Intensity (physics)

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In physics, intensity is the power transferred per unit area. In the SI system, it has units watts per metre squared (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 three dimensions and there is no energy lost to 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,

$P=\int I\,\cdot dA$ ,

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 a point source radiating equally in all directions, 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 an electromagnetic wave, 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 intensity is obtained by multiplying this expression by the velocity of the wave, $c/n$ :

$I={\frac {cn\epsilon _{0}}{2}}|E|^{2}$ ,

where n is the refractive index, $c$ is the speed of light in vacuum and $\epsilon _{0}$ is the vacuum permittivity.

The treatment above does not hold for electromagnetic fields that are not radiating, such as for an evanescent wave. In these cases, the intensity can be defined as the magnitude of the Poynting vector.^[1]

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
v
t
e
Quantity		Unit		Dimension	Notes
Name	Symbol^{[nb 1]}	Name	Symbol	Symbol	Notes
Luminous energy	Q_v ^{[nb 2]}	lumen second	lm⋅s	T⋅J ^{[nb 3]}	units are sometimes called talbots
Luminous flux	Φ_v ^{[nb 2]}	lumen (= cd⋅sr)	lm	J ^{[nb 3]}	also called luminous power
Luminous intensity	I_v	candela (= lm/sr)	cd	J ^{[nb 3]}	an SI base unit, luminous flux per unit solid angle
Luminance	L_v	candela per square metre	cd/m²	L⁻²⋅J	units are sometimes called nits
Illuminance	E_v	lux (= lm/m²)	lx	L⁻²⋅J	used for light incident on a surface
Luminous emittance	M_v	lux (= lm/m²)	lx	L⁻²⋅J	used for light emitted from a surface
Luminous exposure	H_v	lux second	lx⋅s	L⁻²⋅T⋅J
Luminous energy density	ω_v	lumen second per metre³	lm⋅s⋅m⁻³	L⁻³⋅T⋅J
Luminous efficacy	η ^{[nb 2]}	lumen per watt	lm/W	M⁻¹⋅L⁻²⋅T³⋅J	ratio of luminous flux to radiant flux
Luminous efficiency	V			1	also called luminous coefficient
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
↑ 2.0 2.1 2.2 Alternative symbols sometimes seen: W for luminous energy, P or F for luminous flux, and ρ or K for luminous efficacy.
↑ 3.0 3.1 3.2 "J" here is the symbol for the dimension of luminous intensity, not the symbol for the unit joules.

Table 2. SI radiometry units
v
t
e
Quantity		Unit		Dimension	Notes
Name	Symbol^{[nb 1]}	Name	Symbol	Symbol	Notes
Radiant energy	Q_e^{[nb 2]}	joule	J	M⋅L²⋅T⁻²	energy
Radiant flux	Φ_e^{[nb 2]}	watt	W or J/s	M⋅L²⋅T⁻³	radiant energy per unit time, also called radiant power.
Spectral power	Φ_eλ^{[nb 2]}^{[nb 3]}	watt per metre	W⋅m⁻¹	M⋅L⋅T⁻³	radiant power per wavelength.
Radiant intensity	I_e	watt per steradian	W⋅sr⁻¹	M⋅L²⋅T⁻³	power per unit solid angle.
Spectral intensity	I_eλ^{[nb 3]}	watt per steradian per metre	W⋅sr⁻¹⋅m⁻¹	M⋅L⋅T⁻³	radiant intensity per wavelength.
Radiance	L_e	watt per steradian per square metre	W⋅sr⁻¹⋅m⁻²	M⋅T⁻³	power per unit solid angle per unit projected source area. confusingly called "intensity" in some other fields of study.
Spectral radiance	L_eλ^{[nb 3]} or L_eν^{[nb 4]}	watt per steradian per metre³ or watt per steradian per square metre per hertz	W⋅sr⁻¹⋅m⁻³ or W⋅sr⁻¹⋅m⁻²⋅Hz⁻¹	M⋅L⁻¹⋅T⁻³ or M⋅T⁻²	commonly measured in W⋅sr⁻¹⋅m⁻²⋅nm⁻¹ with surface area and either wavelength or frequency.
Irradiance	E_e^{[nb 2]}	watt per square metre	W⋅m⁻²	M⋅T⁻³	power incident on a surface, also called radiant flux density. sometimes confusingly called "intensity" as well.
Spectral irradiance	E_eλ^{[nb 3]} or E_eν^{[nb 4]}	watt per metre³ or watt per square metre per hertz	W⋅m⁻³ or W⋅m⁻²⋅Hz⁻¹	M⋅L⁻¹⋅T⁻³ or M⋅T⁻²	commonly measured in W⋅m⁻²⋅nm⁻¹ or 10⁻²² W⋅m⁻²⋅Hz⁻¹, known as solar flux unit.^{[nb 5]}
Radiant exitance / Radiant emittance	M_e^{[nb 2]}	watt per square metre	W⋅m⁻²	M⋅T⁻³	power emitted from a surface.
Spectral radiant exitance / Spectral radiant emittance	M_eλ^{[nb 3]} or M_eν^{[nb 4]}	watt per metre³ or watt per square metre per hertz	W⋅m⁻³ or W⋅m⁻²⋅Hz⁻¹	M⋅L⁻¹⋅T⁻³ or M⋅T⁻²	power emitted from a surface per unit wavelength or frequency.
Radiosity	J_e	watt per square metre	W⋅m⁻²	M⋅T⁻³	emitted plus reflected power leaving a surface.
Spectral radiosity	J_eλ^{[nb 3]}	watt per metre³	W⋅m⁻³	M⋅L⁻¹⋅T⁻³	emitted plus reflected power leaving a surface per unit wavelength
Radiant exposure	H_e	joule per square metre	J⋅m⁻²	M⋅T⁻²	also referred to as fluence
Radiant energy density	ω_e	joule per metre³	J⋅m⁻³	M⋅L⁻¹⋅T⁻²
See also: SI · Radiometry · Photometry · (Compare)

↑ Standards organizations recommend that radiometric quantities should be denoted with a suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
↑ 2.0 2.1 2.2 2.3 2.4 Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant emittance.
↑ 3.0 3.1 3.2 3.3 3.4 3.5 Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek) to indicate a spectral concentration. Spectral functions of wavelength are indicated by "(λ)" in parentheses instead, for example in spectral transmittance, reflectance and responsivity.
↑ 4.0 4.1 4.2 Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek)—not to be confused with the suffix "v" (for "visual") indicating a photometric quantity.
↑ NOAA / Space Weather Prediction Center includes a definition of the solar flux unit (SFU).

References

↑ Paschotta, Rüdiger. "Optical Intensity". Encyclopedia of Laser Physics and Technology. RP Photonics.

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Intensity (physics)

Mathematical description

Alternative definitions of "intensity"

See also

References