Inverse-square law

In physics, an inverse-square law is any physical law stating that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity.

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.

Contents

Justification

The inverse-square law generally applies when some force, energy, or other conserved quantity is radiated outward radially in three-dimensional space from a point source. Since the surface area of a sphere (which is 4πr2 ) is proportional to the square of the radius, as the 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

Gravitation is the attraction of two objects with mass. This 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 was suggested in 1645 by Ismael Bullialdus. But Bullialdus did not accept 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[1] (Hooke’s lecture “On gravity” at the Royal Society, London, on 21 March; Borelli’s "Theory of the Planets", published later in 1666). 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,[2] as well as giving some credit to Bullialdus.

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 1015.[3]

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 one-quarter the energy (in the same time period).

More generally, the irradiance, i.e., the intensity (or power per unit area in the direction of propagation), of a spherical wavefront varies inversely with the square of the distance from the source (assuming there are no losses caused by absorption or scattering).

For example, the intensity of radiation from the Sun is 9126 watts per square meter at the distance of Mercury (0.387 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.

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;[4] 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%.[5]

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 4\pi r^2 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 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 = 4πr 2, then intensity I (power per unit area) of radiation at distance r is

I = \frac{P}{A} = \frac{P}{4 \pi r^2}. \,

The energy or intensity decreases (divided by 4) as the distance r is doubled; measured in dB it would decrease by 6.02 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.[6] 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 spherical wavefront radiating from a point source decreases by 50% as the distance r is doubled; measured in dB, the decreases is still 6.02 dB, since dB represents an intensity ratio. The behaviour is not inverse-square, but is inverse-proportional (inverse distance law):

p \ \propto \ \frac{1}{r} \,

The same is true for the component of particle velocity v \, that is in-phase with the instantaneous sound pressure p \,:

v \ \propto \frac{1}{r} \ \,

In the near field is a 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 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:

I \ = \ p v \ \propto \ \frac{1}{r^2}. \,

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 − 1)th power law

I\propto \frac{1}{r^{n-1}},

given that the space outside the source is divergence free.

See also

Notes

  1. ^ 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.
  2. ^ 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', at page 66.
  3. ^ Williams, Faller, Hill, E.; Faller, J.; Hill, H. (1971), "New Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass", Physical Review Letters 26 (12): 721–724, Bibcode 1971PhRvL..26..721W, doi:10.1103/PhysRevLett.26.721 
  4. ^ Millerson,G. (1991) Lighting for Film and Television - 3rd Edition p.27
  5. ^ Ryer,A. (1997) "The Light Measurement Handbook", ISBN 0-9658356-9-3 p.26
  6. ^ Inverse-Square law for sound

External links

 This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".