Inverse-square law
The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.
Radar energy expands during both the signal transmission and the reflected return, so the inverse square for both paths means that the radar will receive energy according to the inverse fourth power of the range.
To prevent dilution of energy while propagating a signal, certain methods can be used such as a waveguide, which acts like a canal does for water, or how a gun barrel restricts hot gas expansion to one dimension in order to prevent loss of energy transfer to a bullet.
In mathematical notation the inverse square law can be expressed as an intensity (I) varying as a function of distance (d) from some centre.
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.
Gauss's law for gravity is similarly applicable, and can be used with any physical quantity that acts in accordance with the inverse-square relationship.
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.
Robert Hooke and Giovanni Alfonso Borelli both expounded gravitation in 1666 as an attractive force.
[3] 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:[4] my supposition is that the attraction always is in duplicate proportion to the distance from the center reciprocall.
[5] Hooke remained bitter about Newton claiming the invention of this principle, even though Newton's 1686 Principia acknowledged that Hooke, along with Wren and Halley, had separately appreciated the inverse square law in the solar system,[6] as well as giving some credit to Bullialdus.
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 non-isotropic radiators such as parabolic antennas, headlights, and lasers, the effective origin is located far behind the beam aperture.
[10] 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.
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.
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.
The curvature present in these spaces alters physical laws, influencing a variety of fields such as cosmology, general relativity, and string theory.
[11] John D. Barrow, in his 2020 paper "Non-Euclidean Newtonian Cosmology," expands on the behavior of force (F) and potential (Φ) within hyperbolic 3-space (H3).
The concept of spatial dimensionality, first proposed by Immanuel Kant, remains a topic of debate concerning the inverse-square law.
[12] Dimitria Electra Gatzia and Rex D. Ramsier, in their 2021 paper, contend that the inverse-square law is more closely related to force distribution symmetry than to the dimensionality of space.
[13] 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.
[14] 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:[15][16] 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.
7. tantundem lucis est in angustiori sphaerica superficie, quantum in fusiore, tanto ergo illie stipatior & densior quam hic.
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 (1605–1694) refuted Johannes Kepler's suggestion that "gravity"[17] weakens as the inverse of the distance; instead, Bullialdus argued, "gravity" weakens as the inverse square of the distance:[18][19] 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.
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 (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.
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.
