Inverse-square law 41288 215423909 2008-05-28T03:19:25Z 99.237.155.237 [[Image:Inverse square law.svg|thumb|500px|This diagram shows how the law works. The lines represent the [[flux]] emanating from the source. The total number of flux lines 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 some physical [[quantity]] or strength is [[Inverse (mathematics)|inverse]]ly [[proportionality (mathematics)|proportional]] to the [[square (algebra)|square]] of the [[distance]] from the source of that physical quantity. == Areas of application == In particular the inverse-square law applies in the following cases: doubling the distance between the light and the subject results in one quarter of the light hitting the subject. === Gravitation === [[Gravity|Gravitation]] refers to the attraction between two objects with mass. This law states:<br /> ''The gravitation 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''.<br /> 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.<br />This [[Law of universal gravitation|law]] was first suggested by [[Ismael Bullialdus]] but put on a firm basis by [[Isaac Newton]] after [[Robert Hooke]] proposed the idea in a letter to Newton. Hooke later accused Newton of plagiarism. === 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 |url=http://prola.aps.org/abstract/PRL/v26/i12/p721_1 |journal=[[Physical Review Letters]] |volume=26 |pages=721-724 |year=1971}}</ref> This implies a limit on the [[photon]] rest mass. === Light and other electromagnetic radiation=== 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 ¼ 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 9140 [[watt]]s per square meter at the distance of [[Mercury (planet)|Mercury]] (0.387AU); but only 1370 watts per square meter at the distance of [[Earth]] (1AU)&mdash;a threefold increase in distance results in a ninefold decrease in intensity of radiation. Photographers and theatrical lighting professionals use the inverse-square law to determine optimal location of the [[light source]] for proper illumination of the subject. 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 planing, though this proportionality does not hold in practical situations unless source dimensions are much smaller than the distance '''r'''. === Acoustics === The inverse-square law is used in [[acoustics]] in measuring the [[sound intensity]] at a given distance from the source.<ref name="Inverse square sound intensity">[http://hyperphysics.phy-astr.gsu.edu/hbase/acoustic/invsqs.html Inverse-Square law for sound]</ref> == Examples == === Electromagnetic radiation === Let the total power radiated from a point source, ''e.g.,'' an omnidirectional [[isotropic antenna]], be <math> P \ </math>. 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 <math> r \ </math> is <math> A = 4 \pi r^2 \ </math>, then [[intensity]] <math> I \ </math> of radiation at distance <math> r \ </math> is :<math> I = \frac{P}{A} = \frac{P}{4 \pi r^2}. \, </math> :<math> I \propto \frac{1}{r^2} \, </math> :<math>\frac{I_1} {I_2 } = \frac{{r_2}^2}{{r_1}^2} \, </math> :<math> I_1 = I_{2} \cdot {r_{2}^2} \cdot \frac{1}{{r_1}^2} \, </math> The energy or intensity decreases by a factor of ¼ as the distance <math>r</math> is doubled, or measured in [[Decibel|dB]] it would decrease by 6.02 dB. This is the fundamental reason why [[intensity]] of [[radiation]], whether it is [[electromagnetic]] or [[acoustics|acoustic]] radiation, follows the inverse-square behaviour, at least in the ideal 3 dimensional context (propagation in 2 dimensions would follow a just an inverse-proportional distance behaviour and propagation in one dimension, the [[plane wave]], remains constant in amplitude even as distance from the source changes). === Acoustics === In [[acoustics]], the [[Sound#Sound_pressure|sound pressure]] of a [[sphere|spherical]] [[wavefront]] radiating from a point source decreases by 50% as the distance <math> r</math> is doubled, or measured in [[Decibel|dB]] it decreases by 6.02 dB. The behaviour is not inverse-square, but is inverse-proportional: :<math> p \propto \frac{1}{r} \, </math> :<math> \frac{p_1} {p_2 } = \frac{r_2}{r_1} \, </math> :<math> p_1 = p_2 \cdot r_2 \cdot \frac{1}{r_1} \, </math> However the same is also true for the component of [[particle velocity]] <math> v \,</math> that is [[quadrature|in-phase]] to the instantaneous sound pressure <math>p \,</math>. :<math> v \propto \frac{1}{r} \, </math> Only in the [[near field]] the [[quadrature]] component of the particle velocity is 90° out of phase with the sound pressure and thus does not contribute to the time-averaged energy or the intensity of the sound. This quadrature component happens to be inverse-square. The [[sound intensity]] is the product of the [[root mean square|RMS]] sound pressure and the RMS particle velocity (the in-phase component), both which are inverse-proportional, so the intensity follows an inverse-square behaviour as is also indicated above: :<math> I = p \cdot v \propto \frac{1}{r^2}. \, </math> The inverse-square law pertained to sound intensity. Because sound pressures are more accessible to us, the same law can be called the "inverse-distance law". == Field theory interpretation == For an [[irrotational vector field]] in three-dimensional space the law corresponds to the property that the [[divergence]] is zero outside the source. Generally, for irrotational vector field in ''n''-dimensional [[Euclidean space]], inverse (''n''&nbsp;&minus;&nbsp;1)<sup>th</sup> potention law corresponds to the property of zero divergence outside the source. ==See also== *[[Flux]] *[[Gauss's law]] *[[Kepler's first law]] *[[Telecommunications]] == 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] ==Notes== {{reflist}} {{FS1037C}} [[Category:Philosophy of physics]] [[Category:Scientific method]] [[Category:Mathematical terminology]] [[de:Abstandsgesetz]] [[es:Ley de la inversa del cuadrado]] [[fr:Loi en carré inverse]] [[ja:逆2乗の法則]] [[ko:거꿀제곱법칙]] [[zh:反平方定律]]