# Intensity (physics)

In physics and many other areas of science and engineering the intensity or flux of radiant energy is the power transferred per unit area, where the area is measured on the plane perpendicular to the direction of propagation of the energy. In the SI system, it has units watts per square metre (W/m$^2$), or kg⋅s$^{−3}$ in base units. Intensity is used most frequently with waves such as acoustic waves (sound), matter waves such as electrons in electron microscopes, and electromagnetic waves such as light or radio waves, 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). The intensity of a wave is proportional to the square of its amplitude. For example, the intensity of an electromagnetic wave is proportional to the square of the wave's electric field amplitude.

## 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 the 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, $P=\int \mathbf {I} \,\cdot d\mathbf {A} ,$ where
* P is the net power radiated;
* I is the intensity vector as a function of position;
* the magnitude |I| is the intensity as a function of position;
* dA is a differential element of a closed surface that contains the source.
If one integrates a uniform intensity, |I| = const., over a surface that is perpendicular to the intensity vector, 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;
* 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 electromagnetic 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, travelling in a non-magnetic material, is given by: $\left\langle U\right\rangle ={\frac {n^{2}\varepsilon _{0}}{2}}|E|^{2},$ and the local intensity is obtained by multiplying this expression by the wave velocity, ⁠${\tfrac {\mathrm {c} }{n}}\!:$⁠ $I={\frac {\mathrm {c} n\varepsilon _{0}}{2}}|E|^{2},$ where
* n is the refractive index;
* c is the speed of light in vacuum;
* ε$_0$ 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.

## Electron beams

For electron beams, intensity is the probability of electrons reaching some particular position on a detector (e.g. a charge-coupled device) which is used to produce images that are interpreted in terms of both microstructure of inorganic or biological materials, as well as atomic scale structure. The map of the intensity of scattered electrons or x-rays as a function of direction is also extensively used in crystallography.

## Alternative definitions

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.