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[[Image:Inverse square law.svg|thumb|420px|The lines represent the [[flux]] emanating from the source. The total number of [[flux line]]s depends on the strength of the source and is constant with increasing distance. A greater density of flux lines (lines per unit area) means a stronger field. The density of flux lines is inversely proportional to the square of the distance from the source because the surface area of a sphere increases with the square of the radius. Thus the strength of the field is inversely proportional to the square of the distance from the source.]]


In [[physics]], an '''inverse-square law''' is any [[physical law]] stating that a specified physical [[quantity]] or intensity is [[Proportionality_(mathematics)#Inverse_proportionality|inversely proportional]] to the [[square (algebra)|square]] of the [[distance]] from the source of that physical quantity. In equation form:


:<math>\mbox{Intensity} \ \propto \ \frac{1}{\mbox{distance}^2} \, </math>
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The divergence of a [[vector field]] which is the resultant of radial inverse-square law fields with respect to one or more sources is everywhere proportional to the strength of the local sources, and hence zero outside sources. [[Newton's law of universal gravitation]] follows an inverse-square law, as do the effects of [[electricity|electric]], [[magnetism|magnetic]], [[light]], [[sound]], and [[radiation]] phenomena.
After you're done, go ahead and take free end of coaxial cable and fix it on the input on CCTV converter box. Like Proline UK in the obsolete DVR's like PR 777, PR 1004, PR 1008, PR 1016 used the identical compression technology. CCTV were initially used only in leading places of security alerts like the airport, banks and government establishments. Employee Performance: Every activities with the employee can be tracked and about the basis of the work and gratifaction, any action could be taken.
 
==Justification==
The inverse-square law generally applies when some force, energy, or [[flux|other conserved quantity]] is evenly radiated outward from a [[point source]] in [[three-dimensional space]].  Since the [[surface area]] of a [[sphere]] (which is&nbsp;4π''r''<sup>2</sup>&nbsp;) is proportional to the square of the radius, as the [[flux density|emitted radiation]] gets farther from the source, it is spread out over an area that is increasing in proportion to the square of the distance from the source. Hence, the intensity of radiation passing through any unit area (directly facing the point source) is inversely proportional to the square of the distance from the point source. [[Gauss's law]] applies to, and can be used with any physical quantity that acts in accord to, the inverse-square relationship.
 
==Occurrences==
 
===Gravitation===
[[Gravity|Gravitation]] is the attraction of two objects with mass. Newton's law states:
:''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 from their center.
 
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 point mass while calculating the gravitational force.
 
As the law of gravitation, this [[Law of universal gravitation|law]] was suggested in 1645 by [[Ismael 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 Curtis Wilson (1989), "The Newtonian achievement in astronomy", ch.13 (pages 233–274) in "Planetary astronomy from the Renaissance to the rise of astrophysics: 2A: Tycho Brahe to Newton", CUP 1989.</ref> (Hooke’s lecture "On gravity" at the Royal Society, London, on 21 March;<ref>Thomas Birch, ''The History of the Royal Society of London'', … (London, England:  1756), vol. 2, [http://books.google.com/books?id=lWEVAAAAQAAJ&pg=PA68#v=onepage&q&f=false pages 68-73]; see especially pages 70-72.</ref> Borelli's "Theory of the Planets", published later in 1666<ref>Giovanni Alfonso Borelli, [http://books.google.com/books?id=YZk_AAAAcAAJ&pg=PT4#v=onepage&q&f=false ''Theoricae Mediceorum Planetarum ex Causius Physicis Deductae''] [Theory [of the motion] of the Medicean planets [i.e., moons of Jupiter] deduced from physical causes] (Florence, (Italy):  1666).</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]]. Hooke remained bitter about Newton claiming the invention of this principle, even though Newton's ''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 the 1729 English translation of the ''Principia'', [http://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA66#v=onepage&q=&f=false at page 66].</ref> as well as giving some credit to Bullialdus.<ref>In a letter to Edmund Halley dated June 20, 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:  I. Bernard Cohen and George E. Smith, ed.s, ''The Cambridge Companion to Newton'' (Cambridge, England:  Cambridge University Press, 2002), [http://books.google.com/books?id=3wIzvqzfUXkC&pg=PA204#v=onepage&q&f=false page 204].
</ref>
 
===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>{{citation | last=Williams, Faller, Hill |title=New Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass |journal=[[Physical Review Letters]] |volume=26 | issue=12 |pages=721–724 |year=1971 | doi=10.1103/PhysRevLett.26.721 | first1=E. | last2=Faller | first2=J. | last3=Hill | first3=H. | bibcode=1971PhRvL..26..721W}}</ref>
 
===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 new 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 [[theatrical 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>Millerson,G. (1991) ''Lighting for Film and Television - 3rd Edition'' p.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>Ryer,A. (1997) "The Light Measurement Handbook", ISBN 0-9658356-9-3 p.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.
 
====Example====
 
Let the total power radiated from a point source, for example, an omnidirectional [[isotropic antenna]], be&nbsp;''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''&nbsp;=&nbsp;4''πr''<sup>&nbsp;2</sup>, then [[intensity (physics)|intensity]] ''I'' (power per unit area) of radiation at distance ''r'' is
:<math>
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; measured in [[Decibel|dB]] it would decrease by 6.02&nbsp;dB per doubling of distance.
 
===Acoustics===
 
In [[acoustics]] one usually measures the [[sound pressure]] at a given distance ''r'' from the source using the 1/r law.<ref name="Inverse square sound intensity">[http://hyperphysics.phy-astr.gsu.edu/hbase/acoustic/invsqs.html Inverse-Square law for sound]</ref>  Since intensity is proportional to the square of pressure amplitude, this is just a variation on the inverse-square law.
 
====Example====
 
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 dB, since dB represents an intensity ratio. The behaviour is not inverse-square, but is inverse-proportional (inverse distance law):
 
:<math> 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> 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> I \ = \ p v \ \propto \ \frac{1}{r^2}. \, </math>
 
==Field theory interpretation==
For an [[irrotational vector field]] in three-dimensional space the inverse-square law corresponds to the property that the [[divergence]] is zero outside the source. This can be generalized to higher dimensions. Generally, for an irrotational vector field in ''n''-dimensional [[Euclidean space]], the intensity "I" of the vector field falls off with the distance "r" following the inverse (''n''&nbsp;−&nbsp;1)<sup>th</sup> power law
:<math>I\propto \frac{1}{r^{n-1}}</math>,
given that the space outside the source is divergence free. {{Citation needed|date=March 2011}}
 
==History==
 
[[John Dumbleton]] of the 14th-century [[Oxford Calculators]], was one of the first to express functional relationships in graphical form. He gave a proof of the [[mean speed theorem]] stating that "the latitude of a uniformly difform movement corresponds to the degree of the midpoint" and used this method to study the quantitative decrease in intensity of illumination  in his ''Summa logicæ et philosophiæ naturalis'' (ca. 1349), stating that it was not linearly proportional to the distance, but was unable to expose the Inverse-square law.<ref>John Freely, ''Before Galileo: The Birth of Modern Science in Medieval Europe'' (2012)</ref>
 
In proposition 9 of Book 1 in his book ''Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur'' (1604), the astronomer [[Johannes Kepler]] argued that the spreading of light from a point source obeys an inverse square law:<ref>Johannes Kepler, ''Ad Vitellionem Paralipomena, quibus astronomiae pars optica traditur'' (Frankfurt, (Germany):  Claude de Marne & heir Jean Aubry, 1604), [http://daten.digitale-sammlungen.de/~db/bsb00007828/images/index.html?id=00007828&fip=eayaxsewqxsfsdreayasdasxdsydeayaxsewq&no=32&seite=30 page 10.]</ref><ref>Translation of the Latin quote from Kepler's ''Ad Vitellionem paralipomena'' is from:  Gal, O. & Chen-Morris, R.(2005) [http://articles.adsabs.harvard.edu//full/2005HisSc..43..391G/0000398.000.html "The Archaeology of the Inverse Square Law: (1) Metaphysical Images and Mathematical Practices,"] ''History of Science'', '''43''' :  391-414 ; see especialy p. 397.</ref>
 
<blockquote>Original:  ''Sicut se habent spharicae superificies, quibus origo lucis pro centro est, amplior ad angustiorem:  ita se habet fortitudo seu densitas lucis radiorum in angustiori, ad illamin in laxiori sphaerica, hoc est, conversim.  Nam per 6. 7. tantundem lucis est in angustiori sphaerica superficie, quantum in fusiore, tanto ergo illie stipatior & densior quam hic.''</blockquote>
 
<blockquote>''Translation'':  Just as [the ratio of] spherical surfaces, for which the source of light is the center, [is] from the wider to the narrower, so the density or fortitude of the rays of light in the narrower [space], towards the more spacious spherical surfaces, that is, inversely.  For according to [propositions] 6 & 7, there is as much light in the narrower spherical surface, as in the wider, thus it is as much more compressed and dense here than there.</blockquote>
 
In 1645 in his book ''Astronomia Philolaica'' … , the French astronomer [[Ismaël Bullialdus]] (1605 – 1694) refuted Johannes Kepler's suggestion that "gravity"<ref>Note:  Kepler's notion of "gravity" was not the modern notion:  Kepler believed that the Sun exerted a force on the planets which propelled them in their orbits around the Sun.  Like beams of light from a light house, this force rotated with the Sun as it turned.</ref> weakens as the inverse of the distance; instead, Bullialdus argued, "gravity" weakens as the inverse square of the distance:<ref>Ismail Bullialdus, ''Astronomia Philolaica'' … (Paris, France:  Piget, 1645), [http://diglib.hab.de/drucke/2-1-4-astron-2f-1/start.htm?image=00005 page 23.]</ref><ref>Translation of the Latin quote from Bullialdus' ''Astronomia Philolaica'' … is from: O'Connor, John J. and Roberson, Edmund F.  (2006)  [http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Boulliau.html "Ismael Boulliau"], The MacTutor History of Mathematics Archive,  School of Mathematics and Statistics, University of Saint Andrews, Scotland. </ref>
 
<blockquote>Original:  ''Virtus autem illa, qua Sol prehendit seu harpagat planetas, corporalis quae ipsi pro manibus est, lineis rectis in omnem mundi amplitudinem emissa quasi species solis cum illius corpore rotatur:  cum ergo sit corporalis imminuitur, & extenuatur in maiori spatio & intervallo, ratio autem huius imminutionis eadem est, ac luminus, in ratione nempe dupla intervallorum, sed eversa.''</blockquote>
 
<blockquote>''Translation'':  As for the power by which the Sun seizes or holds the planets, and which, being corporeal, functions in the manner of hands, it is emitted in straight lines throughout the whole extent of the world, and like the species of the Sun, it turns with the body of the Sun; now, seeing that it is corporeal, it becomes weaker and attenuated at a greater distance or interval, and the ratio of its decrease in strength is the same as in the case of light, namely, the duplicate proportion, but inversely, of the distances [that is, 1/d²].</blockquote>
 
In England, the Anglican bishop [[Seth Ward (bishop of Salisbury)|Seth Ward]] (1617 – 1689) publicized the ideas of Bullialdus in his critique ''In Ismaelis Bullialdi astronomiae philolaicae fundamenta inquisitio brevis'' (1653) and publicized the planetary astronomy of Kepler in his book ''Astronomia geometrica'' (1656).
 
In 1663-1664, the English scientist [[Robert Hooke]] was writing his book ''Micrographia'' (1666) in which he discussed, among other things, the relation between the height of the atmosphere and the barometric pressure at the surface.  Since the atmosphere surrounds the earth, which itself is a sphere, the volume of atmosphere bearing on any unit area of the earth's surface is a truncated cone (which extends from the earth's center to the vacuum of space ; obviously only the section of the cone from the earth's surface to space bears on the earth's surface).  Although the volume of a cone is proportional to the cube of its height, Hooke argued that the air's pressure at the earth's surface is instead proportional to the height of the atmosphere because gravity diminishes with altitude.  Although Hooke did not explicitly state so, the relation that he proposed would be true only if gravity decreases as the inverse square of the distance from the earth's center.<ref>(Gal & Chen-Morris, 2005), pp. 391-392.</ref><ref>Robert Hooke, ''Micrographia'' … (London, England:  John Martyn, 1667), [http://digicoll.library.wisc.edu/cgi-bin/HistSciTech/HistSciTech-idx?type=goto&id=HistSciTech.HookeMicro&isize=M&submit=Go+to+page&page=227 page 227:]  "[I say a ''Cylinder'', not a piece of a ''Cone'', because, as I may elsewhere shew in the Explication of Gravity, that ''triplicate'' proportion of the shels of a Sphere, to their respective diameters, I suppose to be removed in this case by the decrease of the power of Gravity.]"</ref>
 
==See also==
*[[Flux]]
*[[Gauss's law]]
*[[Kepler's first law]]
*[[Telecommunications]], particularly:
**[[William Thomson, 1st Baron Kelvin#Calculations on data rate|William Thomson, 1st Baron Kelvin]]
**[[List of ad hoc routing protocols#Power-aware routing protocols|Power-aware routing protocols]]
*[[Inverse proportion#Inverse proportionality|Inverse proportionality]]
*[[Multiplicative inverse]]
 
==References==
{{FS1037C}}
{{reflist}}
 
==External links==
*[http://www.sengpielaudio.com/calculator-distance.htm Damping of sound level with distance]
*[http://www.sengpielaudio.com/calculator-distancelaw.htm Sound pressure p and the inverse distance law 1/r]
*[http://www.ionactive.co.uk/multi-media_video.html?m=6 Inverse Square Law & Radiation Protection by Ionactive (Animation)]
 
{{Use dmy dates|date=March 2011}}
 
{{DEFAULTSORT:Inverse-Square Law}}
[[Category:Philosophy of physics]]
[[Category:Scientific method]]

Latest revision as of 02:43, 10 March 2014


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The usage of CCTV afterwards became common in banks and stores to discourage theft, by recording evidence of criminal activity. Why does one need rest and pleasurable, physically and mentally. There are lots of free benefits which might be included to suit your needs when purchasing CCTV products. This helps in enhancing security therefore in the broader way at premises.

You want anyone who has a nice relationship with his parents. Its unique design allows them to be mounted either around the wall or ceiling. Information such as time stamping was usually not available, along with the fidelity of the playback quality was poor.

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Security in general and CCTV particularly helps prevent hooligans from taking law into their particular hands and in the same time keeps criminals at bay. There are countless online portals which cater assistance for the parents to pick the preeminent school because of their child.

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After you're done, go ahead and take free end of coaxial cable and fix it on the input on CCTV converter box. Like Proline UK in the obsolete DVR's like PR 777, PR 1004, PR 1008, PR 1016 used the identical compression technology. CCTV were initially used only in leading places of security alerts like the airport, banks and government establishments. Employee Performance: Every activities with the employee can be tracked and about the basis of the work and gratifaction, any action could be taken.