In physics, an inverse-square law is any physical law stating that a specified physical quantity
or intensity is inversely proportional to the square of the distance from the source
of that physical quantity. In equation form:
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 electric, magnetic, light, sound, and radiation
phenomena.
Justification The inverse-square law generally applies when
some force, energy, or other conserved quantity is evenly radiated outward from a point source
in three-dimensional space. Since the surface area of a sphere 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 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. 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.
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. 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.
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, 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.
Light and other electromagnetic radiation The intensity of light or other linear waves
radiating from a point source is inversely proportional to the square of the distance
from the source; so an object twice as far away, receives only one-quarter the energy.
More generally, the irradiance, i.e., the intensity, of a spherical wavefront varies
inversely with the square of the distance from the source.
For example, the intensity of radiation from the Sun is 9126 watts per square meter at
the distance of Mercury; but only 1367 watts per square meter at the distance of Earth—an
approximate threefold increase in distance results in an approximate ninefold decrease
in intensity of radiation. For non isotropic radiators such as parabolic
antennas, headlights, and lasers, 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 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; or similarly, to halve the illumination increase
the distance by a factor of 1.4, and to double illumination, reduce the distance to 0.7.
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%.
The fractional reduction in electromagnetic fluence 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 4πr 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. 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 the total power radiated from a point
source, for example, an omnidirectional isotropic antenna, be P. At large distances from 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 of radiation at distance r is
The energy or intensity decreases 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. 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 decrease is still 6.02 dB, since dB represents an intensity ratio. The behaviour
is not inverse-square, but is inverse-proportional:
The same is true for the component of particle velocity that is in-phase with the instantaneous
sound pressure :
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:
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 inverseth power law
, given that the space outside the source is
divergence free. 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, stating that it was not linearly
proportional to the distance, but was unable to expose the Inverse-square law.
In proposition 9 of Book 1 in his book Ad Vitellionem paralipomena, quibus astronomiae
pars optica traditur, the astronomer Johannes Kepler argued that the spreading of light
from a point source obeys an inverse square law:
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.
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.
In 1645 in his book Astronomia Philolaica …, the French astronomer Ismaël Bullialdus
refuted Johannes Kepler's suggestion that "gravity" weakens as the inverse of the distance;
instead, Bullialdus argued, "gravity" weakens as the inverse square of the distance:
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.
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].
In England, the Anglican bishop Seth Ward publicized the ideas of Bullialdus in his
critique In Ismaelis Bullialdi astronomiae philolaicae fundamenta inquisitio brevis and
publicized the planetary astronomy of Kepler in his book Astronomia geometrica.
In 1663-1664, the English scientist Robert Hooke was writing his book Micrographia 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. 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.
See also Flux
Gauss's law Kepler's first law
Telecommunications, particularly: William Thomson, 1st Baron Kelvin
Power-aware routing protocols
Inverse proportionality Multiplicative inverse
References This article incorporates public domain
material from the General Services Administration document "Federal Standard 1037C".
External links Damping of sound level with distance
Sound pressure p and the inverse distance law 1/r
Inverse Square Law & Radiation Protection by Ionactive