Intensity (physics)

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

The intensity or flux of electromagnetic radiation is equal to the time average of the Poynting vector over the wave's period. For radiation propagating through a typical medium the energy density of the radiation, ${\displaystyle u}$, is related to the Poynting vector ${\displaystyle \mathbf{S}}$ by

$${\displaystyle -\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\mathrm{\nabla }\cdot \mathbf{S},}$$ which is derived from Poynting's theorem.

Integrating over a volume of space gives $${\displaystyle -\iiint \frac{\mathrm{\partial }}{\mathrm{\partial }t}\frac{dU}{dV}dV=\iiint (\mathrm{\nabla }\cdot \mathbf{S})dV}$$ where ${\displaystyle U}$ is the energy of the electromagnetic radiation.

Applying the divergence theorem, the rate of flow of energy out of the volume is seen to be related to the surface integral of the Poynting vector over the surface of the volume of space:

${\displaystyle \frac{dU}{dt}=-}$${\displaystyle A}$${\displaystyle \mathbf{S}\cdot d\mathbf{A},}$

Point sources

A common example is the intensity or flux of a point source of given power output ${\displaystyle P}$. Considering a spherical volume centered on the source, the formula above becomes

${\displaystyle P=⟨-\frac{dU}{dt}⟩=}$${\displaystyle ⟨S⟩}$${\displaystyle A}$${\displaystyle d\mathbf{A},}$

where the angle brackets denote a time average over the period of the waves. Since the surface area of a sphere of radius ${\displaystyle r}$ is $A=4\pi r^2$ this gives $${\displaystyle P=⟨S⟩\cdot 4\pi r^2,}$$ therefore the intensity from the point source at distance ${\displaystyle r}$ is $${\displaystyle I=\frac{P}{4\pi r^2}.}$$ This is known as the inverse-square law.

Electromagnetic waves

For a monochromatic propagating electromagnetic wave such as a plane wave or a Gaussian beam travelling in a non-magnetic medium, the time-averaged Poynting vector is related to the amplitude of the electric field, E, by $${\displaystyle ⟨\mathsf{S}⟩=\frac{cn{ϵ}_0}{2}{E}^2,}$$ where c is the speed of light in vacuum, n is the refractive index of the medium, and ${\displaystyle {ϵ}_0}$ is the vacuum permittivity.

The relationship to intensity can also be seen by considering the time-averaged energy density of the wave: $${\displaystyle ⟨U⟩=\frac{n^2{ϵ}_0}{2}{E}^2.}$$ The local intensity is just the energy density times the wave velocity ⁠${\displaystyle \frac{c}{n}}$⁠: $${\displaystyle I=\frac{\mathrm{c}n{ϵ}_0}{2}{E}^2.}$$

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, but it is still often true that the magnitude of the time-averaged Poynting vector is proportional to the time-averaged energy density by a factor ${\displaystyle c}$:

$${\displaystyle I=⟨S⟩\propto c⟨U⟩}$$

An evanescent wave may have a finite electrical amplitude while not transferring any power. The intensity of an evanescent wave can be defined as the magnitude of the Poynting vector.