Irradiance

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In radiometry, irradiance and spectral irradiance of a surface are the radiant flux per unit area received by that surface. The SI unit of irradiance is the watt per square metre (W/m2), while that of spectral irradiance is the watt per square metre per hertz (W·m−2·Hz−1) or the watt per square metre per metre (W·m−3)—commonly the watt per square metre per nanometre (W·m−2·nm−1)—, depending on whether the spectrum is taken as a function of frequency or of wavelength. The CGS unit erg per square centimeter per second (erg·cm−2·s−1) is often used in astronomy. Irradiance is often called intensity in branches of physics other than radiometry, but in radiometry this usage leads to confusion with radiant intensity.

Definitions[edit]

Irradiance[edit]

Irradiance of a surface, denoted Ee ("e" for "energetic", to avoid confusion with photometric quantities) and measured in W/m2, is given by

E_\mathrm{e} = \frac{\partial \Phi_\mathrm{e}}{\partial A},

where

  • ∂ is the partial derivative symbol;
  • Φe is the radiant flux received by that surface, measured in W;
  • A is the area of that surface, measured in m2.

Spectral irradiance[edit]

Irradiance of a surface per unit frequency, denoted Ee,ν and measured in W·m−2·Hz−1, is given by

E_{\mathrm{e},\nu} = \frac{\partial E_\mathrm{e}}{\partial \nu},

where ν is the frequency, measured in Hz.

Irradiance of a surface per unit wavelength, denoted Ee,λ and measured in W/m3 (commonly in W·m−2·nm−1), is given by

E_{\mathrm{e},\lambda} = \frac{\partial E_\mathrm{e}}{\partial \lambda},

where λ is the wavelength, measured in m (commonly in nm).

Alternative definition[edit]

Irradiance of a surface is also defined as the time-average of the component of the Poynting vector perpendicular to that surface:

E_\mathrm{e} = \langle \mathbf{S} \cdot \mathbf{\hat n} \rangle,

where

  • S is the Poynting vector;
  • \mathbf{\hat n} is the normal vector to that surface;

In a propagating sinusoidal linearly polarized electromagnetic plane wave, the Poynting vector always points in the direction of propagation while oscillating in magnitude. The irradiance of a surface perpendicular to the direction of propagation is then given by:[1]

E_\mathrm{e} = \frac{n}{2 \mu_0 \mathrm{c}} E_\mathrm{m}^2 = \frac{n \epsilon_0 \mathrm{c}}{2} E_\mathrm{m}^2,

where

This formula assumes that the magnetic susceptibility is negligible, i.e. that μr ≈ 1 where μr is the magnetic permeability of the propagation medium. This assumption is typically valid in transparent media in the optical frequency range.

Solar energy[edit]

The global irradiance on a horizontal surface on Earth consists of the direct irradiance Ee,dir and diffuse irradiance Ee,diff. On a tilted plane, there is another irradiance component, Ee,refl, which is the component that is reflected from the ground. The average ground reflection is about 20% of the global irradiance. Hence, the irradiance Ee on a tilted plane consists of three components:[2]

E_\mathrm{e} = E_{\mathrm{e},\mathrm{dir}} + E_{\mathrm{e},\mathrm{diff}} + E_{\mathrm{e},\mathrm{refl}}.

The integral of solar irradiance over a time period is called solar irradiation or solar exposure or insolation.[2][3]

SI radiometry units
Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol Symbol
Radiant energy Qe[nb 2] joule J ML2T−2 Energy received, emitted, reflected, or transmitted by a system in form of electromagnetic radiation.
Radiant energy density we joule per cubic metre J/m3 ML−1T−2 Radiant energy of a system per unit volume at a given location.
Radiant flux / Radiant power Φe[nb 2] watt W or J/s ML2T−3 Radiant energy of a system per unit time at a given time.
Spectral flux / Spectral power Φe,ν[nb 3]
or
Φe,λ[nb 4]
watt per hertz
or
watt per metre
W/Hz
or
W/m
ML2T−2
or
MLT−3
Radiant power of a system per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1.
Radiant intensity Ie,Ω[nb 5] watt per steradian W/sr ML2T−3 Radiant power of a system per unit solid angle around a given direction. It is a directional quantity.
Spectral intensity Ie,Ω,ν[nb 3]
or
Ie,Ω,λ[nb 4]
watt per steradian per hertz
or
watt per steradian per metre
W⋅sr−1⋅Hz−1
or
W⋅sr−1⋅m−1
ML2T−2
or
MLT−3
Radiant intensity of a system per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. It is a directional quantity.
Radiance Le,Ω[nb 5] watt per steradian per square metre W⋅sr−1⋅m−2 MT−3 Radiant power of a surface per unit solid angle around a given direction per unit projected area of that surface along that direction. It is a directional quantity. It is sometimes also confusingly called "intensity".
Spectral radiance Le,Ω,ν[nb 3]
or
Le,Ω,λ[nb 4]
watt per steradian per square metre per hertz
or
watt per steradian per square metre, per metre
W⋅sr−1⋅m−2⋅Hz−1
or
W⋅sr−1⋅m−3
MT−2
or
ML−1T−3
Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. It is a directional quantity. It is sometimes also confusingly called "spectral intensity".
Irradiance Ee[nb 2] watt per square metre W/m2 MT−3 Radiant power received by a surface per unit area. It is sometimes also confusingly called "intensity".
Spectral irradiance Ee,ν[nb 3]
or
Ee,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Irradiance of a surface per unit frequency or wavelength. The former is commonly measured in 10−22 W⋅m−2⋅Hz−1, known as solar flux unit, and the latter in W⋅m−2⋅nm−1.[nb 6] It is sometimes also confusingly called "spectral intensity".
Radiosity Je[nb 2] watt per square metre W/m2 MT−3 Radiant power leaving (emitted, reflected and transmitted by) a surface per unit area. It is sometimes also confusingly called "intensity".
Spectral radiosity Je,ν[nb 3]
or
Je,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. It is sometimes also confusingly called "spectral intensity".
Radiant exitance Me[nb 2] watt per square metre W/m2 MT−3 Radiant power emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. It is sometimes also confusingly called "intensity".
Spectral exitance Me,ν[nb 3]
or
Me,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. It is sometimes also confusingly called "spectral intensity".
Radiant exposure He joule per square metre J/m2 MT−2 Irradiance of a surface times exposure time. It is sometimes also called fluence.
See also: SI · Radiometry · Photometry
  1. ^ Standards organizations recommend that radiometric quantities should be denoted with a 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 Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek)—not to be confused with the suffix "v" (for "visual") indicating a photometric quantity.
  4. ^ a b c d e f Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek) to indicate a spectral concentration. Spectral functions of wavelength are indicated by "(λ)" in parentheses instead, for example in spectral transmittance, spectral reflectance and spectral responsivity.
  5. ^ a b The two directional quantities, radiant intensity and radiance, are denoted with suffix "Ω" (Greek) to indicate a directional concentration.
  6. ^ NOAA / Space Weather Prediction Center includes a definition of the solar flux unit (SFU).

See also[edit]

References[edit]

  1. ^ Griffiths, David J. (1999). Introduction to electrodynamics (3. ed., reprint. with corr. ed.). Upper Saddle River, NJ [u.a.]: Prentice-Hall. ISBN 0-13-805326-X. 
  2. ^ a b Quaschning, Volker (2003). "Technology fundamentals—The sun as an energy resource". Renewable Energy World 6 (5): 90–93. 
  3. ^ Liu, B. Y. H.; Jordan, R. C. (1960). "The interrelationship and characteristic distribution of direct, diffuse and total solar radiation". Solar Energy 4 (3): 1. doi:10.1016/0038-092X(60)90062-1.  edit