Why all forces are subjected 1/r^2?

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Discussion Overview

The discussion revolves around the nature of forces and their relationship to distance, specifically why many forces follow an inverse square law (1/r²) rather than a linear relationship (1/r). Participants explore the implications of dimensionality and geometry in understanding these relationships.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question why all forces are subject to an inverse square law rather than a linear relationship, suggesting a need for a logical explanation or conceptual model.
  • One viewpoint proposes that the geometry of space may influence these relationships, speculating that in a four-dimensional space, the relationship could be cubic.
  • Another participant points out that not all forces follow the inverse square law, citing the strong nuclear force as an example that increases with distance up to a certain point.
  • Participants discuss the concept of "flux" to explain the inverse square relationship in electromagnetic forces, emphasizing how the density of field lines decreases with distance.
  • There is a suggestion that gravitational force might need to be re-evaluated in terms of its dependence on the area of a sphere, proposing a modification to the gravitational constant.
  • One participant notes that central potentials yield inverse-square forces and connects this to the dimensionality of space, mentioning modern theories that may deviate from this at small scales.

Areas of Agreement / Disagreement

Participants express differing views on the universality of the inverse square law for all forces, with some asserting exceptions exist. The discussion remains unresolved regarding the implications of dimensionality and the nature of forces.

Contextual Notes

Some claims depend on specific definitions of forces and potential energy functions, and there are unresolved questions about the applicability of these relationships at different scales.

Michael F. Dmitriyev
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Is it not strange thing that all forces are subjected 1/r^2 but not 1/r?
Why their relationships are nonlinear? It must have the some logical explanation or imaginary picture .
 
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Originally posted by russ_watters
Its likely just due to the geometry of space. If space were 4 dimensional, maybe it would be a cube function.
But 3-d already assumes a linear dependence, square dependence and cubic dependence.
 
FIrst of all, all forces are NOT inverse square proportional to distance. THe strong nuclear force actually increases with distance (up to a point).

THe E-M force's inverse-square proportionality is easily visualized with the concept of "flux": Imagine a particle that has a force field around it. Imagnine this force field as lines that radiate from the particle (This is the model of electric field lines, for example). The strength of the force on another particle depends onthe density of the field lines from the first particle.

As you move away from the first particle, the same total number of lines will exist but they will spread further and further apart. As distance increases, the density of the lines gets smaller and smaller. Density is "total number of lines" divided by "the surface area of the sphere with a radius equal to the distance."

Area of a sphere is proportional to the square of the radius, density (and thus strength of the force field) is proportional to the inverse square of the distance.
 
Originally posted by Michael F. Dmitriyev
Is it not strange thing that all forces are subjected 1/r^2 but not 1/r?
Why their relationships are nonlinear?
1/r is nonlinear, too.
 
Originally posted by Chi Meson
FIrst of all, all forces are NOT inverse square proportional to distance. THe strong nuclear force actually increases with distance (up to a point).

THe E-M force's inverse-square proportionality is easily visualized with the concept of "flux": Imagine a particle that has a force field around it. Imagnine this force field as lines that radiate from the particle (This is the model of electric field lines, for example). The strength of the force on another particle depends onthe density of the field lines from the first particle.

As you move away from the first particle, the same total number of lines will exist but they will spread further and further apart. As distance increases, the density of the lines gets smaller and smaller. Density is "total number of lines" divided by "the surface area of the sphere with a radius equal to the distance."

Area of a sphere is proportional to the square of the radius, density (and thus strength of the force field) is proportional to the inverse square of the distance.
May be a Gravity force, for example, must be equal to
G (4pi) M1*M2/(4pi)r^2
I.e. a gravity force must be inversely to area of sphere with radius r. Then G is not correct and should be multiplied on 4pi.
 


Originally posted by turin
1/r is nonlinear, too.
D'oh - can't believe I missed that.
But 3-d already assumes a linear dependence, square dependence and cubic dependence.
I don't understand what you mean: the diagram I linked shows a square relationship between area and distance in 3d space. That geometric relationship is likely the reason we see square (or inverse square) relationships so often in physical laws.
 
Originally posted by Michael F. Dmitriyev
Is it not strange thing that all forces are subjected 1/r^2 but not 1/r?
Why their relationships are nonlinear? It must have the some logical explanation or imaginary picture .

Forces are subject to whatever potential energy function defines them, via a gradient:

\vec{F} = -\vec{\nabla} U

Central potentials (1/r) yield inverse-square forces.

But yes, it actually has to do with the dimensionality of the spaces as well. In fact, in modern theories of large extra dimensions, the Newtonian potential/force is expected to deviate from inverse/inverse-square at submillimeter scales.

moderator edit: fixed TeX
 
Last edited by a moderator:
Originally posted by Michael F. Dmitriyev
May be a Gravity force, for example, must be equal to
G (4pi) M1*M2/(4pi)r^2
I.e. a gravity force must be inversely to area of sphere with radius r. Then G is not correct and should be multiplied on 4pi.

Similar to the way that the original "Coulomb constant", k, turned out to be a variation of the permeability of free space: k = 1/(4 pi epsilon)
 

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