Editing Inverse-square law (section)

==Occurrences==

===Gravitation===
[[Gravity|Gravitation]] is the attraction between objects that have mass. Newton's law states:

{{quote|The gravitational attraction force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of their separation distance. The force is always attractive and acts along the line joining them.<ref name=Newton1>Proposition 75, Theorem 35: p. 956 – I.Bernard Cohen and Anne Whitman, translators: [[Isaac Newton]], ''The Principia'': [[Mathematical Principles of Natural Philosophy]]. Preceded by ''A Guide to Newton's Principia'', by I.Bernard Cohen. University of California Press 1999 {{ISBN|0-520-08816-6}} {{ISBN|0-520-08817-4}}</ref>}}

<math display="block">F=G\frac{m_1 m_2}{r^2}</math>

If the distribution of matter in each body is spherically symmetric, then the objects can be treated as point masses without approximation, as shown in the [[shell theorem]].  Otherwise, if we want to calculate the attraction between massive bodies, we need to add all the point-point attraction forces vectorially and the net attraction might not be exact inverse square. However, if the separation between the massive bodies is much larger compared to their sizes, then to a good approximation, it is reasonable to treat the masses as a point mass located at the object's [[center of mass]] while calculating the gravitational force.

As the law of gravitation, this [[Law of universal gravitation|law]] was suggested in 1645 by [[Ismaël Bullialdus]]. But Bullialdus did not accept [[Kepler's laws of planetary motion|Kepler's second and third laws]], nor did he appreciate [[Christiaan Huygens]]'s solution for circular motion (motion in a straight line pulled aside by the central force).  Indeed, Bullialdus maintained the sun's force was attractive at aphelion and repulsive at perihelion. [[Robert Hooke]] and [[Giovanni Alfonso Borelli]] both expounded gravitation in 1666 as an attractive force.<ref>Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses. See page 239 in: {{Cite book |title=General history of astronomy |date=2009 |publisher=[[Cambridge University Press]] |isbn=978-0-521-54205-0 |editor-last=Taton |editor-first=René |editor-link=René Taton |edition=1. |volume=2 Pt. A: Planetary astronomy from the Renaissance to the rise of astrophysics Tycho Brahe to Newton |location=Cambridge |pages=233–274 |chapter=The Newtonian achievement in astronomy |editor-last2=Wilson |editor-first2=Curtis |editor-last3=Hoskin |editor-first3=Michael A.}}</ref> Hooke's lecture "On gravity" was at the Royal Society, in London, on 21 March.<ref>{{Cite book |last=Birch |first=Thomas |author-link=Thomas Birch |url=https://books.google.com/books?id=lWEVAAAAQAAJ |title=The History of the Royal Society of London |date=1756 |volume=2 |pages=68–73}}; see especially pages 70–72.</ref> Borelli's "Theory of the Planets" was published later in 1666.<ref>{{Cite book |last=Borelli |first=Giovanni Alfonso |author-link=Giovanni Alfonso Borelli |url=https://books.google.com/books?id=YZk_AAAAcAAJ |title=Theoricae mediceorum planetarum ex causis physicis deductae |date=1666 |publisher=Ex typographia S.M.D. |bibcode=1666tmpe.book.....B |language=la |trans-title=Theory [of the motion] of the Medicean planets [i.e., moons of Jupiter] deduced from physical causes}}</ref>  Hooke's 1670 Gresham lecture explained that gravitation applied to "all celestiall bodys" and added the principles that the gravitating power decreases with distance and that in the absence of any such power bodies move in straight lines. By 1679, Hooke thought gravitation had inverse square dependence and communicated this in a letter to [[Isaac Newton]]:<ref name="Hooke1680">{{Cite journal |last=Koyré |first=Alexandre |author-link=Alexandre Koyré |year=1952 |title=An Unpublished Letter of Robert Hooke to Isaac Newton |journal=[[Isis (journal)|Isis]] |volume=43 |issue=4 |pages=312–337 |doi=10.1086/348155 |jstor=227384 |pmid=13010921 }}</ref> 
''my supposition is that the attraction always is in duplicate proportion to the distance from the center reciprocall''.<ref>Hooke's letter to Newton of 6 January 1680 (Koyré 1952:332).</ref>

Hooke remained bitter about Newton claiming the invention of this principle, even though Newton's 1686 ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' acknowledged that Hooke, along with Wren and Halley, had separately appreciated the inverse square law in the [[Solar System]],<ref>Newton acknowledged Wren, Hooke and Halley in this connection in the Scholium to Proposition 4 in Book 1 (in all editions): See for example: {{Cite book |last=Newton |first=Isaac |author-link=Isaac Newton |url=https://books.google.com/books?id=Tm0FAAAAQAAJ |title=The Mathematical Principles of Natural Philosophy |date=1729 |publisher=B. Motte |page=66}}</ref> as well as giving some credit to Bullialdus.<ref>In a letter to Edmund Halley dated 20 June 1686, Newton wrote:  "Bullialdus wrote that all force respecting ye Sun as its center & depending on matter must be reciprocally in a duplicate ratio of ye distance from ye center." See: {{Cite book |last1=Cohen |first1=I. Bernard |author-link=I. Bernard Cohen |url=https://books.google.com/books?id=3wIzvqzfUXkC |title=The Cambridge Companion to Newton |last2=Smith |first2=George E. |author-link2=George E. Smith |date=2002 |publisher=Cambridge University Press |isbn=978-0-521-65696-2 |pages=204}}</ref>

===Electrostatics===
{{Main|Electrostatics}}
The force of attraction or repulsion between two electrically charged particles, in addition to being directly proportional to the product of the electric charges, is inversely proportional to the square of the distance between them; this is known as [[Coulomb's law]]. The deviation of the exponent from 2 is less than one part in 10<sup>15</sup>.<ref>{{Cite journal |last1=Williams |first1=E. R. |last2=Faller |first2=J. E. |last3=Hill |first3=H. A. |year=1971 |title=New Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass |journal=[[Physical Review Letters]] |language=en |volume=26 |issue=12 |pages=721–724 |bibcode=1971PhRvL..26..721W |doi=10.1103/PhysRevLett.26.721 }}</ref>

<math display="block">F=k_\text{e}\frac{q_1 q_2}{r^2}</math>

===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.

===Sound in a gas===

In [[acoustics]], the [[sound pressure]] of a [[sphere|spherical]] [[wavefront]] radiating from a point source decreases by 50% as the distance ''r'' is doubled; measured in [[Decibel|dB]], the decrease is still 6.02&nbsp;dB, since dB represents an intensity ratio. The pressure ratio (as opposed to power ratio) is not inverse-square, but is inverse-proportional (inverse distance law):

<math display="block"> p \ \propto \ \frac{1}{r} \,
 </math>

The same is true for the component of [[particle velocity]] <math> v \,</math> that is [[Phase (waves)#In-phase and quadrature (I&Q) components|in-phase]] with the instantaneous sound pressure <math>p \,</math>:

<math display="block"> v \ \propto \frac{1}{r} \ \, </math>

In the [[Near and far field|near field]] is a [[quadrature phase|quadrature component]] of the particle velocity that is 90° out of phase with the sound pressure and does not contribute to the time-averaged energy or the intensity of the sound.  The [[sound intensity]] is the product of the [[root mean square|RMS]] sound pressure and the ''in-phase'' component of the RMS particle velocity, both of which are inverse-proportional. Accordingly, the intensity follows an inverse-square behaviour:

<math display="block"> I \ = \ p v \ \propto \ \frac{1}{r^2}. \, </math>