Inverse-square law: Wikis

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Encyclopedia  The lines represent the flux emanating from the source. The total number of flux lines depends on the strength of the source and is constant with increasing distance. A greater density of flux lines (lines per unit area) means a stronger field. The density of flux lines is inversely proportional to the square of the distance from the source because the surface area of a sphere increases with the square of the radius. Thus the strength of the field is inversely proportional to the square of the distance from the source.

In physics, an inverse-square law is any physical law stating that some 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.

Justification

The inverse-square law generally applies when some force, energy, or other conserved quantity is radiated outward radially from a source. Since the surface area of a sphere (which is 4πr 2) is proportional to the square of the radius, as the emitted radiation gets farther from the source, it must spread out over an area that is proportional to the square of the distance from the source. Hence, the radiation passing through any unit area is inversely proportional to the square of the distance from the source.

Occurrences

Gravitation

Gravitation is the attraction between 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.

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, instead of the guess of Johannes Kepler at inverse distance dependence. 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 (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 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. This implies a limit on the photon rest mass.

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 9140 watts per square meter at the distance of Mercury (0.387AU); but only 1370 watts per square meter at the distance of Earth (1AU)—a threefold increase in distance results in a ninefold decrease in intensity of radiation.

Photographers and theatrical lighting professionals use the inverse-square law to determine optimal location of the light source for proper illumination of the subject. The inverse-square law can be used only on point source light; a fluorescent lamp is not a point source and therefore one can not use the inverse-square law, as is possible with most other light sources, with a fluorescent lamp.

For an infinite linear-source the equation is: $E = \frac{I}{d}$
For an infinite planar-source the equation is: $E = \frac{I}{1}$ (E is invariant with d).

A plasma light bulb is as close to a point source as is practical for most lamps. A "point source" (subject to 1% error) is obtained from a "pseudo-point-source" (not LED or laser) at a distance 10 times the source radius (5 times the diameter). A four-foot fluorescent lamp is (almost) a point source (subject to 1% error) at a distance of 20 feet. Similarly, as one gets closer to a fluorescent lamp the apparent brightness will increase up to a certain distance (because the viewing angle (sampled area) remains constant and closeup (one foot away) the entire fluorescent tube can not be viewed), after which the intensity will not continue to increase (as it would with a (pseudo) point source).

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 4πr2 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 (radius) r.

The inverse square law in radiography is: $\frac{I_1}{I_2} = \left(\frac{r_2}{r_1}\right)^2$

where I is intensity and r is distance (radius).

Example

Let the total power radiated from a point source, e.g., 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 of radiation at distance r is $I = \frac{P}{A} = \frac{P}{4 \pi r^2}. \,$ $I \propto \frac{1}{r^2} \,$ $\frac{I_1} {I_2} = \frac{r_2^2}{r_1^2} \,$ $I_1 = I_2 r_2^2 \frac{1}{r_1^2} \,$

The energy or intensity decreases (divided by 4) as the distance r is doubled, or measured in dB it would decrease by 6.02 dB. This is the fundamental reason why intensity of radiation, whether it is electromagnetic or acoustic radiation, follows the inverse-square behaviour (assuming it originates from a point-source), at least in the ideal three-dimensional context (propagation in two dimensions would just follow an inverse-proportional distance behaviour and propagation in one dimension, the plane wave, remains constant in amplitude even as distance from the source changes).

Acoustics

The inverse-square law is used in acoustics in measuring the sound intensity at a given distance from the source.

Example

In acoustics, the sound pressure of a spherical wavefront radiating from a point source decreases by 50% as the distance r is doubled, or measured in dB it decreases by 6.02 dB. The behaviour is not inverse-square, but is inverse-proportional (inverse distance law): $p \propto \frac{1}{r} \,$ $\frac{p_1} {p_2 } = \frac{r_2}{r_1} \,$ $p_1 = p_2 r_2 \frac{1}{r_1} \,$

However the same is also true for the component of particle velocity $v \,$ that is in-phase to the instantaneous sound pressure $p \,$. $v \propto \frac{1}{r} \,$

Only in the near field the quadrature component of the particle velocity is 90° out of phase with the sound pressure and thus does not contribute to the time-averaged energy or the intensity of the sound. This quadrature component happens to be inverse-square. The sound intensity is the product of the RMS sound pressure and the RMS particle velocity (the in-phase component), both which are inverse-proportional, so the intensity follows an inverse-square behaviour as is also indicated above: $I = pv \propto \frac{1}{r^2}. \,$

The inverse-square law pertained to sound intensity. Because sound pressures are more accessible to us, the same law can be called the "inverse-distance law".

Field theory interpretation

For an irrotational vector field in three-dimensional space the law corresponds to the property that the divergence is zero outside the source. Generally, for an irrotational vector field in n-dimensional Euclidean space, an inverse (n − 1)th potention law corresponds to the property of zero divergence outside the source.

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. ^
4. ^ (WayBack link to "Point Source Approximation")
5. ^ Inverse-Square law for sound This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".