- Inverse-square law
In

physics , an**inverse-square law**is anyphysical law stating that some physicalquantity or strength is inversely proportional to the square of thedistance 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**Gravitation is the attraction between two objects with mass. This law states:

: "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".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.

This law was first suggested byIsmael Bullialdus but put on a firm basis byIsaac Newton afterRobert 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^{15}. [*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=*] This implies a limit on thePhysical Review Letters |volume=26 |pages=721-724 |year=1971photon rest mass.**Light and other electromagnetic radiation**The

intensity (orilluminance orirradiance ) oflight or other linear waves radiating from apoint 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 theenergy (in the same time period).More generally, the

irradiance , "i.e.," theintensity (or power per unit area in the direction of propagation), of a sphericalwavefront varies inversely with the square of the distance from the source (assuming there are no losses caused by absorption orscattering ).For example, the intensity of radiation from the

Sun is 9140watt s per square meter at the distance of Mercury (0.387AU); but only 1370 watts per square meter at the distance ofEarth (1AU)—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**^{2}where**r**is the radial distance from the center.The law is particularly important in diagnostic

radiography andradiotherapy 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 thesound intensity at a given distance from the source. [*http://hyperphysics.phy-astr.gsu.edu/hbase/acoustic/invsqs.html Inverse-Square law for sound*] ]**Examples****Electromagnetic radiation**Let the total power radiated from a point source, "e.g.," an omnidirectional

isotropic antenna , 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 $A\; =\; 4\; pi\; r^2$, thenintensity $I$ of radiation at distance $r$ is:$I\; =\; frac\{P\}\{A\}\; =\; frac\{P\}\{4\; pi\; r^2\}.\; ,$:$I\; propto\; frac\{1\}\{r^2\}\; ,$:$frac\{I\_1\}\; \{I\_2\; \}\; =\; frac$r_2}^2}r_1}^2} ,:$I\_1\; =\; I\_\{2\}\; cdot$r_2}^2} cdot frac{1}r_1}^2} ,The energy or intensity decreases by a factor of ¼ as the distance $r$ is doubled, or measured in dB it would decrease by 6.02 dB. This is the fundamental reason why

intensity ofradiation , whether it iselectromagnetic or acoustic radiation, follows the inverse-square behaviour, at least in the ideal 3 dimensional context (propagation in 2 dimensions would just follow an inverse-proportional distance behaviour and propagation in one dimension, theplane wave , remains constant in amplitude even as distance from the source changes).**Acoustics**In

acoustics , the sound pressure of a sphericalwavefront radiating from a point source decreases by 50% as the distance $r$ is doubled, or measured in dB it decreases by 6.02 dB. The behaviour is not inverse-square, but is inverse-proportional::$p\; propto\; frac\{1\}\{r\}\; ,$:$frac\{p\_1\}\; \{p\_2\; \}\; =\; frac\{r\_2\}\{r\_1\}\; ,$:$p\_1\; =\; p\_2\; cdot\; r\_2\; cdot\; frac\{1\}\{r\_1\}\; ,$However the same is also true for the component of

particle velocity $v\; ,$ that is in-phase to the instantaneous sound pressure $p\; ,$.:$v\; propto\; frac\{1\}\{r\}\; ,$Only in the

near field thequadrature 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. Thesound intensity is the product of the 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::$I\; =\; p\; cdot\; v\; propto\; frac\{1\}\{r^2\}.\; ,$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 thedivergence is zero outside the source. Generally, for irrotational vector field in "n"-dimensionalEuclidean space , inverse ("n" − 1)^{th}potention law corresponds to the property of zero divergence outside the source.**ee 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**

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