In

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

,

**intensity** is a

measureMeasurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...

of the

energy fluxEnergy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context:# Rate of energy transfer per unit area...

, averaged over the

periodIn mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech. For example, "the intensity of pressure" is meaningless, since the parameters of those variables do not match.

To find the intensity, take the

energy densityEnergy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...

(that is, the energy per unit volume) and multiply it by the

velocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

at which the energy is moving. The resulting vector has the units of

powerIn physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

divided by

areaArea is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

(i.e. W/m²). It is possible to define the intensity of the water coming from a garden sprinkler, but intensity is used most frequently with

waveIn physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

s (i.e.

soundSound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...

or

lightLight or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

).

## 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 due to physics and geometry. Physically,

conservation of energyThe nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

applies. The consequence of this is that the net power coming from the source must be constant, thus:

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. That

*P* is a constant. If we integrate 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:

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*, we get:

If the medium is damped, then the intensity drops off more quickly than the above equation suggests.

Anything that can carry energy can have an intensity associated with it. For an electromagnetic wave, if

*E* is the

complexA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

amplitudeAmplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

of the

electric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

, then the time-averaged

energy densityEnergy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...

of the wave is given by

,

and the intensity is obtained by multiplying this expression by the velocity of the wave,

:

,

where

*n* is the

refractive indexIn optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

,

is the

speed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

in

vacuumIn everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

and

is the vacuum permittivity.

The treatment above does not hold for electromagnetic fields that are not radiating, such as for an

evanescent waveAn evanescent wave is a nearfield standing wave with an intensity that exhibits exponential decay with distance from the boundary at which the wave was formed. Evanescent waves are a general property of wave-equations, and can in principle occur in any context to which a wave-equation applies...

. In these cases, the intensity can be defined as the magnitude of the

Poynting vectorIn physics, the Poynting vector can be thought of as representing the directional energy flux density of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside and Nikolay Umov independently co-invented the Poynting vector...

.

## Alternative definitions of "intensity"

In

photometryPhotometry is the science of the measurement of light, in terms of its perceived brightness to the human eye. It is distinct from radiometry, which is the science of measurement of radiant energy in terms of absolute power; rather, in photometry, the radiant power at each wavelength is weighted by...

and

radiometryIn optics, radiometry is a set of techniques for measuring electromagnetic radiation, including visible light. Radiometric techniques characterize the distribution of the radiation's power in space, as opposed to photometric techniques, which characterize the light's interaction with the human eye...

*intensity* has a different meaning: it is the luminous or radiant power

*per unit solid angle*The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

. This can cause confusion in

opticsOptics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, where

*intensity* can mean any of

radiant intensityIn radiometry, radiant intensity is a measure of the intensity of electromagnetic radiation. It is defined as power per unit solid angle. The SI unit of radiant intensity is watts per steradian . Radiant intensity is distinct from irradiance and radiant exitance, which are often called intensity...

,

luminous intensityIn photometry, luminous intensity is a measure of the wavelength-weighted power emitted by a light source in a particular direction per unit solid angle, based on the luminosity function, a standardized model of the sensitivity of the human eye...

or

irradianceIrradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...

, depending on the background of the person using the term.

RadianceRadiance and spectral radiance are radiometric measures that describe the amount of radiation such as light or radiant heat that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from...

is also sometimes called

*intensity*, especially by astronomers and astrophysicists, and in

heat transferHeat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...

.

## See also

- Sound intensity
Sound intensity or acoustic intensity is defined as the sound power Pac per unit area A. The usual context is the noise measurement of sound intensity in the air at a listener's location.-Acoustic intensity:...

- Magnitude (astronomy)
Magnitude is the logarithmic measure of the brightness of an object, in astronomy, measured in a specific wavelength or passband, usually in optical or near-infrared wavelengths.-Background:...