1363:(1666) 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 (which extends from the Earth's center to the vacuum of space; obviously only the section of the cone from the Earth's surface to space bears on the Earth's surface). 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.
33:
1262:
447:. 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 a point mass located at the object's
1454:
685:, 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 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
704:, 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 (the
1223:, 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.
708:), 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%.
1323:
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
1226:
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. 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
478:
both expounded gravitation in 1666 as an attractive force. Hooke's lecture "On gravity" was at the Royal
Society, in London, on 21 March. Borelli's "Theory of the Planets" was published later in 1666. Hooke's 1670 Gresham lecture explained that gravitation applied to "all celestiall bodys" and added
1489:
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:
1331:
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,
1289:
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
1907:
If the moon and earth were not retained in their orbits by their animate force or some other equivalent, the earth would mount to the moon by a fifty-fourth part of their distance, and the moon fall towards the earth through the other fifty-three parts, and they would there meet, assuming, however,
1297:
Just as spherical surfaces, for which the source of light is the center, from the wider to the narrower, so the density or fortitude of the rays of light in the narrower , towards the more spacious spherical surfaces, that is, inversely. For according to 6 & 7, there is as much light in the
413:
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
44:
depends on the strength of the light source and is constant with increasing distance, where a greater density of flux lines (lines per unit area) means a stronger energy field. The density of flux lines is inversely proportional to the square of the distance from the source because the surface area
1613:
In a letter to Edmund Halley dated 20 June 1686, Newton wrote: "Bullialdus wrote that all force respecting ye Sun as its center & depending on matter must be reciprocally in a duplicate ratio of ye distance from ye center." See: I. Bernard Cohen and George E. Smith, ed.s,
259:
345:
715:(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
849:
would decrease by 6.02 dB per doubling of distance. When referring to measurements of power quantities, a ratio can be expressed as a level in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to the reference value.
1230:
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
510:
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
438:
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
479:
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
751:). 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
171:
178:
1899:
If two stones were placed in any part of the world near each other, and beyond the sphere of influence of a third cognate body, these stones, like two magnetic needles, would come together in the intermediate point,
1332:
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 .
266:
1880:
Note: Both Kepler and
William Gilbert had nearly anticipated the modern conception of gravity, lacking only the inverse-square law in their description of "gravitas". On page 4 of chapter 1, Introductio, of
613:(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
1894:
Gravity is a mutual affection between cognate bodies towards union or conjunction (similar in kind to the magnetic virtue), so that the earth attracts a stone much rather than the stone seeks the earth.
837:
1890:
Every corporeal substance, so far forth as it is corporeal, has a natural fitness for resting in every place where it may be situated by itself beyond the sphere of influence of a body cognate with it.
1103:
881:, the decrease is still 6.02 dB, since dB represents an intensity ratio. The pressure ratio (as opposed to power ratio) is not inverse-square, but is inverse-proportional (inverse distance law):
584:
470:'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.
1458:
1254:
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
1981:
1315:(1605–1694) 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:
1181:
1014:
921:
1845:
735:
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”.
77:
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.
357:
which is the resultant of radial inverse-square law fields with respect to one or more sources is proportional to the strength of the local sources, and hence zero outside sources.
1596:
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
118:
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 intensity is proportional (see
496:
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.
125:
1026:
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
972:
947:
731:
treatment planning, though this proportionality does not hold in practical situations unless source dimensions are much smaller than the distance. As stated in
1235:. General relativity instead interprets gravity as a distortion of spacetime, causing freely falling particles to traverse geodesics in this curved spacetime.
443:
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
45:
of a sphere increases with the square of the radius. Thus the field intensity is inversely proportional to the square of the distance from the source.
1408:
777:
1951:
1038:
component of the RMS particle velocity, both of which are inverse-proportional. Accordingly, the intensity follows an inverse-square behaviour:
1403:
1043:
520:
492:
1632:
Williams, E.; Faller, J.; Hill, H. (1971), "New
Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass",
254:{\displaystyle {\frac {{\text{intensity}}_{1}}{{\text{intensity}}_{2}}}={\frac {{\text{distance}}_{2}^{2}}{{\text{distance}}_{1}^{2}}}}
1947:
Translation of the Latin quote from
Bullialdus' 'Astronomia Philolaica' … is from: O'Connor, John J. and Roberson, Edmund F. (2006)
66:
1137:
979:
886:
358:
418:
is similarly applicable, and can be used with any physical quantity that acts in accordance with the inverse-square relationship.
1463:
340:{\displaystyle {\text{intensity}}_{1}\times {\text{distance}}_{1}^{2}={\text{intensity}}_{2}\times {\text{distance}}_{2}^{2}}
1258:(ca. 1349), stating that it was not linearly proportional to the distance, but was unable to expose the Inverse-square law.
670:(1 AU)—an approximate threefold increase in distance results in an approximate ninefold decrease in intensity of radiation.
1958:, The MacTutor History of Mathematics Archive, School of Mathematics and Statistics, University of Saint Andrews, Scotland.
1387:
463:
950:
1120:
is zero outside the source. This can be generalized to higher dimensions. Generally, for an irrotational vector field in
17:
1685:
485:
my supposition is that the attraction always is in duplicate proportion to the distance from the center reciprocall
1469:
1341:
1861:
2017:
1415:
87:
return, so the inverse square for both paths means that the radar will receive energy according to the inverse
1298:
narrower spherical surface, as in the wider, thus it is as much more compressed and dense here than there.
455:
639:
varies inversely with the square of the distance from the source (assuming there are no losses caused by
2022:
490:
Hooke remained bitter about Newton claiming the invention of this principle, even though Newton's 1686
415:
1936:
1948:
1250:, was one of the first to express functional relationships in graphical form. He gave a proof of the
1113:
1766:
1205:. The inherent curvature in these spaces impacts physical laws, underpinning various fields such as
1740:
475:
1634:
1440:
394:
102:, which acts like a canal does for water, or how a gun barrel restricts hot gas expansion to one
1862:"The Archaeology of the Inverse Square Law: (1) Metaphysical Images and Mathematical Practices,"
1420:
1198:
1128:, the intensity "I" of the vector field falls off with the distance "r" following the inverse (
36:
S represents the light source, while r represents the measured points. The lines represent the
1619:
1601:
1504:
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1724:
1643:
84:
2002:
1312:
459:
8:
1435:
1202:
956:
931:
764:
640:
594:
1728:
1647:
1116:
in three-dimensional space, the inverse-square law corresponds to the property that the
747: be the total power radiated from a point source (for example, an omnidirectional
723:
is the radial distance from the center. The law is particularly important in diagnostic
1997:
1902:
each approaching the other by a space proportional to the comparative mass of the other
1714:
1570:
1554:
1397:
1251:
1247:
1232:
1210:
1194:
1019:
748:
690:
674:
467:
1803:
1532:
1324:
imminutionis eadem est, ac luminus, in ratione nempe dupla intervallorum, sed eversa.
1681:
1562:
926:
678:
663:
62:
1574:
1281:
argued that the spreading of light from a point source obeys an inverse square law:
689:
in the direction of the narrow beam relative to a wide beam in all directions of an
1778:
1732:
1651:
1546:
1473:
1031:
1023:
705:
659:
629:
512:
70:
166:{\displaystyle {\text{intensity}}\ \propto \ {\frac {1}{{\text{distance}}^{2}}}\,}
1955:
1522:
of the
Medicean planets deduced from physical causes] (Florence, (Italy): 1666)
1377:
1278:
1266:
1125:
1027:
625:
1736:
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58:
2011:
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1214:
444:
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discussed the inverse-square law and how it affects the intensity of light.
728:
686:
610:
480:
471:
410:
398:
390:
354:
88:
1346:
In
Ismaelis Bullialdi astronomiae philolaicae fundamenta inquisitio brevis
724:
697:
598:
362:
1844:(Frankfurt, (Germany): Claude de Marne & heir Jean Aubry, 1604),
1117:
644:
602:
434:
is the attraction between objects that have mass. Newton's law states:
350:
1912:
the earth attracts a stone much rather than the stone seeks the earth"
1558:
841:
The energy or intensity decreases (divided by 4) as the distance
1914:
Kepler is breaking away from the Aristotelian tradition that objects
1206:
870:
859:
636:
385:
The inverse-square law generally applies when some force, energy, or
374:
103:
99:
41:
32:
1842:
Ad Vitellionem Paralipomena, quibus astronomiae pars optica traditur
1348:(1653) and publicized the planetary astronomy of Kepler in his book
1275:
Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur
515:. The deviation of the exponent from 2 is less than one part in 10.
1719:
1702:
1550:
74:
98:
while propagating a signal, certain methods can be used such as a
1535:(1952). "An Unpublished Letter of Robert Hooke to Isaac Newton".
878:
846:
712:
431:
50:
1887:"The true theory of gravity is founded on the following axioms:
1344:(1617–1689) publicized the ideas of Bullialdus in his critique
873:
radiating from a point source decreases by 50% as the distance
867:
633:
614:
402:
107:
95:
1829:
Before Galileo: The Birth of Modern Science in Medieval Europe
1261:
1185:
given that the space outside the source is divergence free.
682:
667:
606:
370:
366:
119:
80:
40:
emanating from the sources and fluxes. The total number of
1587:
Hooke's letter to Newton of 6 January 1680 (Koyré 1952:332).
1290:
fusiore, tanto ergo illie stipatior & densior quam hic.
588:
1519:
Theoricae Mediceorum Planetarum ex Causis Physicis Deductae
1372:
832:{\displaystyle I={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.\,}
666:); but only 1367 watts per square meter at the distance of
655:
386:
83:
energy expands during both the signal transmission and the
37:
1618:(Cambridge, England: Cambridge University Press, 2002),
1098:{\displaystyle I\ =\ pv\ \propto \ {\frac {1}{r^{2}}}.\,}
651:
579:{\displaystyle F=k_{\text{e}}{\frac {q_{1}q_{2}}{r^{2}}}}
122:) to the reciprocal of the square of the distance thus:
409:) is proportional to the square of the radius, as the
1767:"Dimensionality, symmetry and the Inverse Square Law"
1140:
1046:
982:
959:
934:
889:
780:
523:
269:
181:
128:
61:
stating that the observed "intensity" of a specified
1760:
1758:
361:
follows an inverse-square law, as do the effects of
1908:
that the substance of both is of the same density."
1804:"Introduction to Non-Euclidean General Relativity"
1765:Gatzia, Dimitria Electra; Ramsier, Rex D. (2021).
1631:
1175:
1097:
1008:
966:
941:
915:
831:
578:
339:
253:
175:It can also be mathematically expressed as :
165:
2003:Sound pressure p and the inverse distance law 1/r
1755:
1680:Ryer,A. (1997) “The Light Measurement Handbook”,
650:For example, the intensity of radiation from the
263:or as the formulation of a constant quantity:
106:in order to prevent loss of energy transfer to a
2009:
1531:
1871: : 391–414 ; see especially p. 397.
770:(power per unit area) of radiation at distance
1885:, Kepler sets out his description as follows:
1860:is from: Gal, O. & Chen-Morris, R.(2005)
1669:Lighting for Film and Television – 3rd Edition
1176:{\displaystyle I\propto {\frac {1}{r^{n-1}}},}
1107:
1856:Translation of the Latin quote from Kepler's
1764:
1696:
1694:
1188:
1009:{\displaystyle v\ \propto {\frac {1}{r}}\ \,}
916:{\displaystyle p\ \propto \ {\frac {1}{r}}\,}
711:The fractional reduction in electromagnetic
1967:(Gal & Chen-Morris, 2005), pp. 391–392.
1918:to be in their natural place, that a stone
451:while calculating the gravitational force.
1691:
1501:The History of the Royal Society of London
1980:… (London, England: John Martyn, 1667),
1718:
1094:
1005:
963:
938:
912:
828:
589:Light and other electromagnetic radiation
162:
1525:
1260:
31:
1273:In proposition 9 of Book 1 in his book
1193:The inverse-square law, fundamental in
609:or other linear waves radiating from a
14:
2010:
1700:
1503:, … (London, England: 1756), vol. 2,
953:with the instantaneous sound pressure
925:The same is true for the component of
1256:Summa logicæ et philosophiæ naturalis
359:Newton's law of universal gravitation
1998:Damping of sound level with distance
1355:In 1663–1664, the English scientist
1317:
1283:
658:per square meter at the distance of
1703:"Non-Euclidean Newtonian Cosmology"
24:
628:per unit area in the direction of
389:is evenly radiated outward from a
25:
2034:
1991:
1935:… (Paris, France: Piget, 1645),
1616:The Cambridge Companion to Newton
1490:Tycho Brahe to Newton", CUP 1989.
1404:William Thomson, 1st Baron Kelvin
1388:Kepler's laws of planetary motion
853:
499:
1457: This article incorporates
1452:
1340:In England, the Anglican bishop
1227:to the dimensionality of space.
620:More generally, the irradiance,
462:. But Bullialdus did not accept
454:As the law of gravitation, this
380:
1970:
1961:
1941:
1925:
1874:
1850:
1834:
1821:
1796:
1674:
1470:General Services Administration
1661:
1625:
1607:
1590:
1581:
1510:
1493:
1483:
464:Kepler's second and third laws
426:
421:
13:
1:
1707:Classical and Quantum Gravity
1507:; see especially pages 70–72.
1446:
1409:Power-aware routing protocols
7:
1858:Ad Vitellionem paralipomena
1366:
1311:..., the French astronomer
1108:Field theory interpretation
845:is doubled; if measured in
617:(in the same time period).
10:
2039:
1809:. MIT OpenCourseWare. 2018
1656:10.1103/PhysRevLett.26.721
1516:Giovanni Alfonso Borelli,
1238:
1189:Non-Euclidean implications
1132: − 1) power law
738:
503:
113:
1114:irrotational vector field
458:was suggested in 1645 by
1954:30 November 2016 at the
1737:10.1088/1361-6382/ab8437
1701:Barrow, John D. (2020).
1321:
1287:
1199:non-Euclidean geometries
1197:spaces, also applies to
877:is doubled; measured in
476:Giovanni Alfonso Borelli
466:, nor did he appreciate
387:other conserved quantity
1910:Notice that in saying "
1635:Physical Review Letters
1441:Principle of similitude
1416:Inverse proportionality
1277:(1604), the astronomer
1034:sound pressure and the
416:Gauss's law for gravity
395:three-dimensional space
94:To prevent dilution of
1783:10.1098/rsnr.2019.0044
1465:Federal Standard 1037C
1459:public domain material
1421:Multiplicative inverse
1329:
1295:
1270:
1177:
1099:
1030:is the product of the
1010:
968:
943:
917:
833:
580:
441:
341:
255:
167:
67:inversely proportional
46:
2018:Philosophy of physics
1933:Astronomia Philolaica
1922:to be with the earth.
1359:was writing his book
1350:Astronomia geometrica
1309:Astronomia Philolaica
1307:In 1645, in his book
1264:
1178:
1100:
1011:
969:
944:
918:
834:
581:
436:
342:
256:
168:
35:
1667:Millerson,G. (1991)
1246:of the 14th-century
1138:
1044:
1024:quadrature component
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957:
932:
887:
778:
521:
267:
179:
126:
1931:Ismail Bullialdus,
1743:on 25 February 2020
1729:2020CQGra..37l5007B
1648:1971PhRvL..26..721W
1476:on 22 January 2022.
967:{\displaystyle p\,}
942:{\displaystyle v\,}
675:isotropic radiators
336:
301:
248:
231:
1865:History of Science
1398:Telecommunications
1271:
1265:German astronomer
1252:mean speed theorem
1248:Oxford Calculators
1233:special relativity
1211:general relativity
1173:
1095:
1006:
964:
939:
913:
829:
749:isotropic radiator
681:, headlights, and
679:parabolic antennas
624:the intensity (or
576:
468:Christiaan Huygens
414:the point source.
337:
320:
285:
251:
232:
215:
163:
55:inverse-square law
47:
18:Inverse square law
2023:Scientific method
1949:"Ismael Boulliau"
1840:Johannes Kepler,
1771:Notes and Records
1338:
1337:
1313:Ismaël Bullialdus
1304:
1303:
1168:
1089:
1073:
1067:
1058:
1052:
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927:particle velocity
910:
901:
895:
823:
795:
691:isotropic antenna
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537:
460:Ismaël Bullialdus
411:emitted radiation
405:(which is 4π
324:
309:
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274:
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208:
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136:
132:
63:physical quantity
16:(Redirected from
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1739:. Archived from
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1533:Koyré, Alexandre
1529:
1523:
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1508:
1497:
1491:
1487:
1478:
1477:
1472:. Archived from
1456:
1455:
1400:, particularly:
1318:
1284:
1203:hyperbolic space
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706:square root of 2
585:
583:
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541:
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1956:Wayback Machine
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1642:(12): 721–724,
1630:
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1526:
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1498:
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1436:Square–cube law
1378:Antenna (radio)
1369:
1334:
1326:
1306:
1300:
1292:
1279:Johannes Kepler
1267:Johannes Kepler
1241:
1191:
1156:
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1139:
1136:
1135:
1126:Euclidean space
1110:
1083:
1079:
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1028:sound intensity
992:
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591:
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1993:
1992:External links
1990:
1987:
1986:
1976:Robert Hooke,
1969:
1960:
1940:
1924:
1873:
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1833:
1820:
1795:
1777:(2): 333–347.
1754:
1713:(12): 125007.
1690:
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1551:10.1086/348155
1545:(4): 312–337.
1524:
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1499:Thomas Birch,
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1426:Distance decay
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1393:Kepler problem
1390:
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864:sound pressure
855:
854:Sound in a gas
852:
827:
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808:
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791:
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759: = 4
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733:Fourier theory
702:stage lighting
590:
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571:
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560:
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550:
546:
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506:Electrostatics
504:Main article:
501:
500:Electrostatics
498:
449:center of mass
428:
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91:of the range.
59:scientific law
26:
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1234:
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1215:string theory
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558:
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524:
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513:Coulomb's law
507:
497:
495:
494:
488:
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477:
473:
469:
465:
461:
457:
452:
450:
446:
445:shell theorem
440:
435:
433:
419:
417:
412:
408:
404:
400:
397:. Since the
396:
392:
388:
381:Justification
378:
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34:
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1978:Micrographia
1977:
1972:
1963:
1943:
1932:
1927:
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1915:
1911:
1906:
1901:
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1876:
1868:
1864:
1857:
1852:
1841:
1836:
1828:
1823:
1811:. Retrieved
1798:
1786:. Retrieved
1774:
1770:
1745:. Retrieved
1741:the original
1710:
1706:
1676:
1668:
1663:
1639:
1633:
1627:
1615:
1609:
1597:
1592:
1583:
1542:
1536:
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1518:
1512:
1500:
1495:
1485:
1474:the original
1464:
1450:
1361:Micrographia
1360:
1357:Robert Hooke
1354:
1349:
1345:
1339:
1330:
1322:
1308:
1305:
1296:
1288:
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752:
744:
742:
729:radiotherapy
720:
716:
710:
695:
687:antenna gain
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619:
611:point source
592:
517:
509:
491:
489:
484:
481:Isaac Newton
472:Robert Hooke
453:
442:
437:
430:
406:
399:surface area
391:point source
384:
355:vector field
348:
262:
174:
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89:fourth power
79:
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48:
29:
27:Physical law
1505:pages 68–73
1383:Gauss's law
725:radiography
698:photography
630:propagation
599:illuminance
432:Gravitation
427:Gravitation
422:Occurrences
377:phenomena.
2012:Categories
1720:2002.10155
1602:at page 66
1447:References
1118:divergence
1020:near field
645:scattering
641:absorption
603:irradiance
351:divergence
42:flux lines
1982:page 227:
1598:Principia
1342:Seth Ward
1207:cosmology
1195:Euclidean
1161:−
1145:∝
1069:∝
990:∝
897:∝
871:wavefront
868:spherical
860:acoustics
810:π
765:intensity
637:wavefront
634:spherical
595:intensity
493:Principia
375:radiation
318:×
308:intensity
283:×
273:intensity
199:intensity
187:intensity
138:∝
131:intensity
104:dimension
100:waveguide
85:reflected
1952:Archived
1937:page 23.
1846:page 10.
1620:page 204
1575:41626961
1567:13010921
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951:in-phase
949:that is
677:such as
673:For non-
654:is 9126
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363:electric
323:distance
288:distance
235:distance
218:distance
151:distance
75:distance
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1788:30 July
1747:30 July
1725:Bibcode
1644:Bibcode
1239:History
1112:For an
1018:In the
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713:fluence
662:(0.387
660:Mercury
114:Formula
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69:to the
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71:square
1920:seeks
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1715:arXiv
1571:S2CID
1555:JSTOR
1461:from
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866:of a
668:Earth
656:watts
626:power
622:i.e.,
607:light
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439:them.
401:of a
371:sound
367:light
353:of a
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1916:seek
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1790:2023
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1688:p.26
1682:ISBN
1671:p.27
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727:and
700:and
597:(or
593:The
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1779:doi
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