Intensity (physics)

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, $${\displaystyle P=\int \mathbf {I} \,\cdot d\mathbf {A} ,}$$  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 $${\displaystyle P=|I|\cdot A_{\mathrm {surf} }=|I|\cdot 4\pi r^{2},}$$  where

• |I| is the intensity at the surface of the sphere;
• r is the radius of the sphere;
• ${\displaystyle A_{\mathrm {surf} }=4\pi r^{2}}$  is the expression for the surface area of a sphere.

Solving for |I| gives $${\displaystyle |I|={\frac {P}{A_{\mathrm {surf} }}}={\frac {P}{4\pi r^{2}}}.}$$ 

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: $${\displaystyle \left\langle U\right\rangle ={\frac {n^{2}\varepsilon _{0}}{2}}|E|^{2},}$$  and the local intensity is obtained by multiplying this expression by the wave velocity, ${\displaystyle {\tfrac {\mathrm {c} }{n}}\!:}$  $${\displaystyle I={\frac {\mathrm {c} n\varepsilon _{0}}{2}}|E|^{2},}$$  where

• n is the refractive index;
• c is the speed of lightinvacuum;
• ε₀ 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.^[1]

Inphotometry 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 intensityorirradiance, 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.

• Field strength
• Sound intensity
• Magnitude (astronomy)