Gravity, Coulomb, EM

[moonlord]

Why are all these forces proportional to the inverse square of the distance (1/r^2)? We do live in 3D, don't we?

[jinydu]

It follows from Gauss' Law; I've explained it in depth in some other threads.

But the short answer is that a point object radiates field lines outwards uniformly in all directions. The set of all points at an equal distance from the point is a sphere. The number of field lines passing through a fixed cross-sectional area of the sphere is inversely proportional to the surface area of the sphere. But the surface area of a sphere is proportional to the square of its radius. Thus, the number of field lines per unit area decreases according to an inverse square law.

In general, omnidirectional fields in n dimensions drop off according to an inverse (n-1) law.

The same reasoning can used to explain why the magnetic field strength from an infinitely long straight wire is inversely proportional to the distance from the wire.

[wendy]

The radiant flux model works like this.

Suppose from a sourse there is a flow of particles. These radiate in all directions at constant velocity v. At a given distance, vt, the flux will cover a surface (usually a sphere), such that the density * surface is constant (ie the flux produced at -t.)

Since the model of flux produces a constant S*d, regardless of radius, we have, eg d = 1/S. For euclidean geometries in 3d, we have S = 4pi r², which leads to d = Q/4pi r^2. (Q = power of source)

Electric and magnetic fields have a permeance/permittivity, which converts flux to force at the rate, eg E = D / epsilon. This means that the electric field is then E = D / epsilon = Q / 4 pi epsilon r^2.

For the sun, we find that the mass is derived from the siderial year of the year, and the AU, by the ratio GM = AU^3 / (yr/2pi)^2. Because in higher dimensions, the solar mass is Surface * acceleration, we have in N dimensions, GM = AU^n (2pi/yr)^2. G can be treated constant.

For a planet, we have a = GM/R^(n-1), giving, GdR^n/R^(n-1) = GdR, where d = density of the planet. That is, an earth-like 4d world will have a similar density to an earthlike 3d world.

In higher euclidean dimensions, this takes on different numbers, eg

4D E = Q / 2 pi^2 r^3.

In hyperbolic space, S is essentially [k^(R/r)+k^(-R/r)]/2 It does not typically follow an inverse law, unless R is typically very small.

W