Intensity (physics) - Mathematical Description

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:

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 complex amplitude of the electric field, then the time-averaged energy density 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 index, is the speed of light in vacuum and is the vacuum permittivity.

The treatment above does not hold for electromagnetic fields that are not radiating, such as for an evanescent wave. In these cases, the intensity can be defined as the magnitude of the Poynting vector.