Inverse-square law
In science, an inverse-square law is any scientific law stating that the observed "intensity" of a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. 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.
Justification[edit]
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 (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 for gravity is similarly applicable, and can be used with any physical quantity that acts in accordance with the inverse-square relationship.
Occurrences[edit]
Gravitation[edit]
Gravitation is the attraction between objects that have mass. Newton's law states:
Non-Euclidean implications[edit]
The inverse-square law, fundamental in Euclidean spaces, also applies to non-Euclidean geometries, including hyperbolic space. The inherent curvature in these spaces impacts physical laws, underpinning various fields such as cosmology, general relativity, and string theory.[11]
John D. Barrow, in his 2020 paper "Non-Euclidean Newtonian Cosmology," elaborates on the behavior of force (F) and potential (Φ) within hyperbolic 3-space (H3). He illustrates that F and Φ obey the formulas F ∝ 1 / R^2 sinh^2(r/R) and Φ ∝ coth(r/R), where R and r represent the curvature radius and the distance from the focal point, respectively.[11]
The concept of the dimensionality of space, first proposed by Immanuel Kant, is an ongoing topic of debate in relation to the inverse-square law.[12] Dimitria Electra Gatzia and Rex D. Ramsier, in their 2021 paper, argue that the inverse-square law pertains more to the symmetry in force distribution than to the dimensionality of space.[12]
Within the realm of non-Euclidean geometries and general relativity, deviations from the inverse-square law might not stem from the law itself but rather from the assumption that the force between bodies depends instantaneously on distance, contradicting special relativity. General relativity instead interprets gravity as a distortion of spacetime, causing freely falling particles to traverse geodesics in this curved spacetime.[13]