (5,67 ⋅ 10-8) A = area of object (m2) T = temperature of object (K) How can I get to know the...
  9. 2

    Stefan-Boltzmann law, luminosity, brightness and magnitude?

    From what I understand, in the equation P=\sigma AT^4, P is the power output of the star which is the energy radiated per second in EM radiation of all frequencies, and I think luminosity is also defined as the energy radiated per second in EM radiation of all frequencies. Therefore luminosity...
  10. G

    Is Manipulating the Stefan-Boltzmann Law with Metamaterials Possible?

    So I saw some papers on arXiv that explore the notion of playing around with the Stefan-Boltzmann law using metamaterials: http://arxiv.org/abs/1109.5444 http://arxiv.org/abs/1411.1360 I know enough physics that it looks rather interesting, but I don't know enough math to judge how credible it...
  11. M

    Questions About Stefan-Boltzmann Law

    Hi, I have a few questions about Stefan-Boltzmann's law. Let's say that the temerature of an object increases for 2 degrees Celsius, that means that the emmited energy of a surface of that object will increase 16 times. Is this happening because the number of photons is increased 16 times or is...
  12. Q

    The Stefan-Boltzmann Law on Pure Proton Plasma

    Hi guys. I had a question concerning a pure proton plasma with a given temperature and its loss of energy due to the Stefan-Boltzmann law. From my understanding, the Stefan-Boltzmann law applies to objects that contain electrons. I don't think that's true but I would like to know if it is. If...
  13. T

    Calculating Heat Transfer (Thermal Radiation) to a Shipping Container

    Hello, I am trying to calculate the thermal radiation heat transfer to a shipping container which would be painted white. The purpose is to size an air conditioner for equipment, not living conditions. I have already calculated heat from the other sources. The desired internal temperature is...
  14. T

    Derive Stefan-Boltzmann law from Wien’s law.

    Homework Statement Derive Stefan-Boltzmann law from Wien’s law Hint: you can use (without proof) R(T) = -∞∫∞ R(λ,T)dλ , ρ(λ,T)=(4/c)*R(λ,T) Homework Equations wien's law: p(λ,T) = f(λ,T)/^5 Stefan's law : R(T) = σT^4 The Attempt at a Solution Honestly I am having trouble just...
  15. P

    Derivation of Stefan-Boltzmann Law from Wien's Law

    Homework Statement Derive Stefan-Boltzmann Law from Wien's Law. Hint: You can use (without proof) R(T)=∫(-∞ to ∞) R(λ,T)dλ, p(λ,T)= 4/c R(λ,T). Homework Equations Stefan-Boltzmann Law:P=AσT^4 Wien's Law: λmax=(2.898*10^-3 m*K)/T. The Attempt at a Solution Let λmax=(2.898*10^-3...
  16. M

    Use Planck’s radiation law to derive the Stefan-Boltzmann

    Homework Statement a) Use Planck’s radiation law to derive the Stefan-Boltzmann law for the case of zero background temperature (i.e., T0 = 0). Homework Equations http://csep10.phys.utk.edu/astr162/lect/light/plancklaw.gif The Attempt at a Solution So I know we are suppose to...
  17. L

    Heat transfer stefan-Boltzmann

    Homework Statement Determine the temperature of the filament of a 60watt lamp, if the filament has an area of 10mm^2 and an absorptivity of 0.35. Assume stefan-Boltzmann constant is 5.67*10^-8 W/(m.K^4). Homework Equations The Attempt at a Solution The equation I have attempted...
  18. B

    Stephan-Boltzmann Law Help for Science Fair Project - Needed Now!

    Hello, I am in DIRE need of help with my science fair project and am completely stuck. All/most relevant information should be able to be found here: http://sciencebuddies.org/science-fair-projects/project_ideas/Phys_p073.shtml?fave=no&isb=c2lkOjEsaWE6UGh5cyxwOjUscmlkOjEyNDE1MzMw&from=TSW 1. I...
  19. S

    The Stefan-Boltzmann Law Demonstrator Device

    Hello Physics Forum! There's an interesting experiment to verify the Stefan-Boltzmann law using an incandescent lamp. Which treats the filament as an approximate blackbody, approximates that the input power fully goes to the radiating channel (i.e. no conduction or gas loss), measures the...
  20. J

    Stefan-boltzmann solved for temperature?

    My physics is extremely rusty (I was bored to death back in school and had no practical application of the stuff to make it interesting and/or truly sink it.) Now, I have an application, and while I have a long way to go, it is pleasurable to be comprehending things as I go. To a point, and...
  21. L

    Stefan-Boltzmann law question (astronomy)

    Homework Statement in about 6X10^7 y from now, the sun will be a red giant star with a temperature 1/2 its present value and a radius 100 times larger. a) what will be the luminosity of the red giant sun be compared to its current value? b) assume that the new luminosity gives you the...
  22. T

    Derivation of Stefan-Boltzmann law from Thermodynamics

    I've been trying to derive the Stefan-Boltzmann law using thermodynamics, and have resorted to looking up the derivation in the feynman lectures and on wikipedia, and I'm confused by both. I think the wikipedia derivation is the best one to look at, it's here...
  23. O

    Derive Stefan-Boltzmann Law from Planck Distribution for blackbody radiation

    Homework Statement Starting with the Planck distribution R(\lambda,T) for blackbody radiation. (a) Derive the blackbody Stefan-Boltzmann law (ie total flux is proportional to T4) by integrating the above expression over all wavelengths. Thus show that R(T) =...
  24. D

    Integral in Stefan-Boltzmann law

    Hi! I'm wondering if anybody can recommend me a book where it's explained how to solve (analytically) integral that appears in Stefan-Boltzmann's law: \int_0^\infty \frac{x^n}{(e^x-1)^m}dx Thanx!
  25. S

    Radiation - solving for Q - Stefan-Boltzmann law

    Homework Statement A person is standing outdoors in the shade where the temperature is 27°C. (a) What is the radiant energy absorbed per second by his head when it is covered with hair? The surface area of the hair (assumed to be flat) is 160 cm2 and its emissivity is 0.85. (b) What...
  26. L

    Why Can't Stefan-Boltzmann Law Be Explained in Classical Mechanics?

    I've looked all over the internet, and can't find a decent explanation. Could someone please explain why the stefan-boltzmann radiation law of I = oT^4 be explained in classical mechanics rather then quantum? It's urgent, sorry to be pushy. Cheers.
  27. ideasrule

    Stefan-Boltzmann Lab: Exploring Exponent of R vs T

    Homework Statement A few weeks ago I did an experiment that involved empirically determining the exponent on the Stefan-Boltzmann law. I used a light bulb and measured the voltage and current across it for different voltages. Since P=k*Tn and R=cT (i.e. resistance of the filament is...
  28. F

    What are the conditions of a ideal Stefan-Boltzmann gas?

    Does anybody know the conditions of a ideal Stefan Boltzmann gas? And the EoS? thanks a lot florian
  29. J

    Linearizing Stefan-Boltzmann equation

    Homework Statement I want to linearize both the Stefan-Boltzmann equation and the adiabatic relation to draw a relationship between δL/L(0) and δR/R(0). Homework Equations Stefan-Boltzmann equation in the form of L=4πσ(R^2)(T^4) Adiabatic relation TV^(γ-1) = constant The Attempt at...
  30. O

    Proving Wien's Law Includes Stefan-Boltzmann & Wien's Displacement Laws

    Hi, I'm supposed to prove that Wien's Law: P(\lambda,T) = \frac{f(\lambda T)}{\lambda^5} includes Stefan-Botlzmann's Law R(T) = \sigma T^4 and Wien's Displacement Law: \lambda_{max} T = b For Wien's Displacement Law: I know that I would have to find when P(\lambda ,T) graphed against...
  31. M

    Understand Stefan-Boltzmann Law: Where Does the T^4 Come From?

    I've been reading about the Stefan-Boltzmann law, but there is one thing that I don't understand. Why is it T^4? I can't think of anywhere that the 4 is coming from and can't find anything about this with Google searches.