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Intensity (physics)

From Wikipedia, the free encyclopedia
Power transferred per unit area
For other uses, seeIntensity (disambiguation).

Inphysics and many other areas of science and engineering theintensity orflux ofradiant energy is thepower transferred per unitarea, where the area is measured on the plane perpendicular to the direction of propagation of the energy.[a] In theSI system, it has unitswatts persquare metre (W/m2), orkg⋅s−3 inbase units. Intensity is used most frequently withwaves such as acoustic waves (sound),matter waves such as electrons inelectron microscopes, andelectromagnetic waves such aslight orradio waves, in which case theaverage power transfer over oneperiod of the wave is used.Intensity can be applied to other circumstances where energy is transferred. For example, one could calculate the intensity of thekinetic energy carried by drops of water from agarden 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 theenergy density (energy per unit volume) at a point in space and multiplying it by thevelocity at which the energy is moving. The resultingvector 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'selectric field amplitude.

Mathematical description

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

−∂u∂t=∇⋅S,{\displaystyle -{\frac {\partial u}{\partial t}}=\nabla \cdot \mathbf {S} ,}which is derived fromPoynting's theorem.

Integrating over a volume of space gives−∭∂∂tdUdVdV=∭(∇⋅S)dV{\displaystyle -\iiint {\frac {\partial }{\partial t}}{\frac {dU}{dV}}\,dV=\iiint (\nabla \cdot \mathbf {S} )\,dV}whereU{\displaystyle U} is the energy of the electromagnetic radiation.

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

dUdt=−{\displaystyle {\frac {dU}{dt}}=-}\oiintA{\displaystyle \scriptstyle A}S⋅dA,{\displaystyle \mathbf {S} \cdot d\mathbf {A} ,}

Point sources

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A common example is the intensity or flux of apoint source of given power outputP{\displaystyle P}. Considering a spherical volume centered on the source, the formula above becomes

P=⟨−dUdt⟩={\displaystyle P=\left\langle -{\frac {dU}{dt}}\right\rangle =}⟨S⟩{\displaystyle \langle S\rangle }\oiintA{\displaystyle \scriptstyle A}dA,{\displaystyle d\mathbf {A} ,}

where the angle brackets denote a time average over the period of the waves. Since the surface area of a sphere of radiusr{\displaystyle r} isA=4πr2{\textstyle A=4\pi r^{2}} this givesP=⟨S⟩⋅4πr2,{\displaystyle P=\langle S\rangle \cdot 4\pi r^{2},}therefore the intensity from the point source at distancer{\displaystyle r} isI=P4πr2.{\displaystyle I={\frac {P}{4\pi r^{2}}}.}This is known as theinverse-square law.

Electromagnetic waves

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For a monochromatic propagating electromagnetic wave such as aplane wave or aGaussian beam travelling in a non-magnetic medium, the time-averaged Poynting vector is related to the amplitude of theelectric field,E, by⟨S⟩=cnϵ02E2,{\displaystyle \left\langle {\mathsf {S}}\right\rangle ={\frac {cn\epsilon _{0}}{2}}E^{2},}wherec is thespeed of light invacuum,n is therefractive index of the medium, andϵ0{\displaystyle \epsilon _{0}} is thevacuum permittivity.

The relationship to intensity can also be seen by considering the time-averagedenergy density of the wave:⟨U⟩=n2ϵ02E2.{\displaystyle \left\langle U\right\rangle ={\frac {n^{2}\epsilon _{0}}{2}}E^{2}.}The local intensity is just the energy density times the wave velocity⁠cn{\displaystyle {\tfrac {c}{n}}}⁠:I=cnϵ02E2.{\displaystyle I={\frac {\mathrm {c} n\epsilon _{0}}{2}}E^{2}.}

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 factorc{\displaystyle c}:[1]

I=⟨S⟩∝c⟨U⟩{\displaystyle I=\langle S\rangle \propto c\langle U\rangle }

Anevanescent 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 thePoynting vector.[2]

Electron beams

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Forelectron beams,intensity is the probability of electrons reaching some particular position on a detector (e.g. acharge-coupled device[3]) which is used to produce images that are interpreted in terms of bothmicrostructure of inorganic or biological materials, as well asatomic scale structure.[4] The map of the intensity of scattered electrons or x-rays as a function of direction is also extensively used incrystallography.[4][5]

Alternative definitions

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Inphotometry andradiometryintensity has a different meaning: it is the luminous or radiant powerper unitsolid angle. This can cause confusion inoptics, whereintensity can mean any ofradiant intensity,luminous intensity orirradiance, depending on the background of the person using the term.Radiance is also sometimes calledintensity, especially by astronomers and astrophysicists, and inheat transfer.

See also

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Footnotes

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  1. ^The termsintensity andflux have multiple, inconsistent, definitions in physics and related fields. This article covers the concept of power per unit area, whatever one calls it. Inradiometry the termsintensity andflux have different meanings, not covered here.

References

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  1. ^Klein, Miles; Furtak, Thomas (1985).Optics. John Wiley & Sons, Inc. p. 49.ISBN 0-471-87297-0.
  2. ^Paschotta, Rüdiger (14 January 2008)."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.Bibcode:1988RScI...59.2102S.doi:10.1063/1.1140039.ISSN 0034-6748.
  4. ^abCowley, 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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