Physics:Radiant energy density

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

Mathematical definition

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

[math]\displaystyle{ w_\mathrm{e} = \frac{\partial Q_\mathrm{e}}{\partial V}, }[/math]

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, 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:

[math]\displaystyle{ E_\mathrm{e} = c w_\mathrm{e}, }[/math]

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:

[math]\displaystyle{ L_\mathrm{e} = \frac{c}{4\pi} w_\mathrm{e}. }[/math]

Radiant exitance through a small opening from such a cavity is:

[math]\displaystyle{ M_\mathrm{e} = \pi L_\mathrm{e} = \frac{c}{4} w_\mathrm{e}. }[/math]

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	Name	Symbol	Symbol
Radiant energy	Qe	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	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".
Spectral flux	Φe,ν	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	watt per metre	W/m	M⋅L⋅T−3
Radiant intensity	Ie,Ω	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	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	watt per steradian per metre	W⋅sr−1⋅m−1	M⋅L⋅T−3
Radiance	Le,Ω	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	Le,Ω,ν	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	watt per steradian per square metre, per metre	W⋅sr−1⋅m−3	M⋅L−1⋅T−3
Irradiance Flux density	Ee	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	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	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiosity	Je	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	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	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exitance	Me	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	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	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	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	joule per square metre, per metre	J/m3	M⋅L−1⋅T−2
Hemispherical emissivity	ε	N/A		1	Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity	εν  or ελ	N/A		1	Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity	εΩ	N/A		1	Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity	εΩ,ν  or εΩ,λ	N/A		1	Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance	A	N/A		1	Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance	Aν  or Aλ	N/A		1	Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance	AΩ	N/A		1	Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance	AΩ,ν  or AΩ,λ	N/A		1	Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectance	R	N/A		1	Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance	Rν  or Rλ	N/A		1	Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance	RΩ	N/A		1	Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance	RΩ,ν  or RΩ,λ	N/A		1	Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance	T	N/A		1	Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance	Tν  or Tλ	N/A		1	Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance	TΩ	N/A		1	Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance	TΩ,ν  or TΩ,λ	N/A		1	Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient	μ	reciprocal metre	m−1	L−1	Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient	μν  or μλ	reciprocal metre	m−1	L−1	Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient	μΩ	reciprocal metre	m−1	L−1	Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient	μΩ,ν  or μΩ,λ	reciprocal metre	m−1	L−1	Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
See also: SI · Radiometry · Photometry