Why is force inversely proportional to distance?

[l-1j-cho]

Hello all :)

A random guy on the internet proposed the question above. Why is force inversely proportional to the square of distance? And he gave us a hint that this is not a physics problem but a math problem. Could you guys give me a clue?

I really don't think that dimensional analysis nor experimental values are the answer that he is looking for.

 

[arildno]

Well, it isn't true that forces, in general, are inversely proportional to some squared distance.

And, how some particular force is most accurately represented is a problem within physics, rather than within maths.

 

[Vagn]

Which force do you mean? Gravitational and Electrostatic both do, but magnetic, and the weak/strong interactions don't necessarily.
Think about the inverse square law, and how the geometry of something traveling at a constant speed would disperse when emitted by a point object.

 

[l-1j-cho]

oops my mistake.

 

[l-1j-cho]

He meant gravitational force.

 

[BruceW]

I guess you could argue that since the gravitational field lines go radially inward towards the mass, the density of the field lines would be inversely proportional to the surface over which they are spread over. (And therefore inversely proportional to r^2). And since force is proportional to the density of field lines, the force is inversely proportional to distance squared.
But this argument isn't mathematically or physically solid, it assumes too much. (But it does happen to be true in the case of Newtonian gravity and electrostatics).

 

[l-1j-cho]

the density of the field lines would be inversely proportional to the surface over which they are spread over.

Then, shoudn't it be inversely proportional to the cube of the distance?
I really don;t know :(

 

[BruceW]

the surface of a sphere is proportional to radius squared, so the field lines per area (through a given sphere) is inversely proportional to radius squared.

 

[l-1j-cho]

oh i get it
thanks

 

[vanhees71]

This kind of "why questions" are a bit problematic since science doesn't answer them. Science tries to describe what can be observed as concise as possible and then to make models (or even theories), i.e., finding out the pattern behind the observations, by finding an as small as possible set of fundamental laws to explain as many phenomena as possible.

First of all you have to specify which force you are talking about. Not all forces are inversely proportional to the square (!) of the distance! E.g., the strong force between two nucleons is Yukawa-like, i.e., exponentially falling with distance.

The "reason" behind these laws, i.e., the reason for their specific form, can be traced back to the fundamental symmetries of space-time. The best theory we have about the elementary particles which are the building blocks for all known matter (there's more unknown in the universe than known, but that's another story) is the standard model of elementary particles, which bases on the Minkowski geometry of special-relativistic space-time and its Poincare symmetry. This symmetry governs the possible mathematical forms of field equations, which themselves finally lead to the force laws we observe in nature.

For the electromagnetic force it turns out that it is due to a massless vector field, the electromagnetic field, and in the static case you find Coulomb's Law for the force between two structureless point charges, i.e., a central force falling with the square of the distance between the particles.

 

[arildno]

It is a mathematical fact that inverse square laws for forces, in Euclidean space imply "conservation of force-area over a simply connected surface enveloping the singularity"

However, there is, as yet, no fundamental reason WHY the quantity "force-area" should be conserved, but it is a very pleasing mathematical property with SOME force models, that happens to be true in a few special cases.