Radiant energy density

In radiometry, radiant energy density is the radiant energy per unit volume.[1] The SI unit of radiant energy density is the joule per cubic metre (J/m3).

Mathematical definition

Radiant energy density, denoted we ("e" for "energetic", to avoid confusion with photometric quantities), is defined as[2]

where

Relation to other radiometric quantities

Because radiation always transmits the energy,[2] it is useful to wonder what the speed of the transmission is. If all the radiation at given location propagates in the same direction, then the radiant flux through a unit area perpendicular to the propagation direction is given by the irradiance:[2]

where c is the radiation propagation speed.

Contrarily if the radiation intensity is equal in all directions, like in a cavity in a thermodynamic equilibrium, then the energy transmission is best described by radiance:[3]

Radiant exitance through a small opening from such a cavity is:[4]

These relations can be used for example in the black-body radiation equation's derivation.

SI radiometry units

SI radiometry units
Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol
Radiant energy Qe[nb 2] joule J M⋅L2⋅T−2 Energy of electromagnetic radiation.
Radiant energy density we joule per cubic metre J/m3 M⋅L−1⋅T−2 Radiant energy per unit volume.
Radiant flux Φe[nb 2] watt W = J/s M⋅L2⋅T−3 Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power", and called luminosity in Astronomy.
Spectral flux Φe,ν[nb 3] watt per hertz W/Hz M⋅L2⋅T −2 Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm−1.
Φe,λ[nb 4] watt per metre W/m M⋅L⋅T−3
Radiant intensity Ie,Ω[nb 5] watt per steradian W/sr M⋅L2⋅T−3 Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity Ie,Ω,ν[nb 3] watt per steradian per hertz W⋅sr−1⋅Hz−1 M⋅L2⋅T−2 Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅nm−1. This is a directional quantity.
Ie,Ω,λ[nb 4] watt per steradian per metre W⋅sr−1⋅m−1 M⋅L⋅T−3
Radiance Le,Ω[nb 5] watt per steradian per square metre W⋅sr−1⋅m−2 M⋅T−3 Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity".
Spectral radiance
Specific intensity 		Le,Ω,ν[nb 3] watt per steradian per square metre per hertz W⋅sr−1⋅m−2⋅Hz−1 M⋅T−2 Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity. This is sometimes also confusingly called "spectral intensity".
Le,Ω,λ[nb 4] watt per steradian per square metre, per metre W⋅sr−1⋅m−3 M⋅L−1⋅T−3
Irradiance
Flux density 		Ee[nb 2] watt per square metre W/m2 M⋅T−3 Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral irradiance
Spectral flux density 		Ee,ν[nb 3] watt per square metre per hertz W⋅m−2⋅Hz−1 M⋅T−2 Irradiance of a surface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include jansky (1 Jy = 10−26 W⋅m−2⋅Hz−1) and solar flux unit (1 sfu = 10−22 W⋅m−2⋅Hz−1 = 104 Jy).
Ee,λ[nb 4] watt per square metre, per metre W/m3 M⋅L−1⋅T−3
Radiosity Je[nb 2] watt per square metre W/m2 M⋅T−3 Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral radiosity Je,ν[nb 3] watt per square metre per hertz W⋅m−2⋅Hz−1 M⋅T−2 Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. This is sometimes also confusingly called "spectral intensity".
Je,λ[nb 4] watt per square metre, per metre W/m3 M⋅L−1⋅T−3
Radiant exitance Me[nb 2] watt per square metre W/m2 M⋅T−3 Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity".
Spectral exitance Me,ν[nb 3] watt per square metre per hertz W⋅m−2⋅Hz−1 M⋅T−2 Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity".
Me,λ[nb 4] watt per square metre, per metre W/m3 M⋅L−1⋅T−3
Radiant exposure He joule per square metre J/m2 M⋅T−2 Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence".
Spectral exposure He,ν[nb 3] joule per square metre per hertz J⋅m−2⋅Hz−1 M⋅T−1 Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m−2⋅nm−1. This is sometimes also called "spectral fluence".
He,λ[nb 4] joule per square metre, per metre J/m3 M⋅L−1⋅T−2
See also:
  1. ^ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
  2. ^ a b c d e Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
  3. ^ a b c d e f g Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.)
  4. ^ a b c d e f g Spectral quantities given per unit wavelength are denoted with suffix "λ".
  5. ^ a b Directional quantities are denoted with suffix "Ω".

References

  1. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Radiant energy density". doi:10.1351/goldbook.goldbook.R05040
  2. ^ a b c Karel Rusňák. Přenos energie elektromagnetickým vlněním. Department of Physics, Faculty of Applied Sciences, University of West Bohemia. 2005-11. Visited 2013-10-06
  3. ^ Max Planck. The Theory of Heat Radiation. Equation 21. 1914.
  4. ^ Max Planck. The Theory of Heat Radiation. Equation 7. 1914.