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It seems paradoxical that the strength of so many phenomena (Newtonian gravity, Coulomb force) are calculable by the inverse square of distance.

However, since volume is determined by three dimensions and presumably these phenomena have to travel through all three, how is it possible that their strengths are governed by the inverse of the distance squared?

The gravitational force and intensity of light is merely 4 times weaker at 2 times the distance, but the volume of a sphere between the two is 8 times larger.

Since presumably these phenomena would affect all objects in a spherical shell surrounding the source with equal intensity, they travel in all three dimensions. How come these laws do not obey an inverse-cube relationship while traveling through space?

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This is not paradoxical and it is not necessary for any physical phenomenon to a priori have to obey any particular law. Some phenomena do have to obey inverse-square laws (such as, particularly, the light intensity from a point source) but they are relatively limited (more on them below).

Even worse, gravity and electricity don't even follow this in general! For the latter, it is only point charges in the electrostatic regime that obey an inverse-square law. For more complicated systems you will have magnetic interactions as well as corrections that depend on the shape of the charge distributions. If the systems are (globally) neutral, there will still be electrostatic interactions which will fall off as the inverse cube or faster! The van der Waals forces between molecules, for instance, are electrostatic in origin but go down as $1/r^6$.

It is for systems with a conserved flux that the inverse-square law must hold, at least at large distances. If a point light source emits a fixed amount of energy per unit time, then this energy must go through every imaginary spherical surface we think up. Since their area goes up as $r^2$, the power per unit area (a.k.a. the irradiance) must go down as $1/r^2$. In a simplified picture, this is also true for the electrostatic force, where it is the flow of virtual photons that must be conserved.

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    $\begingroup$ You mention that $1/r^2$ only holds for point-charges in the static regime. You may want to add that, in the case of gravity, $1/r^2$ only holds in the non-relativistic limit (the general case being that of the Schwarzschild potential). $\endgroup$ – AccidentalFourierTransform Jul 19 '17 at 18:53
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The field line picture known from school might be helpful with that:

The surface area of the surrounding sphere (and not it's volume) determines the density of the lines sourced by a point charge, corresponding to the field strength.

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These physical phenomena (gravity, Coulomb force) are forces caused by an object you can consider pointlike. That is, for the inverse square law to hold, the object emits the force uniformly in all directions from one point.

That means that at any distance (call it $R$) from the object, you'll feel the same force as you would anywhere over the surface of a sphere whose radius is that distance.

The surface of a sphere is $2$-dimensional, not $3$-dimensional, and its area goes like $R^2$. The larger the radius, the larger the surface of the sphere, and the further away you are from the source. So the strength of the source is inversely proportional to the surface area of the sphere.

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These kind of forces are coming from a system which is invariant under rotations, so under the SO(3) group (dim space : 3). Therefore, it should exists 3 generators of these rotations, thus 3 gauge transformations. Moreover, if your system is conserved in time, the energy is conserved and these generators are constant of motion.

When we are interested of interactions, we observe interactions which become really small at large distances, and in the case of gravity their is an attractive force.

Then, if you look at a force F = f(r), if I well remember, only in the case f(r) = 1/r^2 you can obtain such gauge generators which are known as 1 component of the angular momentum (imposing a plenary motion, so invariant under a rotation around the angular momentum) and two components of the Laplace-Runge-Lenz vector (imposing the axes of the ellipse to be constant, generating the 2 others rotations).

If you change the geometry of your system, you will study some other symmetries and thus obtain some other group leading to other generators. Then the allowed forces which will conserve the geometry of your problem will be different.

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According to the square cube law, for any positive real number r, a system has the same properties as another system like it except that it's r times bigger occurring at 1/r times the speed. The square cube law is actually not exactly true but is very close to true due to the extremely small size of molecules. If you slowly increase the tension of a brittle material that was etched nanosmooth, there will be a probability distribution of what tension it will break at with a standard deviation much smaller than the strength it breaks at. For a glass rod etched nanosmooth, it will break at very close to 4 times the tension if it's twice as thick because the amount of tension a rod supports is independent of its length and a rod of twice the thickness is like 4 rods of the original thickness. The higher the tension applied to a brittle material that was etched nanosmooth, the higher the rate of homogeneous nucleation of a crack that's large enough to sustain its own growth and then grow at the speed of sound in the material leading to its fracture because the shorter the crack needs to be to sustain its growth. The reason its growth will be sustain after the nucleated crack reaches a critical size is because the longer a crack is, the more tension is magnified at its tip. Let's define the strength of a sphere of glass that was etched nanosmooth to be the tension at which the expected time for it to break is its diameter times 1 s/m. A nanosmooth sphere of twice the size then has a very slightly lower tensile strength because its tensile strength is defined to by that which gives 1/16 the rate of nucleation of a crack that's big enough to sustain its growth. Glass is an unstable substance because the rate of nucleation of its crystalline form is not zero so its strength can't be defined at really large size scales. We can however define the strength of corundum in a similar way at arbitrarily large size scales according to a simplified quantum mechanical theory where electrons and nuclei are point charges with no nuclear chemistry and the gravitation constant is zero because it's an infinitely stable substance according to that theory. I think it turns out that according to that definition, a human sized sphere of it has almost as high a strength as one a few atoms across but once you go past a certain size, its strength approximately varies as the reciprocal of the log of its size. Because of that, the square cube law gets closer to being exactly true as the size approaches infinity because the fraction of the tensile strength lost with a doubling of the size varies as the reciprocal of the log of the size. The strength of a corundum sphere also decreases with temperature at temperatures below room temperature given sufficiently large size because a higher temperature gives a higher rate of nucleation of a crack that's big enough to sustain its growth. At absolute zero however, a corundum sphere will not lose any strength at any size no matter how large.

According to https://en.wikipedia.org/wiki/Size_effect_on_structural_strength, materials get weaker at a larger size by a much larger amount then their theoretical strength decreases with size. That's probably because scratching is far off from following the square cube law. For a tetrahedral diamond tip made by fracturing it along its cleavage planes sliding along nanosmooth glass at a given speed and force, it will probably make a scratch in nanosmooth glass that's more than twice as deep if 4 times the force is applied in the exact same direction because scratching is a process explained at the molecular level. See How does an infinitely hard tip scratch an amorphous brittle material when it slides along it?. Surface tension doesn't follow the square cube law either. The radius of curvature of a meniscus of water will only multiply by square root of 2 if the amount of gravity is reduced by half.

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protected by Qmechanic Jan 12 '13 at 8:17

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