# Intensity (physics)

In physics, **intensity** is the power transferred per unit area, where the area is measured on the plane perpendicular to the direction of propagation of the energy.^{[1]} In the SI system, it has units watts per square metre (W/m^{2}). 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 (i.e., surface power density).

## 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,

- ,

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.

If one integrates over a surface of uniform intensity *I*, for instance over a sphere centered around the point source, the equation becomes

- ,

where *I* is the intensity at the surface of the sphere, and *r* is the radius of the sphere. ( is the expression for the surface area of a sphere).

Solving for *I* gives

- .

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, then the time-averaged energy density of the wave is given by:

- ,

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

- ,

where *n* is the refractive index, c is the speed of light in vacuum and 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.

## See also

Quantity | Unit | Dimension | Notes | |||||
---|---|---|---|---|---|---|---|---|

Name | Symbol^{[nb 1]} |
Name | Symbol | Symbol | ||||

Luminous energy | Q_{v} ^{[nb 2]} |
lumen second | lm⋅s | T⋅J ^{[nb 3]} |
Units are sometimes called talbots. | |||

Luminous flux / luminous power | Φ_{v} ^{[nb 2]} |
lumen (= cd⋅sr) | lm | J ^{[nb 3]} |
Luminous energy per unit time. | |||

Luminous intensity | I_{v} |
candela (= lm/sr) | cd | J ^{[nb 3]} |
Luminous power per unit solid angle. | |||

Luminance | L_{v} |
candela per square metre | cd/m^{2} |
L^{−2}⋅J |
Luminous power per unit solid angle per unit projected source area. Units are sometimes called nits. | |||

Illuminance | E_{v} |
lux (= lm/m^{2}) |
lx | L^{−2}⋅J |
Luminous power incident on a surface. | |||

Luminous exitance / luminous emittance | M_{v} |
lux | lx | L^{−2}⋅J |
Luminous power emitted from a surface. | |||

Luminous exposure | H_{v} |
lux second | lx⋅s | L^{−2}⋅T⋅J |
||||

Luminous energy density | ω_{v} |
lumen second per cubic metre | lm⋅s⋅m^{−3} |
L^{−3}⋅T⋅J |
||||

Luminous efficacy | η ^{[nb 2]} |
lumen per watt | lm/W | M^{−1}⋅L^{−2}⋅T^{3}⋅J |
Ratio of luminous flux to radiant flux or power consumption, depending on context. | |||

Luminous efficiency / luminous coefficient | V |
1 | ||||||

See also: SI · Photometry · Radiometry · (Compare) |

- ↑ 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 - 1 2 3 Alternative symbols sometimes seen:
*W*for luminous energy,*P*or*F*for luminous flux, and*ρ*or*K*for luminous efficacy. - 1 2 3 "
**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 | Q_{e}^{[nb 2]} |
joule | J | M⋅L^{2}⋅T^{−2} |
Energy of electromagnetic radiation. | |||

Radiant energy density | w_{e} |
joule per cubic metre | J/m^{3} |
M⋅L^{−1}⋅T^{−2} |
Radiant energy per unit volume. | |||

Radiant flux | Φ_{e}^{[nb 2]} |
watt | W or J/s |
M⋅L^{2}⋅T^{−3} |
Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power". | |||

Spectral flux | Φ_{e,ν}^{[nb 3]}or Φ _{e,λ}^{[nb 4]} |
watt per hertzorwatt per metre |
W/HzorW/m |
M⋅L^{2}⋅T^{−2}orM⋅L⋅T^{−3} |
Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅sr^{−1}⋅m^{−2}⋅nm^{−1}. | |||

Radiant intensity | I_{e,Ω}^{[nb 5]} |
watt per steradian | W/sr | M⋅L^{2}⋅T^{−3} |
Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity. | |||

Spectral intensity | I_{e,Ω,ν}^{[nb 3]}or I_{e,Ω,λ}^{[nb 4]} |
watt per steradian per hertzorwatt per steradian per metre |
W⋅sr^{−1}⋅Hz^{−1}orW⋅sr ^{−1}⋅m^{−1} |
M⋅L^{2}⋅T^{−2}orM⋅L⋅T^{−3} |
Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr^{−1}⋅m^{−2}⋅nm^{−1}. This is a directional quantity. | |||

Radiance | L_{e,Ω}^{[nb 5]} |
watt per steradian per square metre | W⋅sr^{−1}⋅m^{−2} |
M⋅T^{−3} |
Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity". | |||

Spectral radiance | L_{e,Ω,ν}^{[nb 3]}or L_{e,Ω,λ}^{[nb 4]} |
watt per steradian per square metre per hertzorwatt per steradian per square metre, per metre |
W⋅sr^{−1}⋅m^{−2}⋅Hz^{−1}orW⋅sr ^{−1}⋅m^{−3} |
M⋅T^{−2}orM⋅L^{−1}⋅T^{−3} |
Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr^{−1}⋅m^{−2}⋅nm^{−1}. This is a directional quantity. This is sometimes also confusingly called "spectral intensity". | |||

Irradiance Flux density |
E_{e}^{[nb 2]} |
watt per square metre | W/m^{2} |
M⋅T^{−3} |
Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity". | |||

Spectral irradiance Spectral flux density |
E_{e,ν}^{[nb 3]}or E_{e,λ}^{[nb 4]} |
watt per square metre per hertzorwatt per square metre, per metre |
W⋅m^{−2}⋅Hz^{−1}orW/m ^{3} |
M⋅T^{−2}orM⋅L^{−1}⋅T^{−3} |
Irradiance of a surface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include Jansky = 10^{−26} W⋅m^{−2}⋅Hz^{−1} and solar flux unit (1SFU = 10^{−22} W⋅m^{−2}⋅Hz^{−1}=10^{4}Jy). | |||

Radiosity | J_{e}^{[nb 2]} |
watt per square metre | W/m^{2} |
M⋅T^{−3} |
Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity". | |||

Spectral radiosity | J_{e,ν}^{[nb 3]}or J_{e,λ}^{[nb 4]} |
watt per square metre per hertzorwatt per square metre, per metre |
W⋅m^{−2}⋅Hz^{−1}orW/m ^{3} |
M⋅T^{−2}orM⋅L^{−1}⋅T^{−3} |
Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m^{−2}⋅nm^{−1}. This is sometimes also confusingly called "spectral intensity". | |||

Radiant exitance | M_{e}^{[nb 2]} |
watt per square metre | W/m^{2} |
M⋅T^{−3} |
Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity". | |||

Spectral exitance | M_{e,ν}^{[nb 3]}or M_{e,λ}^{[nb 4]} |
watt per square metre per hertzorwatt per square metre, per metre |
W⋅m^{−2}⋅Hz^{−1}orW/m ^{3} |
M⋅T^{−2}orM⋅L^{−1}⋅T^{−3} |
Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m^{−2}⋅nm^{−1}. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity". | |||

Radiant exposure | H_{e} |
joule per square metre | J/m^{2} |
M⋅T^{−2} |
Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence". | |||

Spectral exposure | H_{e,ν}^{[nb 3]}or H_{e,λ}^{[nb 4]} |
joule per square metre per hertzorjoule per square metre, per metre |
J⋅m^{−2}⋅Hz^{−1}orJ/m ^{3} |
M⋅T^{−1}orM⋅L^{−1}⋅T^{−2} |
Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m^{−2}⋅nm^{−1}. This is sometimes also called "spectral fluence". | |||

Hemispherical emissivity | ε |
1 | Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface. | |||||

Spectral hemispherical emissivity | ε_{ν}or ε_{λ} |
1 | Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface. | |||||

Directional emissivity | ε_{Ω} |
1 | Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface. | |||||

Spectral directional emissivity | ε_{Ω,ν}or ε_{Ω,λ} |
1 | Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface. | |||||

Hemispherical absorptance | A |
1 | Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance". | |||||

Spectral hemispherical absorptance | A_{ν}or A_{λ} |
1 | Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance". | |||||

Directional absorptance | A_{Ω} |
1 | Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance". | |||||

Spectral directional absorptance | A_{Ω,ν}or A_{Ω,λ} |
1 | Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance". | |||||

Hemispherical reflectance | R |
1 | Radiant flux reflected by a surface, divided by that received by that surface. | |||||

Spectral hemispherical reflectance | R_{ν}or R_{λ} |
1 | Spectral flux reflected by a surface, divided by that received by that surface. | |||||

Directional reflectance | R_{Ω} |
1 | Radiance reflected by a surface, divided by that received by that surface. | |||||

Spectral directional reflectance | R_{Ω,ν}or R_{Ω,λ} |
1 | Spectral radiance reflected by a surface, divided by that received by that surface. | |||||

Hemispherical transmittance | T |
1 | Radiant flux transmitted by a surface, divided by that received by that surface. | |||||

Spectral hemispherical transmittance | T_{ν}or T_{λ} |
1 | Spectral flux transmitted by a surface, divided by that received by that surface. | |||||

Directional transmittance | T_{Ω} |
1 | Radiance transmitted by a surface, divided by that received by that surface. | |||||

Spectral directional transmittance | T_{Ω,ν}or T_{Ω,λ} |
1 | Spectral radiance transmitted by a surface, divided by that received by that surface. | |||||

Hemispherical attenuation coefficient | μ |
reciprocal metre | m^{−1} |
L^{−1} |
Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume. | |||

Spectral hemispherical attenuation coefficient | μ_{ν}or μ_{λ} |
reciprocal metre | m^{−1} |
L^{−1} |
Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume. | |||

Directional attenuation coefficient | μ_{Ω} |
reciprocal metre | m^{−1} |
L^{−1} |
Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume. | |||

Spectral directional attenuation coefficient | μ_{Ω,ν}or μ_{Ω,λ} |
reciprocal metre | m^{−1} |
L^{−1} |
Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume. | |||

See also: SI · Radiometry · Photometry · (Compare) |

- ↑ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
- 1 2 3 4 5 Alternative symbols sometimes seen:
*W*or*E*for radiant energy,*P*or*F*for radiant flux,*I*for irradiance,*W*for radiant exitance. - 1 2 3 4 5 6 7 Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek)—not to be confused with suffix "v" (for "visual") indicating a photometric quantity.
- 1 2 3 4 5 6 7 Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
- 1 2 Directional quantities are denoted with suffix "Ω" (Greek).

## References

- ↑ "intensity".
*Merriam-Webster.com*. Retrieved Feb 15, 2015. - ↑ Paschotta, Rüdiger. "Optical Intensity".
*Encyclopedia of Laser Physics and Technology*. RP Photonics.