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Radiant flux per unit area
In radiometry, radiant exitance or radiant emittance is the radiant flux emitted by a surface per unit area, whereas spectral exitance or spectral emittance is the radiant exitance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. This is the emitted component of radiosity. The SI unit of radiant exitance is the watt per square metre (W/m2), while that of spectral exitance in frequency is the watt per square metre per hertz (W·m−2·Hz−1) and that of spectral exitance in wavelength is the watt per square metre per metre (W·m−3)—commonly the watt per square metre per nanometre (W·m−2·nm−1). The CGS unit erg per square centimeter per second (erg·cm−2·s−1) is often used in astronomy. Radiant exitance is often called "intensity" in branches of physics other than radiometry, but in radiometry this usage leads to confusion with radiant intensity.
Mathematical definitions[edit]Radiant exitance of a surface, denoted Me ("e" for "energetic", to avoid confusion with photometric quantities), is defined as[1] M e = ∂ Φ e ∂ A , {\displaystyle M_{\mathrm {e} }={\frac {\partial \Phi _{\mathrm {e} }}{\partial A}},} where ∂ is the partial derivative symbol, Φe is the radiant flux emitted, and A is the surface area.
The radiant flux received by a surface is called irradiance.
The radiant exitance of a black surface, according to the Stefan–Boltzmann law, is equal to: M e ∘ = σ T 4 , {\displaystyle M_{\mathrm {e} }^{\circ }=\sigma T^{4},} where σ is the Stefan–Boltzmann constant, and T is the temperature of that surface. For a real surface, the radiant exitance is equal to: M e = ε M e ∘ = ε σ T 4 , {\displaystyle M_{\mathrm {e} }=\varepsilon M_{\mathrm {e} }^{\circ }=\varepsilon \sigma T^{4},} where ε is the emissivity of that surface.
Spectral exitance in frequency of a surface, denoted Me,ν, is defined as[1]
where ν is the frequency.
Spectral exitance in wavelength of a surface, denoted Me,λ, is defined as[1] M e , λ = ∂ M e ∂ λ , {\displaystyle M_{\mathrm {e} ,\lambda }={\frac {\partial M_{\mathrm {e} }}{\partial \lambda }},} where λ is the wavelength.
The spectral exitance of a black surface around a given frequency or wavelength, according to Lambert's cosine law and Planck's law, is equal to:
where h is the Planck constant, ν is the frequency, λ is the wavelength, k is the Boltzmann constant, c is the speed of light in the medium, T is the temperature of that surface. For a real surface, the spectral exitance is equal to: M e , ν = ε M e , ν ∘ = 2 π h ε ν 3 c 2 1 e h ν k T − 1 , M e , λ = ε M e , λ ∘ = 2 π h ε c 2 λ 5 1 e h c λ k T − 1 . {\displaystyle {\begin{aligned}M_{\mathrm {e} ,\nu }&=\varepsilon M_{\mathrm {e} ,\nu }^{\circ }={\frac {2\pi h\varepsilon \nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}},\\[8pt]M_{\mathrm {e} ,\lambda }&=\varepsilon M_{\mathrm {e} ,\lambda }^{\circ }={\frac {2\pi h\varepsilon c^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda kT}}-1}}.\end{aligned}}}
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