Thermal radiation
Thermal radiation is electromagnetic radiation emitted from the surface of an object which is due to the object's temperature. Infrared radiation from a common household radiator or electric heater is an example of thermal radiation, as is the light emitted by a glowing incandescent light bulb. Thermal radiation is generated when heat from the movement of charged particles within atoms is converted to electromagnetic radiation.
If the object is a black body, the radiation is termed black-body radiation.  The emitted wave frequency of the thermal radiation is a probability distribution depending only on temperature, and for a genuine black body is given by Plancks law of radiation. Wien's law gives the most likely frequency of the emitted radiation, and the StefanBoltzmann law gives the heat intensity.

# Properties
There are three main properties that characterize thermal radiation:
- Thermal radiation, even at a single temperature, occurs at a wide range of frequencies. How much of each frequency is given by Plancks law of radiation (for idealized materials).  This is shown by the curves in the diagram at right.
- The main frequency (or color) of the emitted radiation increases as the temperature increases. For example, a red hot object radiates most in the long wavelengths of the visible band, which is why it appears red. If it heats up further, the main frequency shifts to the middle of the visible band, and the spread of frequencies mentioned in the first point make it appear white. We then say the object is white hot. This is Wien's displacement law.  In the diagram the peak value for each curve moves to the left as the temperature increases.
- The total amount of radiation, of all frequencies, goes up very fast as the temperature rises (it grows as T4, where T is the absolute temperature of the body). An object at the temperature of a kitchen oven (about twice room temperature in absolute terms - 600 K vs. 300 K) radiates 16 times as much power per unit area. An object the temperature of the filament in an incandescent bulb (roughly 3000 K, or 10 times room temperature) radiates 10,000 times as much per unit area. Mathematically, the total power radiated rises as the fourth power of the absolute temperature, the StefanBoltzmann law.  In the plot, the area under each curve rises rapidly as the temperature increases.
These properties apply if the distances considered are much larger than the wavelengths contributing to the spectrum (around 10 micrometres at 300 K). Indeed, thermal radiation here takes only travelling waves into account. A more sophisticated framework involving eletromagnetics has to be used for lower distances (near-field thermal radiation).

# Interchange of energy
Thermal radiation is an important concept in thermodynamics as it is partially responsible for heat exchange between objects, as warmer bodies radiate more heat than colder ones. (Other factors are convection and conduction.)  The interplay of energy exchange is characterized by the following equation:
Here, \alpha \, represents spectral absorption factor, \rho \, spectral reflection factor and \tau \, spectral transmission factor. All these elements depend also on the wavelength \lambda\,. The spectral absorption factor is equal to the emissivity \epsilon \,; this relation is known as Kirchhoff's law of thermal radiation. An object is called a black body if, for all frequencies, the following formula applies:
In a practical situation and room-temperature setting, humans lose considerable energy due to thermal radiation.  However, the energy lost by emitting infrared heat is partially regained by absorbing the heat of surrounding objects (the remainder resulting from generated heat through metabolism).  Human skin has an emissivity of very close to 1.0 .  Using the formulas below then shows a human being, roughly 2 square meter in area, and about 307 kelvins in temperature, continuously radiates about 1000 watts.  However, if people are indoors, surrounded by surfaces at 296 K, they receive back about 900 watts from the wall, ceiling, and other surroundings, so the net loss is only about 100 watts.  These heat transfer estimates are highly dependent on extrinsic variables, such as wearing clothes (decreasing total thermal "circuit" conductivity, therefore reducing total output heat flux.)  Only truly "grey" systems (relative equivalent emissivity/absorptivity and no directional transmissivity dependence in all control volume bodies considered.) can achieve reasonable irradiative flux estimates through the Stefan-Boltzmann law.  However, encountering this "ideally calculable" situation is virtually impossible (although common engineering procedures surrender the dependency of these unknown variables and "assume" this to be the case).  Optimistically, these "grey" approximations will get you close to real solutions, as most divergence from Stefan-Boltzmann solutions is small (especially in most lab controlled environments).
If objects appear white (reflective in the visual spectrum), they are not necessarily equally reflective (and thus non-emissive) in the thermal infrared; e. g. most household radiators are painted white despite the fact that they have to be good thermal radiators.  Acrylic and urethane based white paints have 93% blackbody radiation efficiency at room temperature (meaning the term "black body" does not always correspond to the visually perceived color of an object).  These materials that do not follow the "black color = high emissivity/absorptivity" caveat will most likely have functional spectral emissivity/absorptivity dependence.
Calculation of radiative heat transfer between groups of object, including a 'cavity' or 'surroundings' requires solution of a set of simultaneous equations using the Radiosity method. In these calculations, the geometrical configuration of the problem is distilled to a set of numbers called view factors, which give the proportion of radiation leaving any given surface that hits another specific surface. These calculations are important in the fields of solar thermal energy, boiler and furnace design and raytraced computer graphics.

# Formula
Thermal radiation power of a black body per unit of area, unit of solid angle and unit of frequency \nu is given by Planck's law as:
This formula mathematically follows from calculation of spectral distribution of energy in quantized electromagnetic field which is in complete thermal equilibrium with the radiating object.
Integrating the above equation over \nu the power output given by the StefanBoltzmann law is obtained, as:
Further, the wavelength \lambda \,, for which the emission intensity is highest, is given by Wien's Law as:
For surfaces which are not black bodies, one has to consider the (generally frequency dependent) emissivity correction factor \epsilon(\upsilon). This correction factor has to be multiplied with the radiation spectrum formula before integration. The resulting formula for the power output can be written in a way that contains a temperature dependent correction factor which is (somewhat confusingly) often called \epsilon as well:

## Constants
Definitions of constants used in the above equations:

## Variables
Definitions of variables, with example values: