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/m^3).

Mathematical definition

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

${\displaystyle w_{\mathrm {e} }={\frac {\partial Q_{\mathrm {e} }}{\partial V}},}$

where

• ∂ is the partial derivative symbol;
• Q_e is the radiant energy;
• V is the volume.

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]

${\displaystyle E_{\mathrm {e} }=cw_{\mathrm {e} },}$

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]

${\displaystyle L_{\mathrm {e} }={\frac {c}{4\pi }}w_{\mathrm {e} }.}$

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

${\displaystyle M_{\mathrm {e} }=\pi L_{\mathrm {e} }={\frac {c}{4}}w_{\mathrm {e} }.}$

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

SI radiometry units

SI radiometry units

• v
• t
• e

Quantity		Unit		Dimension	Notes
Name	Symbol[nb 1]	Name	Symbol
Radiant energy	Qe[nb 2]	joule	J	M⋅L^2⋅T^−2	Energy of electromagnetic radiation.
Radiant energy density	we	joule per cubic metre	J/m^3	M⋅L^−1⋅T^−2	Radiant energy per unit volume.
Radiant flux	Φe[nb 2]	watt	W = J/s	M⋅L^2⋅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⋅L^2⋅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⋅L^2⋅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⋅L^2⋅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/m^2	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 = 10^4 Jy).
	Ee,λ[nb 4]	watt per square metre, per metre	W/m^3	M⋅L^−1⋅T^−3
Radiosity	Je[nb 2]	watt per square metre	W/m^2	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/m^3	M⋅L^−1⋅T^−3
Radiant exitance	Me[nb 2]	watt per square metre	W/m^2	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/m^3	M⋅L^−1⋅T^−3
Radiant exposure	He	joule per square metre	J/m^2	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/m^3	M⋅L^−1⋅T^−2
See also: SI Radiometry Photometry

1. Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
2. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
3. 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. Spectral quantities given per unit wavelength are denoted with suffix "λ".
5. 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. 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.