The
intensity (or
illuminance or
irradiance) of
light or other linear waves radiating from a
point source (energy per unit of area perpendicular to the source) is inversely proportional to the square of the distance from the source; so an object (of the same size) twice as far away, receives only one-quarter the
energy (in the same time period).
More generally, the irradiance,
i.e., the intensity (or
power per unit area in the direction of
propagation), of a
spherical wavefront varies inversely with the square of the distance from the source (assuming there are no losses caused by
absorption or
scattering).
For example, the intensity of radiation from the
Sun is 9126
watts per square meter at the distance of
Mercury (0.387
AU); but only 1367 watts per square meter at the distance of
Earth (1 AU)—an approximate threefold increase in distance results in an approximate ninefold decrease in intensity of radiation.
For non-
isotropic radiators such as
parabolic antennas, headlights, and
lasers, the effective origin is located far behind the beam aperture. If you are close to the origin, you don't have to go far to double the radius, so the signal drops quickly. When you are far from the origin and still have a strong signal, like with a laser, you have to travel very far to double the radius and reduce the signal. This means you have a stronger signal or have
antenna gain in the direction of the narrow beam relative to a wide beam in all directions of an
isotropic antenna.
In
photography and
stage lighting, the inverse-square law is used to determine the "fall off" or the difference in illumination on a subject as it moves closer to or further from the light source. For quick approximations, it is enough to remember that doubling the distance reduces illumination to one quarter;
[8] or similarly, to halve the illumination increase the distance by a factor of 1.4 (the
square root of 2), and to double illumination, reduce the distance to 0.7 (square root of 1/2). When the illuminant is not a point source, the inverse square rule is often still a useful approximation; when the size of the light source is less than one-fifth of the distance to the subject, the calculation error is less than 1%.
[9]
The fractional reduction in electromagnetic
fluence (Φ) for indirectly ionizing radiation with increasing distance from a point source can be calculated using the inverse-square law. Since emissions from a point source have radial directions, they intercept at a perpendicular incidence. The area of such a shell is 4π
r 2 where
r is the radial distance from the center. The law is particularly important in diagnostic
radiography and
radiotherapy treatment planning, though this proportionality does not hold in practical situations unless source dimensions are much smaller than the distance. As stated in fourier theory of heat "as the point source is magnification by distances , its radiation is dilute proportional to the sin of the angle, of the increasing circumference arc from the point of origin"
Example[edit]
Let the total power radiated from a point source, for example, an omnidirectional
isotropic radiator, be
P. At large distances from the source (compared to the size of the source), this power is distributed over larger and larger spherical surfaces as the distance from the source increases. Since the surface area of a sphere of radius
r is
A = 4
πr 2, then
intensity I (power per unit area) of radiation at distance
r is
I = P A = P 4 π r 2 . {\displaystyle I={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.\,}
The energy or intensity decreases (divided by 4) as the distance
r is doubled; measured in
dB it would decrease by 6.02 dB per doubling of distance.