Inverse-square law

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In physics, an inverse-square law is any physical law stating that some quantity is inversely proportional to the square of the distance from a point. In particular:

  • 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;
  • 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 Coulomb's law;
  • The energy or intensity of light radiating from a point source is inversely proportional to the square of the distance from the source. An object twice as far away (of the same size), receives only 1/4 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.
For another example, let the total power radiated from a point source, e.g., an omnidirectional isotropic antenna, be <math> P \ <math>. 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 <math> r \ <math> is <math> A = 4 \pi r^2 \ <math>, then intensity of radiation at distance <math> r \ <math> is
<math>

J = \frac{P}{A} = \frac{P}{4 \pi \cdot r^2}. <math>

The energy or intensity, decreases by a factor of 1/4 as the distance <math> r \ <math> 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 behavior, at least in the ideal 3 dimensional context (propagation in 2 dimensions would follow a just an inverse-proportional distance behavior and propagation in 1 dimension, the plane wave, remains constant in amplitude even as distance from the source changes).
<math>

p \sim 1/r.\, <math>

However the same is also true for the component of particle velocity <math> v \ <math> that is in-phase to the instantaneous sound pressure <math> p \ <math>.
<math>

v \sim 1/r.\, <math>

(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 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 behavior as is also indicated above.
<math>

J = p \cdot v \sim \frac{1}{r^2} <math>

Source (partial): from Federal Standard 1037C

See also

ja:逆2乗の法則

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