Editing Intensity (physics) (section)

==Mathematical description==
If a [[point source]] is radiating energy in all directions (producing a [[spherical wave]]), and no energy is absorbed or scattered by the medium, then the intensity decreases in proportion to the distance from the object squared. This is an example of the [[inverse-square law]].

Applying the law of [[conservation of energy]], if the net power emanating is constant,
<math display="block">P = \int \mathbf I\, \cdot d\mathbf A,</math>
where 
*{{mvar|P}} is the net power radiated;
*{{math|'''I'''}} is the intensity vector as a function of position;
*the magnitude {{mvar|{{abs|I}}}} is the intensity as a function of position;
*{{math|''d'''''A'''}} is a [[differential element]] of a closed surface that contains the source.

If one integrates a uniform intensity, {{math|1={{abs|''I''}} = const.}}, over a surface that is perpendicular to the intensity vector, for instance over a sphere centered around the point source, the equation becomes
<math display="block">P = |I| \cdot A_\mathrm{surf} = |I| \cdot 4\pi r^2,</math>
where 
*{{mvar|{{abs|I}}}} is the intensity at the surface of the sphere;
*{{mvar|r}} is the radius of the sphere;
*<math>A_\mathrm{surf} = 4\pi r^2 </math> is the expression for the surface area of a sphere.

Solving for {{mvar|{{abs|I}}}} gives
<math display="block">|I| = \frac{P}{A_\mathrm{surf}} = \frac{P}{4\pi r^2}. </math>

If the medium is damped, then the intensity drops off more quickly than the above equation suggests.

Anything that can transmit energy can have an intensity associated with it. For a monochromatic propagating electromagnetic wave, such as a [[plane wave]] or a [[Gaussian beam]], if {{mvar|E}} is the [[complex amplitude]] of the [[electric field]], then the time-averaged [[energy density]] of the wave, travelling in a non-magnetic material, is given by:
<math display="block">\left\langle U \right \rangle = \frac{n^2 \varepsilon_0}{2} |E|^2,</math>
and the local intensity is obtained by multiplying this expression by the wave velocity, {{tmath|\tfrac{\mathrm c}{n} \! :}}
<math display="block">I = \frac{\mathrm{c} n \varepsilon_0}{2} |E|^2,</math>
where
*{{mvar|n}} is the [[refractive index]];
*{{math|c}} is the [[speed of light]] in [[vacuum]];
*{{math|''&epsilon;''{{sub|0}}}} is the [[vacuum permittivity]].

For non-monochromatic waves, the intensity contributions of different spectral components can simply be added. The treatment above does not hold for arbitrary electromagnetic fields. For example, an [[evanescent wave]] may have a finite electrical amplitude while not transferring any power. The intensity should then be defined as the magnitude of the [[Poynting vector]].<ref>{{cite encyclopedia |encyclopedia=Encyclopedia of Laser Physics and Technology |title=Optical Intensity |url=https://www.rp-photonics.com/optical_intensity.html |publisher=RP Photonics |first=Rüdiger |last=Paschotta}}</ref>