# Intensity (physics)

## Overview

In physics, intensity is a measure of the time-averaged energy flux. 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 density (that is, the energy per unit volume) and multiply it by the velocity at which the energy is moving. The resulting vector has the units of power divided by area (i.e. watt/m²). It is possible to define the intensity of the water coming from a garden sprinkler, but intensity is used most frequently with waves (i.e. sound or light).

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

${\displaystyle P=\int I\,dA}$

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. 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:

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

where I is the intensity at the surface of the sphere, and r is the radius of the sphere. (${\displaystyle A_{surf}=4\pi r^{2}}$ is the expression for the surface area of a sphere). Solving for I, we get:

${\displaystyle |I|={\frac {P}{A_{surf}}}={\frac {P}{4\pi r^{2}}}}$

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

Anything that can carry energy can have an intensity associated with it. For an electromagnetic wave, if E is the complex amplitude of the electric field, then the energy density of the wave is given by

${\displaystyle U={\frac {n^{2}\epsilon _{0}}{2}}|E|^{2}}$,

and the intensity is obtained multiplying this expression by the velocity of the wave, ${\displaystyle c/n}$:

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

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

## Intensity in photometry and radiometry

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.