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## Mathematical definitions

Irradiance of a surface, denoted Ee ("e" for "energetic", to avoid confusion with photometric quantities), is defined as E_\mathrm{e} = \frac{\partial \Phi_\mathrm{e}}{\partial A}, where
If we want to talk about the radiant flux emitted by a surface, we speak of radiant exitance.

Spectral irradiance in frequency of a surface, denoted Ee,ν, is defined as E_{\mathrm{e},\nu} = \frac{\partial E_\mathrm{e}}{\partial \nu}, where ν is the frequency. Spectral irradiance in wavelength of a surface, denoted Ee,λ, is defined as E_{\mathrm{e},\lambda} = \frac{\partial E_\mathrm{e}}{\partial \lambda}, where λ is the wavelength.

## Property

Irradiance of a surface is also, according to the definition of radiant flux, equal to the time-average of the component of the Poynting vector perpendicular to the surface: E_\mathrm{e} = \langle|\mathbf{S}|\rangle \cos \alpha, where
• < • > is the time-average;
• S is the Poynting vector;
• α is the angle between a unit vector normal to the surface and S.
For a propagating sinusoidal linearly polarized electromagnetic plane wave, the Poynting vector always points to the direction of propagation while oscillating in magnitude. The irradiance of a surface is then given by E_\mathrm{e} = \frac{n}{2 \mu_0 \mathrm{c}} E_\mathrm{m}^2 \cos \alpha = \frac{n \varepsilon_0 \mathrm{c}}{2} E_\mathrm{m}^2 \cos \alpha, 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

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: 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 exposure" or " insolation".