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In physics and many other areas of science and engineering the intensity or flux of radiant energy is the power transferred per unit area, where the area is measured on the plane perpendicular to the direction of propagation of the energy.^[a] In the SI system, it has units watts per square metre (W/m²), or kg⋅s⁻³ in base units. Intensity is used most frequently with waves such as acoustic waves (sound), matter waves such as electrons in electron microscopes, and electromagnetic waves such as light or radio waves, in which case the average power transfer over one period of the wave is used. Intensity can be applied to other circumstances where energy is transferred. For example, one could calculate the intensity of the kinetic energy carried by drops of water from a garden sprinkler.

The word "intensity" as used here is not synonymous with "strength", "amplitude", "magnitude", or "level", as it sometimes is in colloquial speech.

Intensity can be found by taking the energy density (energy per unit volume) at a point in space and multiplying it by the velocity at which the energy is moving. The resulting vector has the units of power divided by area (i.e., surface power density). The intensity of a wave is proportional to the square of its amplitude. For example, the intensity of an electromagnetic wave is proportional to the square of the wave's electric field amplitude.

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, $P=\int \mathbf {I} \,\cdot d\mathbf {A} ,$ where

$P$ is the net power radiated;
$I$ is the intensity vector as a function of position;
the magnitude $| I |$ is the intensity as a function of position;
$d A$ is a differential element of a closed surface that contains the source.

If one integrates a uniform intensity, $| I | = const.$ , over a surface that is perpendicular to the intensity vector, for instance over a sphere centered around the point source, the equation becomes $P=|I|\cdot A_{\mathrm {surf} }=|I|\cdot 4\pi r^{2},$ where

$| I |$ is the intensity at the surface of the sphere;
$r$ is the radius of the sphere;
$A_{\mathrm {surf} }=4\pi r^{2}$ is the expression for the surface area of a sphere.

Solving for $| I |$ gives $|I|={\frac {P}{A_{\mathrm {surf} }}}={\frac {P}{4\pi r^{2}}}.$

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 $E$ 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: $\left\langle U\right\rangle ={\frac {n^{2}\varepsilon _{0}}{2}}|E|^{2},$ and the local intensity is obtained by multiplying this expression by the wave velocity, ⁠ ${\tfrac {\mathrm {c} }{n}}\!:$ ⁠ $I={\frac {\mathrm {c} n\varepsilon _{0}}{2}}|E|^{2},$ where

$n$ is the refractive index;
$c$ is the speed of light in vacuum;
$ε 0$ 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.^[1]

Electron beams

For electron beams, intensity is the probability of electrons reaching some particular position on a detector (e.g. a charge-coupled device^[2]) which is used to produce images that are interpreted in terms of both microstructure of inorganic or biological materials, as well as atomic scale structure.^[3] The map of the intensity of scattered electrons or x-rays as a function of direction is also extensively used in crystallography.^[3]^[4]

Alternative definitions

In photometry and radiometry intensity has a different meaning: it is the luminous or radiant power per unit solid angle. This can cause confusion in optics, where intensity can mean any of radiant intensity, luminous intensity or irradiance, depending on the background of the person using the term. Radiance is also sometimes called intensity, especially by astronomers and astrophysicists, and in heat transfer.

Footnotes

^ The terms intensity and flux have multiple, inconsistent, definitions in physics and related fields. This article covers the concept of power per unit area, whatever one calls it. In radiometry the terms intensity and flux have different meanings, not covered here.

References

^ Paschotta, Rüdiger. "Optical Intensity". Encyclopedia of Laser Physics and Technology. RP Photonics.
^ Spence, J. C. H.; Zuo, J. M. (1988-09-01). "Large dynamic range, parallel detection system for electron diffraction and imaging". Review of Scientific Instruments. 59 (9): 2102–2105. doi:10.1063/1.1140039. ISSN 0034-6748.
^ ^a ^b Cowley, J. M. (1995). Diffraction physics. North Holland personal library (3rd ed.). Amsterdam: Elsevier. ISBN 978-0-444-82218-5.
^ Cullity, B. D.; Stock, Stuart R. (2001). Elements of X-ray diffraction (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 978-0-201-61091-8.

[1] The terms intensity and flux have multiple, inconsistent, definitions in physics and related fields. This article covers the concept of power per unit area, whatever one calls it. In radiometry the terms intensity and flux have different meanings, not covered here.

[2] Paschotta, Rüdiger. "Optical Intensity". Encyclopedia of Laser Physics and Technology. RP Photonics.

[3] Spence, J. C. H.; Zuo, J. M. (1988-09-01). "Large dynamic range, parallel detection system for electron diffraction and imaging". Review of Scientific Instruments. 59 (9): 2102–2105. doi:10.1063/1.1140039. ISSN 0034-6748.

[:0-4] Cowley, J. M. (1995). Diffraction physics. North Holland personal library (3rd ed.). Amsterdam: Elsevier. ISBN 978-0-444-82218-5.

[5] Cullity, B. D.; Stock, Stuart R. (2001). Elements of X-ray diffraction (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 978-0-201-61091-8.

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@@ Line 1: / Line 1: @@
+{{Short description|Power transferred per unit area}}
 {{Other uses|Intensity (disambiguation)}}
-In the [[physics]] language, the '''intensity''' or '''flux''' of [[radiant energy]] is the [[Power (physics)|power]] transferred per unit [[area]] in a direction perpendicular to that area. In the [[SI]] system, it has units [[watt]]s per [[square metre]] (W/m<sup>2</sup>), or [[kilogram|kg]]⋅[[second|s]]<sup>−3</sup> in [[SI base unit|base units]]. Intensity is used most frequently with [[wave]]s such as acoustic waves ([[sound]]) or [[electromagnetic wave]]s such as [[light]] or [[radio wave]]s, in which case the [[time averaging|''average'']] power transfer over one [[Period (physics)|period]] of the wave is used. ''Intensity'' can be applied to other circumstances where energy is transferred. For example, one could calculate the intensity of the [[kinetic energy]] carried by drops of water from a [[garden sprinkler]].
+In [[physics]] and many other areas of science and engineering the '''intensity''' or '''flux''' of [[radiant energy]] is the [[Power (physics)|power]] transferred per unit [[area]], where the area is measured on the plane perpendicular to the direction of propagation of the energy.{{efn|The terms ''intensity'' and ''flux'' have multiple, inconsistent, definitions in physics and related fields. This article covers the concept of power per unit area, whatever one calls it. In [[radiometry]] the terms ''intensity'' and ''flux'' have different meanings, not covered here.}} In the [[SI]] system, it has units [[watt]]s per [[square metre]] (W/m<sup>2</sup>), or [[kilogram|kg]]⋅[[second|s]]<sup>−3</sup> in [[SI base unit|base units]]. Intensity is used most frequently with [[wave]]s such as acoustic waves ([[sound]]), [[matter wave]]s such as electrons in [[electron microscope]]s, and [[electromagnetic wave]]s such as [[light]] or [[radio wave]]s, in which case the [[time averaging|''average'']] power transfer over one [[Period (physics)|period]] of the wave is used. ''Intensity'' can be applied to other circumstances where energy is transferred. For example, one could calculate the intensity of the [[kinetic energy]] carried by drops of water from a [[garden sprinkler]].
 The word "intensity" as used here is not synonymous with "[[wikt:strength|strength]]", "[[wikt:amplitude|amplitude]]", "[[wikt:magnitude|magnitude]]", or "[[wikt:level|level]]", as it sometimes is in colloquial speech.
-Intensity can be found by taking the [[energy density]] (energy per unit volume) at a point in space and multiplying it by the [[velocity]] at which the energy is moving. The resulting [[Vector (geometry)|vector]] has the units of power divided by area (i.e., [[surface power density]]).
+Intensity can be found by taking the [[energy density]] (energy per unit volume) at a point in space and multiplying it by the [[velocity]] at which the energy is moving. The resulting [[Vector (geometry)|vector]] has the units of power divided by area (i.e., [[surface power density]]). The intensity of a wave is proportional to the square of its amplitude. For example, the intensity of an electromagnetic wave is proportional to the square of the wave's [[electric field]] amplitude.
 ==Mathematical description==
@@ Line 11: / Line 12: @@
 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
-where ''P'' is the net power radiated, '''I''' is the intensity as a function of position, and ''d'''''A''' is a [[differential element]] of a closed surface that contains the source.
+*{{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 over a surface of uniform intensity ''I'', for instance over a sphere centered around the point source, the equation becomes
+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>
+<math display="block">P = |I| \cdot A_\mathrm{surf} = |I| \cdot 4\pi r^2,</math>
+where
-where ''I'' is the intensity at the surface of the sphere, and ''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).
+*{{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 ''I'' gives
+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 ''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:
+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, c/''n'':
+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
-where ''n'' is the [[refractive index]], c is the [[speed of light]] in [[vacuum]] and <math>\varepsilon_0</math> is the [[vacuum permittivity]].
+*{{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>
+==Electron beams==
+For [[electron beams]], ''intensity'' is the probability of electrons reaching some particular position on a detector (e.g. a [[charge-coupled device]]<ref>{{Cite journal |last=Spence |first=J. C. H. |last2=Zuo |first2=J. M. |date=1988-09-01 |title=Large dynamic range, parallel detection system for electron diffraction and imaging |url=https://doi.org/10.1063/1.1140039 |journal=Review of Scientific Instruments |volume=59 |issue=9 |pages=2102–2105 |doi=10.1063/1.1140039 |issn=0034-6748}}</ref>) which is used to produce images that are interpreted in terms of both [[microstructure]] of inorganic or biological materials, as well as [[atomic scale]] structure.<ref name=":0">{{Cite book |last=Cowley |first=J. M. |url=https://en.wikipedia.org/wiki/Special:BookSources/0-444-82218-6 |title=Diffraction physics |date=1995 |publisher=Elsevier |isbn=978-0-444-82218-5 |edition=3rd |series=North Holland personal library |location=Amsterdam}}</ref> The map of the intensity of scattered electrons or x-rays as a function of direction is also extensively used in [[crystallography]].<ref name=":0" /><ref>{{Cite book |last=Cullity |first=B. D. |url=https://en.wikipedia.org/wiki/Special:BookSources/0-201-61091-4 |title=Elements of X-ray diffraction |last2=Stock |first2=Stuart R. |date=2001 |publisher=Prentice Hall |isbn=978-0-201-61091-8 |edition=3rd |location=Upper Saddle River, NJ}}</ref>
 ==Alternative definitions==
@@ Line 37: / Line 51: @@
 *[[Sound intensity]]
 *[[Magnitude (astronomy)]]
+==Footnotes==
+{{notelist}}
 == References ==
@@ Line 44: / Line 61: @@
 {{DEFAULTSORT:Intensity (Physics)}}
-[[Category:Optics]]
+[[Category:Optical quantities]]
 [[Category:Radiometry]]
 [[Category:Physical quantities]]

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