# 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/m$^3$).

## Mathematical definition

Radiant energy density, denoted w$_e$ ("e" for "energetic", to avoid confusion with photometric quantities), is defined as
    $w_{\mathrm {e} }={\frac {\partial Q_{\mathrm {e} }}{\partial V}},$
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, 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:
    $E_{\mathrm {e} }=cw_{\mathrm {e} },$
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:
    $L_{\mathrm {e} }={\frac {c}{4\pi }}w_{\mathrm {e} }.$
Radiant exitance through a small opening from such a cavity is:
    $M_{\mathrm {e} }=\pi L_{\mathrm {e} }={\frac {c}{4}}w_{\mathrm {e} }.$
These relations can be used for example in the black-body radiation equation's derivation.

## SI radiometry units

SI radiometry unitsvte :
| Quantity | Quantity | Unit | Unit | Dimension | Notes |
| Name | Symbol | Name | Symbol | Dimension | Notes |
| Radiant energy | Q$_e$ | joule | J | M⋅L$^2$⋅T$^{−2}$ | Energy of electromagnetic radiation. |
| Radiant energy density | w$_e$ | joule per cubic metre | J/m$^3$ | M⋅L$^{−1}$⋅T$^{−2}$ | Radiant energy per unit volume. |
| Radiant flux | Φ$_e$ | 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 | Φ$_{e,ν}$ | 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$^{−1}$. |
| Spectral flux | Φ$_{e,λ}$ | watt per metre | W/m | M⋅L⋅T$^{−3}$ | Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm$^{−1}$. |
| Radiant intensity | I$_{e,Ω}$ | 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 | I$_{e,Ω,ν}$ | 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. |
| Spectral intensity | I$_{e,Ω,λ}$ | watt per steradian per metre | W⋅sr$^{−1}$⋅m$^{−1}$ | M⋅L⋅T$^{−3}$ | Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr$^{−1}$⋅nm$^{−1}$. This is a directional quantity. |
| Radiance | L$_{e,Ω}$ | 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 Specific intensity | L$_{e,Ω,ν}$ | 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". |
| Spectral radiance Specific intensity | L$_{e,Ω,λ}$ | watt per steradian per square metre, per metre | W⋅sr$^{−1}$⋅m$^{−3}$ | M⋅L$^{−1}$⋅T$^{−3}$ | 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". |
| Irradiance Flux density | E$_e$ | 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 | E$_{e,ν}$ | 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$^{−1}$ = 10$^4$ Jy). |
| Spectral irradiance Spectral flux density | E$_{e,λ}$ | watt per square metre, per metre | W/m$^3$ | M⋅L$^{−1}$⋅T$^{−3}$ | 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). |
| Radiosity | J$_e$ | 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 | J$_{e,ν}$ | 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". |
| Spectral radiosity | J$_{e,λ}$ | watt per square metre, per metre | W/m$^3$ | M⋅L$^{−1}$⋅T$^{−3}$ | 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". |
| Radiant exitance | M$_e$ | 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 | M$_{e,ν}$ | 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". |
| Spectral exitance | M$_{e,λ}$ | watt per square metre, per metre | W/m$^3$ | M⋅L$^{−1}$⋅T$^{−3}$ | 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". |
| Radiant exposure | H$_e$ | 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 | H$_{e,ν}$ | 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". |
| Spectral exposure | H$_{e,λ}$ | joule per square metre, per metre | J/m$^3$ | M⋅L$^{−1}$⋅T$^{−2}$ | 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". |
| See also: SIRadiometryPhotometry ||||||
* ↑ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
* 1 2 3 4 5 Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
* 1 2 3 4 5 6 7 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.)
* 1 2 3 4 5 6 7 Spectral quantities given per unit wavelength are denoted with suffix "λ".
* 1 2 Directional quantities are denoted with suffix "Ω".