The inverse square law is ubiquitous throughout physics. It refers to the decreasing intensity of the effect of a point source of a spherically symmetrical three-dimensional field as the square of the distance from the source increases.

If \(a\) is the measured quantity and \(x\) is the distance from the source:

\(a\propto {1\over x^{2}}\)

What makes a quantity inversely proportional to the square of the distance? Well, because a sphere's surface area is proportional to the square of its radius, the flux per unit area from any point source will be attenuated as the square of the distance from the source. For example, imagine the influence of a point source as represented by radially emanating flux lines, normal to a closed spherical surface. At twice the distance from the source, the same number of flux lines will pass through four times the original cross-sectional area. Therefore, a test particle at the new radius will experience a quarter of the quantity.

One of the most striking aspects of the inverse square law is the similarity between Newton's universal law of gravitation and Coulomb's law. For the most part, it's relatively easy to picture the intensity of, for example, the light from a star or the volume of a bass drum becoming diluted or weakened. The light and sound are spread, as it were, over an increasingly large area. However, the effect of mass or charge diminishing with distance is somewhat more abstract. 

The inverse square law manifested by Newton and Coulomb has been around for quite some time. In fact, its relation to electric force was first mooted by Joseph Priestley as early as 1766, which is roughly a century after Isaac Newton's claim to have previously deduced the elliptical nature of the orbits of planets: the force of their attraction towards the Sun is proportional to the reciprocal to the square of their distance from it. This (slightly imagined) extract from a conversation between Isaac Newton and Edmund Halley in 1684 ultimately led to Newton publishing his famous Principia Mathematica. Coulomb's law, which appeared in 1785 and describes the electrostatic force between two stationary point charges as being proportional to the product of their magnitudes and inversely proportional to the square of their separation can, of course, be attractive or repulsive, unlike Newton's analogous law of gravity which concerns two bodies of mass, and describes a force which is always attractive.

Both laws are most easily demonstrated by an idealised test particle at the surface of a spherical region enclosing a primary source of gravity, or charge. The effect on the particle, be it an apple or an electron, of the force exerted on it results from its position in a gravitational or electrostatic field. The field strength is the force per unit mass (or charge), on the particle, due to whatever is causing the field, be it a planet or a proton. The gravitational field strength at a point in the gravitational field of a mass is equivalent to the acceleration due to gravity experienced by the test particle.

Of course, in the real world, charged particles possess mass and it's interesting to compare the magnitude of the gravitational and electrical forces between them. For example, the order of magnitude of the ratio of the electrostatic repulsion to the gravitational attraction of two helium nuclei is 10³⁵. This ratio is astonishingly large at the scale of ions. However, gravity dominates on the scale of stars and galaxies since these vast objects are essentially electrically-neutral entities.

Both Coulomb's law and Newton's universal law of gravitation have to some degree been superseded, respectively, by Maxwell's equations of electromagnetism and Einstein's general theory of relativity (the latter giving rise to the idea that the force of gravity is a manifestation of the geometry of space-time). This begs the question: Are electric forces a manifestation of the geometry of something that isn't space-time? Formalising gravitational and electromagnetic fields as arising from disturbances in multidimensional surfaces makes the visualisation of an inverse square law much easier. Our test particles can be imagined as rolling around like marbles on an elastic membrane.

We know that space-time is awash with waves of gravitational and electromagnetic energy, their associated forces mediated by exchange particles that can be described as excited states of their respective fields and essentially informing point sources of mass, or charge, how to behave in each other’s proximity. However, further discussion along these lines risks taking us down the path of quantum field theory, which is very much a topic of study for another day!