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/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 (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,

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

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, 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.

See also

• Field strength
• Sound intensity
• Magnitude (astronomy)

Table 1. SI photometry quantities

Quantity		Unit		Dimension	Notes
Name	Symbol[nb 1]	Name	Symbol	Symbol
Luminous energy	Qv [nb 2]	lumen second	lm⋅s	T⋅J [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/m^2	L^−2⋅J	Luminous power per unit solid angle per unit projected source area. Units are sometimes called nits.
Illuminance	Ev	lux (= lm/m^2)	lx	L^−2⋅J	Luminous power incident on a surface.
Luminous exitance / luminous emittance	Mv	lux	lx	L^−2⋅J	Luminous power emitted from a surface.
Luminous exposure	Hv	lux second	lx⋅s	L^−2⋅T⋅J
Luminous energy density	ωv	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)

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. Alternative symbols sometimes seen: W for luminous energy, P or F for luminous flux, and ρ or K for luminous efficacy.
3. "J" here is the symbol for the dimension of luminous intensity, not the symbol for the unit joules.

Table 2. SI radiometry units

Quantity		Unit		Dimension	Notes
Name	Symbol[nb 1]	Name	Symbol	Symbol
Radiant energy	Qe[nb 2]	joule	J	M⋅L^2⋅T^−2	Energy of electromagnetic radiation.
Radiant energy density	we	joule per cubic metre	J/m^3	M⋅L^−1⋅T^−2	Radiant energy per unit volume.
Radiant flux	Φe[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	Φe,ν[nb 3] or Φe,λ[nb 4]	watt per hertz or watt per metre	W/Hz or W/m	M⋅L^2⋅T^−2 or M⋅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	Ie,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	Ie,Ω,ν[nb 3] or Ie,Ω,λ[nb 4]	watt per steradian per hertz or watt per steradian per metre	W⋅sr^−1⋅Hz^−1 or W⋅sr^−1⋅m^−1	M⋅L^2⋅T^−2 or M⋅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	Le,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	Le,Ω,ν[nb 3] 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	M⋅T^−2 or M⋅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	Ee[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	Ee,ν[nb 3] 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/m^3	M⋅T^−2 or M⋅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=10^4Jy).
Radiosity	Je[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	Je,ν[nb 3] 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/m^3	M⋅T^−2 or M⋅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	Me[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	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/m^3	M⋅T^−2 or M⋅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	He	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	He,ν[nb 3] or He,λ[nb 4]	joule per square metre per hertz or joule per square metre, per metre	J⋅m^−2⋅Hz^−1 or J/m^3	M⋅T^−1 or M⋅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	εν or ελ			1	Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity	εΩ			1	Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity	εΩ,ν or εΩ,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	Aν or Aλ			1	Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance	AΩ			1	Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance	AΩ,ν or AΩ,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	Rν or Rλ			1	Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance	RΩ			1	Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance	RΩ,ν or RΩ,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	Tν or Tλ			1	Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance	TΩ			1	Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance	TΩ,ν or TΩ,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	μν or μλ	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	μΩ	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	μΩ,ν or μΩ,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)

1. Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
2. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
3. 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.
4. Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
5. Directional quantities are denoted with suffix "Ω" (Greek).

References

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