Editing Inverse-square law (section)

===Light and other electromagnetic radiation===
The [[intensity (physics)|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 (physics)|power]] per unit area in the direction of [[wave propagation|propagation]]), of a [[sphere|spherical]] [[wavefront]] varies inversely with the square of the distance from the source (assuming there are no losses caused by [[absorption (optics)|absorption]] or [[scattering]]).

For example, the intensity of radiation from the [[Sun]] is 9126 [[watt]]s per square meter at the distance of [[Mercury (planet)|Mercury]] (0.387 [[Astronomical unit|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 radiator]]s such as [[parabolic antenna]]s, headlights, and [[laser]]s, 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 radiator|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;<ref>{{Cite book |last=Millerson |first=Gerald |url=https://books.google.com/books?id=Kf6XAAAAQBAJ |title=Lighting for TV and Film |date=1999 |publisher=CRC Press |isbn=978-1-136-05522-5 |pages=27}}</ref> 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%.<ref>{{Cite book |last=Ryder |first=Alexander D. |url=https://cgvr.informatik.uni-bremen.de/teaching/cg_literatur/ILT-Light-Measurement-Handbook.pdf |title=The Light Measurement Handbook |date=1997 |publisher=International Light |isbn=978-0-96-583569-5 |pages=26}}</ref>

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'' <sup>2</sup> 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====
Let ''P''&nbsp; be the total power radiated from a point source (for example, an omnidirectional [[isotropic radiator]]). 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''&nbsp;=&nbsp;4''πr''<sup>&nbsp;2</sup>, the [[intensity (physics)|intensity]] ''I'' (power per unit area) of radiation at distance ''r'' is
<math display="block">
I = \frac P A = \frac P {4 \pi r^2}. \,
</math>

The energy or intensity decreases (divided by&nbsp;4) as the distance ''r'' is doubled; if measured in [[Decibel|dB]] would decrease by 6.02&nbsp;dB per doubling of distance. When referring to measurements of power quantities, a ratio can be expressed as a level in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to the reference value.