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==Mathematical description==
==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]].
The intensity or flux of electromagnetic radiation is equal to the time average of the [[Poynting vector]] over the wave's period. For radiation propagating through a typical medium the energy density of the radiation, <math>u</math>, is related to the Poynting vector <math>\mathbf{S}</math> by


Applying the law of [[conservation of energy]], if the net power emanating is constant,
<math display="block">  
<math display="block">P = \int \mathbf I\, \cdot d\mathbf A,</math>
-\frac{\partial u}{\partial t}=\nabla\cdot\mathbf{S},
where
</math>
*{{mvar|P}} is the net power radiated;
which is derived from [[Poynting's theorem]].  
*{{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
Integrating over a volume of space gives
<math display="block">P = |I| \cdot A_\mathrm{surf} = |I| \cdot 4\pi r^2,</math>
<math display="block">
where  
-\iiint\frac{\partial}{\partial t}\frac{dU}{dV}\, dV=\iiint(\nabla\cdot\mathbf{S})\, dV
*{{mvar|{{abs|I}}}} is the intensity at the surface of the sphere;
</math>
*{{mvar|r}} is the radius of the sphere;
where <math>U</math> is the energy of the electromagnetic radiation.
*<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
Applying the [[divergence theorem]], the rate of flow of energy out of the volume is seen to be related to the [[surface integral]] of the Poynting vector over the surface of the volume of space:
<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.
{{block indent |<math>
\frac{dU}{dt} = -</math>{{oiint
| intsubscpt = <math>\scriptstyle A</math>
| integrand = <math>\mathbf{S}\cdot d\mathbf{A},</math>
}}}}


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:
===Point sources===
<math display="block">\left\langle U \right \rangle = \frac{n^2 \varepsilon_0}{2} |E|^2,</math>
A common example is the intensity or flux of a [[point source]] of given power output <math>P</math>. Considering a spherical volume centered on the source, the formula above becomes
and the local intensity is obtained by multiplying this expression by the wave velocity, {{tmath|\tfrac{\mathrm c}{n} \! :}}
{{block indent |<math>
<math display="block">I = \frac{\mathrm{c} n \varepsilon_0}{2} |E|^2,</math>
P=\left \langle -\frac{dU}{dt} \right \rangle = </math>{{oiint
where
| preintegral = <math>\langle S \rangle</math>
*{{mvar|n}} is the [[refractive index]];
| intsubscpt = <math>\scriptstyle A</math>
*{{math|c}} is the [[speed of light]] in [[vacuum]];
| integrand = <math>d\mathbf{A},</math>
*{{math|''&epsilon;''{{sub|0}}}} is the [[vacuum permittivity]].
}}}}
where the angle brackets denote a time average over the period of the waves. Since the surface area of a sphere of radius <math>r</math> is <math display=inline>A = 4\pi r^2</math> this gives
<math display=block>
P = \langle S \rangle \cdot 4\pi r^2,  
</math>
therefore the intensity from the point source at distance <math>r</math> is
<math display="block">  
I=\frac{P}{4\pi r^2}.
</math>
This is known as the [[inverse-square law]].


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>
===Electromagnetic waves===
For a monochromatic propagating electromagnetic wave such as a [[plane wave]] or a [[Gaussian beam]] travelling in a non-magnetic medium, the time-averaged Poynting vector is related to the amplitude of the [[electric field]], {{mvar|E}}, by
<math display="block">\left\langle\mathsf{S}\right\rangle = \frac{cn\epsilon_0}{2}  E^2,</math>
where {{mvar|c}} is the [[speed of light]] in [[vacuum]], {{mvar|n}} is the [[refractive index]] of the medium, and <math>\epsilon_0</math> is the [[vacuum permittivity]].
 
The relationship to intensity can also be seen by considering the time-averaged [[energy density]] of the wave:
<math display="block">\left\langle U \right \rangle = \frac{n^2 \epsilon_0}{2} E^2.</math>
The local intensity is just the energy density times the wave velocity {{tmath|\tfrac{c}{n} }}:
<math display="block">I = \frac{\mathrm{c} n \epsilon_0}{2} E^2.</math>
 
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, but it is still often true that the magnitude of the time-averaged Poynting vector is proportional to the time-averaged energy density by a factor <math>c</math>:<ref>{{cite book |last1=Klein |first1=Miles |last2=Furtak |first2=Thomas |date=1985 |title=Optics |publisher=John Wiley & Sons, Inc |page=49 |isbn=0-471-87297-0}} </ref>
 
<math display="block">I = \langle S\rangle\propto c\langle U\rangle</math>
 
An [[evanescent wave]] may have a finite electrical amplitude while not transferring any power. The intensity of an evanescent wave can 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==
==Electron beams==

Revision as of 21:34, 14 June 2025

Template:Short description Script error: No such module "other uses". 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.Template:Efn 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

The intensity or flux of electromagnetic radiation is equal to the time average of the Poynting vector over the wave's period. For radiation propagating through a typical medium the energy density of the radiation, u, is related to the Poynting vector 𝐒 by

−∂u∂t=∇⋅𝐒, which is derived from Poynting's theorem.

Integrating over a volume of space gives −∭∂∂tdUdVdV=∭(∇⋅𝐒)dV where U is the energy of the electromagnetic radiation.

Applying the divergence theorem, the rate of flow of energy out of the volume is seen to be related to the surface integral of the Poynting vector over the surface of the volume of space:

Template:Block indent

Point sources

A common example is the intensity or flux of a point source of given power output P. Considering a spherical volume centered on the source, the formula above becomes Template:Block indent where the angle brackets denote a time average over the period of the waves. Since the surface area of a sphere of radius r is A=4πr2 this gives P=⟨S⟩⋅4πr2, therefore the intensity from the point source at distance r is I=P4πr2. This is known as the inverse-square law.

Electromagnetic waves

For a monochromatic propagating electromagnetic wave such as a plane wave or a Gaussian beam travelling in a non-magnetic medium, the time-averaged Poynting vector is related to the amplitude of the electric field, Template:Mvar, by ⟨S⟩=cnϵ02E2, where Template:Mvar is the speed of light in vacuum, Template:Mvar is the refractive index of the medium, and ϵ0 is the vacuum permittivity.

The relationship to intensity can also be seen by considering the time-averaged energy density of the wave: ⟨U⟩=n2ϵ02E2. The local intensity is just the energy density times the wave velocity Template:Tmath: I=cnϵ02E2.

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, but it is still often true that the magnitude of the time-averaged Poynting vector is proportional to the time-averaged energy density by a factor c:[1]

I=⟨S⟩∝c⟨U⟩

An evanescent wave may have a finite electrical amplitude while not transferring any power. The intensity of an evanescent wave can be defined as the magnitude of the Poynting vector.[2]

Electron beams

For electron beams, intensity is the probability of electrons reaching some particular position on a detector (e.g. a charge-coupled device[3]) 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.[4] The map of the intensity of scattered electrons or x-rays as a function of direction is also extensively used in crystallography.[4][5]

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.

See also

Footnotes

Template:Notelist

References

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