# Inverse-square law

We've seen some occurences of the inverse-square law in many forces in physics. For ex: the gravity and the Coulomb's force.

But my question is: what does that mean? What does the inverse-square law mean?

Gravity is caused by curvature of space-time due to the presence of a mass (or energy). I read in some book (that I don't remember the name) that other properties of elementary particles could cause curvature in other recurved dimensions of our universe. Then when a force is caused by a curvature in some dimension (or dimensions) it always obbeys the inverse-square law?

Thank you.

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If a field comes from a point source radially outwards (or at least can be thought of as one), then it's like a sphere made of field lines. Because a sphere's surface area is $$\pi$$r^2, as r increases the surface area also increases, and the field lines would therefore decrease in an inverse square relation.

The picture here will help:
http://en.wikipedia.org/wiki/Inverse-square_law

DaveC426913
Gold Member
Yes, you should realize that the effect is nothing more than geometry; it is not a property of the physics involved.

You'd get the same result if you set it up an array of fruit crates into which you were tossing cranberries.

tom.stoer
One can try to understand the inverse square law from field theory. A massless field (e.g. the electromagnetic field) has a propagator which behaves as 1/p² where p is the n-momentum of the particle. The related potential (e.g. the Coulomb potential) is the Fourier transform of 1/p². For D>2 the behavour is always ~ 1/rD-2. So for D=3 one gets 1/r for the Coulomb potential. The same reasoning applies to the gravitational field.

It is interesting that the classical reasoning with the surface of the sphere and the field theoretic calculation both lead to the same result.

DaveC426913
Gold Member
One can try to understand the inverse square law from field theory.
Again though, for the sake the OP's understanding of the issue, the inverse square law is more fundamental than any field or particle physics.

(If he were asking about, say, the relationship between the diameter of a circle and its circumference, we wouldn't explain it in terms of planetary motion or somesuch, we'd say pi is the fundamental relationship. Full stop.)

The inverse square law is simply a property of the surface of an expanding sphere Any expanding sphere. As mentioned, I could toss berries into (strategically-arranged) buckets and it would obey the same law.

tom.stoer
I don't think that you are right. In order to apply the argument regarding the surface of a sphere you have to have something in mind like field lines. So you end up with fields again ...

diazona
Homework Helper
But it applies equally well to something like shockwave intensity. Or the berries. In either case, no field lines are involved. It's just that in 3 spatial dimensions, anything which "propagates" outwards in all directions equally will obey an inverse-square law.

And in any case, invoking field theory seems like overkill for this question.

DaveC426913
Gold Member
I don't think that you are right. In order to apply the argument regarding the surface of a sphere you have to have something in mind like field lines. So you end up with fields again ...
Throwing berries through a line of fence-grids at 2 yards, 4 yards and 6 yards will result in 9 berries per grid at 2 yards, 4 berries per grid at 4 yards and 1 berry per grid at 6 yards.

The beam from a flashlight or BBs from a BB gun firing random shots will obey the same law.

tom.stoer