# Intensity (physics)

For other uses, see Intensity (disambiguation).

In physics, intensity is the power transferred per unit area, which is transmitted through an imagined surface perpendicular to the propagation direction.[1] In the SI system, it has units watts per square metre (W/m2). It is used most frequently with waves (e.g. sound or light), 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.

## 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 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,

$P = \int \bold I\, \cdot \mathrm{d}\bold A$,

where P is the net power radiated, I is the intensity as a function of position, and dA 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

$P = |I| \cdot A_\mathrm{surf} = |I| \cdot 4\pi r^2 \,$,

where I is the intensity at the surface of the sphere, and 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 wave, such as a plane wave or a Gaussian beam, if E is the complex amplitude of the electric field via, then the time-averaged energy density of the wave is given by:

$\left\langle U \right \rangle = \frac{n^2 \epsilon_0}{2} |E|^2$,

and the local intensity is obtained by multiplying this expression by the wave velocity, c/n:

$I = \frac{\mathrm{c} n \epsilon_0}{2} |E|^2$,

where n is the refractive index, c is the speed of light in vacuum and $\epsilon_0$ 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.[2]

## Alternative definitions of "intensity"

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.

Table 1. SI photometry units
Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol Symbol
Luminous energy Qv [nb 2] lumen second lm⋅s TJ [nb 3] Units are sometimes called talbots.
Luminous flux / Luminous power Φv [nb 2] lumen (= cd⋅sr) lm J [nb 3] Luminous energy per unit time.
Luminous intensity Iv candela (= lm/sr) cd J [nb 3] Luminous power per unit solid angle.
Luminance Lv candela per square metre cd/m2 L−2J Luminous power per unit solid angle per unit projected source area. Units are sometimes called nits.
Illuminance Ev lux (= lm/m2) lx L−2J Luminous power incident on a surface.
Luminous exitance / Luminous emittance Mv lux lx L−2J Luminous power emitted from a surface.
Luminous exposure Hv lux second lx⋅s L−2TJ
Luminous energy density ωv lumen second per cubic metre lm⋅s⋅m−3 L−3TJ
Luminous efficacy η [nb 2] lumen per watt lm/W M−1L−2T3J Ratio of luminous flux to radiant flux.
Luminous efficiency / Luminous coefficient V 1
1. ^ Standards organizations recommend that photometric quantities be denoted with a suffix "v" (for "visual") to avoid confusion with radiometric or photon quantities. For example: USA Standard Letter Symbols for Illuminating Engineering USAS Z7.1-1967, Y10.18-1967
2. ^ a b c Alternative symbols sometimes seen: W for luminous energy, P or F for luminous flux, and ρ or K for luminous efficacy.
3. ^ a b c "J" here is the symbol for the dimension of luminous intensity, not the symbol for the unit joules.

Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol Symbol
Radiant energy Qe[nb 2] joule J ML2T−2 Energy received, emitted, reflected, or transmitted by a system in form of electromagnetic radiation.
Radiant energy density we joule per cubic metre J/m3 ML−1T−2 Radiant energy of a system per unit volume at a given location.
Radiant flux / Radiant power Φe[nb 2] watt W or J/s ML2T−3 Radiant energy of a system per unit time at a given time.
Spectral flux / Spectral power Φe,ν[nb 3]
or
Φe,λ[nb 4]
watt per hertz
or
watt per metre
W/Hz
or
W/m
ML2T−2
or
MLT−3
Radiant power of a system per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1.
Radiant intensity Ie,Ω[nb 5] watt per steradian W/sr ML2T−3 Radiant power of a system per unit solid angle around a given direction. It is a directional quantity.
Spectral intensity Ie,Ω,ν[nb 3]
or
Ie,Ω,λ[nb 4]
or
W⋅sr−1⋅Hz−1
or
W⋅sr−1⋅m−1
ML2T−2
or
MLT−3
Radiant intensity of a system per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. It is a directional quantity.
Radiance Le,Ω[nb 5] watt per steradian per square metre W⋅sr−1⋅m−2 MT−3 Radiant power of a surface per unit solid angle around a given direction per unit projected area of that surface along that direction. It is a directional quantity. It is sometimes also confusingly called "intensity".
or
Le,Ω,λ[nb 4]
watt per steradian per square metre per hertz
or
watt per steradian per square metre, per metre
W⋅sr−1⋅m−2⋅Hz−1
or
W⋅sr−1⋅m−3
MT−2
or
ML−1T−3
Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. It is a directional quantity. It is sometimes also confusingly called "spectral intensity".
Irradiance Ee[nb 2] watt per square metre W/m2 MT−3 Radiant power received by a surface per unit area. It is sometimes also confusingly called "intensity".
or
Ee,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Irradiance of a surface per unit frequency or wavelength. The former is commonly measured in 10−22 W⋅m−2⋅Hz−1, known as solar flux unit, and the latter in W⋅m−2⋅nm−1.[nb 6] It is sometimes also confusingly called "spectral intensity".
Radiosity Je[nb 2] watt per square metre W/m2 MT−3 Radiant power leaving (emitted, reflected and transmitted by) a surface per unit area. It is sometimes also confusingly called "intensity".
or
Je,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. It is sometimes also confusingly called "spectral intensity".
Radiant exitance Me[nb 2] watt per square metre W/m2 MT−3 Radiant power emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. It is sometimes also confusingly called "intensity".
Spectral exitance Me,ν[nb 3]
or
Me,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. It is sometimes also confusingly called "spectral intensity".
Radiant exposure He joule per square metre J/m2 MT−2 Irradiance of a surface times exposure time. It is sometimes also called fluence.