Inverse-square law

This diagram shows how the law works. 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.

Areas of application

In particular the inverse square law applies in the following cases: doubling the distance between the light and the subject results in a quarter reduction of the light.

Gravitation

The gravitational attraction between two massive objects, in addition to being directly proportional to the product of their masses, is inversely proportional to the square of the distance between them; this law was first suggested by Ismael Bullialdus but put on a firm basis by Isaac Newton after Robert Hooke proposed the idea in a letter to Newton. Hooke later accused Newton of plagiarism.

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.

Light and other electromagnetic radiation

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 ¼ 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 three-fold increase in distance results in a nine-fold 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 law is also particularly important in diagnostic radiography and radiotherapy treatment planing.

Acoustics

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

Examples

Electromagnetic radiation

For another 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=4\pi r^{2}\$, then intensity I of radiation at distance r is

$I={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.\,$

$I\propto {\frac {1}{r^{2}}}\,$

${\frac {I_{1}}{I_{2}}}={\frac {{r_{2}}^{2}}{{r_{1}}^{2}}}\,$

$I_{1}=I_{2}\cdot {r_{2}^{2}}\cdot {\frac {1}{{r_{1}}^{2}}}\,$

The energy or intensity decreases by a factor of ¼ 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, at least in the ideal 3 dimensional context (propagation in 2 dimensions would follow a just 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

In acoustics, the sound pressure of a spherical wavefront radiating from a point source decreases by a factor of ½ as the distance $r$ is doubled, or measured in dB it would decrease by 6.02 dB. The behaviour is not inverse-square, but is inverse-proportional:

$p\propto {\frac {1}{r}}\,$

${\frac {p_{1}}{p_{2}}}={\frac {r_{2}}{r_{1}}}\,$

$p_{1}=p_{2}\cdot r_{2}\cdot {\frac {1}{r_{1}}}\,$

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:

$I=p\cdot v\propto {\frac {1}{r^{2}}}.\,$

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 the law corresponds to the property that the divergence is zero outside the source.

 This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22.

See also

• Flux
• Gauss's law
• Kepler's first law
• Telecommunications