Why the nature has chosen the inverse square law. For instance, the gravitational force as well as the Coulomb force is inversely proportional to the square of distances. Why not these forces are inversely related with power 1 or 3 or may be some higher power of distance?
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$\begingroup$ "Why" questions are always tricky to answer, but en.wikipedia.org/wiki/Gauss's_law may be what you're looking for. The electric/gravitational field spreads out as the surface area of a sphere, and the surface area of a sphere is proportional to $r^2$. $\endgroup$– JoldCommented Feb 14, 2017 at 4:01
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$\begingroup$ Possible duplicate of Why are so many forces explainable using inverse squares when space is three dimensional? $\endgroup$– FlorisCommented Feb 14, 2017 at 4:10
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2 Answers
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Inverse square law is a a consequence of there being three space dimensions. Surface area of a sphere in n dimensions, has power (n-1). The forces that have to be present at every point around the source, do diminish along the surface of the sphere which gives inverse square law for a three dimensional space.
The inverse square law also proves that gravity is only three dimensional in normal cases (barring singularity etc.). For example, if there were 4 spacial dimensions, gravity would have followed inverse cube law.
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$\begingroup$ I don't think the electroweak or strong forces are $1/r^2$.... Indeed the potential between quark has a part basically linear. $\endgroup$ Commented Feb 14, 2017 at 5:50
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$\begingroup$ The inverse square law is not determined only by the dimension of space. It depends on the nature of the interacrtion. For example, a massive analog of the electromagnetic field produces the Yukawa potential, which is not an inverse square law. $\endgroup$– Rd BashaCommented Sep 3, 2022 at 17:22
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While it is true that the Coulomb (electrostatic) and gravitational forces have a $1/r^2$ behaviour, the other two fundamental forces of nature, the weak and string forces, do not vary with distance in that way. In particular, the strong force has a constant part to it, meaning its potential $\sim r$. The residual strong interaction is described by a Yukawa-type potential: $e^{-\lambda r}/r$, thus producing a force that is not $1/r^2$.
So Nature has NOT chosen $1/r^2$ in 50% of the cases.
The $1/r^2$ is remarkable because of the geometrical relations that follows from this (basically Gauss's law), but it seems a coincidence rather than anything fundamental.
answered Feb 14, 2017 at 6:05