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Newton's law of universal gravitation: "Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them."

Coulomb's law: "The magnitude of the Electrostatics force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of charges and inversely proportional to the square of the distances between them."

How did Sir Isaac Newton and Sir Charles Augustine De Coulomb come to know that the force, gravitational or coulomb's, is inversely proportional to the square of the distance between two point masses or charges, why didn't they just say that the force is inversely proportional to the distance of two bodies from each other or two charges? There must have been something that made them formulate these inverse-square laws.

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Related: physics.stackexchange.com/q/22010/10851 –  Chris White Mar 22 '13 at 6:25
The earliest high-precision tests of the exponent -2 for electricity were done by verifying the shell theorem empirically for the zero field on the interior of a spherical shell. This is discussed in Purcell and Morin, Electricity and Magnetism, ch. 1. –  Ben Crowell Mar 22 '13 at 15:23
Related: physics.stackexchange.com/q/32779 –  Dimensio1n0 Jun 6 '13 at 13:02
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3 Answers

up vote 5 down vote accepted

The short answer is "observations".

In the case of the gravitational law, the orbits of the planets around the sun, the moon around the earth fit mathematically a force with an inverse square law for the distance. An inverse law does not.

In the case of electricity this article points out the observational history:

Early investigators who suspected that the electrical force diminished with distance as the gravitational force did (i.e., as the inverse square of the distance) included Daniel Bernoulli 1 and Alessandro Volta, both of whom measured the force between plates of a capacitor, and Aepinus who supposed the inverse-square law in 1758.

and then others took it from there to end up with the comprehensive publications of Coulomb, based on measurements.

Finally, in 1785, the French physicist Charles Augustin de Coulomb published his first three reports of electricity and magnetism where he stated his law. This publication was essential to the development of the theory of electromagnetism.[8] He used a torsion balance to study the repulsion and attraction forces of charged particles and determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

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Historically, Isaac Newton was of course the first man who discovered the inverse square law and Coulomb just copied it from Newton. Electrostatic and gravitational fields both follow the $1/r^2$ law because in some regime, they're described by the same mathematical equation $\Delta \Phi = \rho$. In plain English, these laws indicate the field lines have to be shared by (i.e. diluted to) the area $4\pi r^2$ of a sphere which is why the intensity has to drop as $1/4\pi r^2$. This is really the "heuristic" explanation why the law is $1/r^2$ in a three-dimensional space. In a 9-dimensional space, it would be $1/r^8$ and so on.

Isaac Newton originally determined the inverse square law from two independent but closely related considerations, almost at the same moment. One of them was a comparison of the motion of bodies near the Earth and the motion of the Moon. The other one were Kepler's laws for planetary orbits.

Concerning the Moon-ball analogy, he could have figured out that the Moon is 60 times further from the Earth's center than objects on the Earth's surface (360,000 km vs 6,000 km). He translated this ratio to a ratio of the forces that must act on the objects at these two distances to yield the right ratio of periodicities, and figured out that the forces' ratio is 1-to-3,600 and the law is therefore $1/r^2$.

The other method – which is nearly equivalent (and mainly differs by having the Sun instead of the Earth as the source of gravity at the center) – used Kepler's third law. Kepler was able to deduce the exact orbits of the planets (including the timing) from Tycho Brahe's meticulous observations and extract the phenomenological laws. The third one says that $T^2\sim a^3$: the squared period of the orbit is proportional to the third power of the greater semi-axis of the elliptical orbit. When one studies circular orbits, it's straightforward to prove that this power law relating the period and the radius is equivalent to the $1/r^2$ power law in the force. You're invited to check it yourself; if you're missing some maths to do so, I am afraid that my reproduction of the proof wouldn't be any helpful, anyway.

Let me give you the derivation, anyway. The centripetal force (and acceleration), like the opposite centrifugal one, goes like $r\omega^2\sim r/T^2$ where $T$ is the period and $r$ is the radius of the circular orbit. Because Kepler determined $T^2\sim r^{3}$ in his third law, $r/T^2$ goes like $r/r^3=1/r^2$ and that's it: I derived that the acceleration (and therefore force) that has to act on the planet has to go as $1/r^2$.

(Note that if you wanted $1/r$, the third Kepler's law would have to say $T^2\sim r^2$ i.e. $T\sim r$. This proportionality would be equivalent to constant velocities of the planets, regardless of their distance from the Sun. That's indeed how things would work if space had 2 spatial dimensions but the real-world Solar System and similar systems just don't work like that: the speed of nearby planets is higher.)

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Actually, the idea of the inverse-square law predated Newton (or at least Newton's publications on the topic) by at least a little bit. Robert Hooke proposed that gravity obeyed it before Newton did so. His reasoning was inspired by geometric considerations and light. If you have a point source of light, the power passing through a sphere centered on the point will be constant. If the point is emitting equally in all directions, you can conclude that the power per unit area, or intensity, will fall off as $1/r^2$, and this was known for a while. Basically, the power of $2$ comes from the fact that spheres' areas grow with that power of radius, which comes from the fact that space has $2+1 = 3$ dimensions.

It is somewhat natural then to imagine a conserved quantity coming from a source, evenly distributed over a sphere. It's influence on a test particle is just determined by what intercepts the test particle, so the influence drops off as the quantity is distributed over larger and larger surfaces.

By the way, Newton never gave Hooke much credit for this idea. Hooke, not having the tools available to do anything with it (unlike Newton, who proved that Kepler's empirical laws follow immediately from an inverse-square law), received less recognition than he thought he deserved. This rivalry led to Hooke teaming up with Leibniz against Newton, and the debate grew to encompass calculus, philosophy, and an entire continent's worth of academics.

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Come on, Chris, this is silly. You can't give Hooke or anyone before Newton credit for the inverse-square law for the force because Newton was the first man who introduced the force itself, including $F=ma$ and calculus. If Hooke was talking about "something" that goes like the inverse square, it was clearly not the force, and this claim by Hooke is clearly inequivalent to - has nothing essential to do with - Newton's insight. –  Luboš Motl Mar 22 '13 at 7:08
it seems there were even earlier people than Hook who had insight into an inverse square dependance of gravitational attraction, lots of references in wiki en.wikipedia.org/wiki/… –  anna v Mar 22 '13 at 7:27
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