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Why do both Coulomb and Gravity follow 1/r^2 laws?

  1. Oct 1, 2009 #1
    I am wondering why both Coulomb and Classical Gravity follow 1/r^2 laws. Weak and Nuclear force fall off at much faster distance scales! Does anyone know why it is that both Coulomb and Gravity have 1/r^2 strength, while the other forces have different distance scales?
  2. jcsd
  3. Oct 1, 2009 #2

    D H

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    The Coulomb force and Newtonian gravity follow 1/r2 laws because classically, space is three dimensional.

    The weak force does not follow a 1/r2 law because W and Z bosons decay. Photons don't decay. The nuclear force is a bit like Van der Waals forces. Van der Waals forces are second-order, or residual, electromagnetic effects that result from things like dipoles. The nuclear force is a residual effect of the strong interaction. These residual forces naturally fall off much quicker than 1/r2.
  4. Oct 1, 2009 #3
    Thanks, DH.

    My first followup would be: why does 3D space imply 1/r^2? But I am going to take a guess from your answer about W and Z Bosons:

    3D space implies 1/r^2 laws is because the idea is that the interactions are mediated by force carriers, and the number of force carriers that smacks a thing falls off by 1/r^2 in 3D space.

    Is that correct? If so, it would seem to imply that a electron, for example, is constantly (or almost constantly) sending out clouds of some kind of force carrier particles symmetrically in the radial direction. Is that true too?

    What is the strong interaction--is that what holds quarks together?

    Thanks again.

  5. Oct 1, 2009 #4
    Hi Djinn-
    The Coulomb 1/r2 law holds up for distances larger then the electron Compton wavelength (~3.86 x 10-11cm). For distances shorter than this, the bare (unrenormalized) electron charge starts to become visible. The lowest order correction to the Coulomb field at short distances is a virtual positron electron pair (a bubble diagram).
    Bob S
  6. Oct 2, 2009 #5


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    Your supposition hits the mark. We can choose to assume that electromagnetic interaction has a mediator. As you know we call that mediator 'electric field'.
    The surface area of a sphere is proportional to the square of its radius. At larger distance to the origin the capacity for interaction is thought of as "diluted" over a larger area.

    The field itself is not thought of as discontinuous. By contrast: matter is an assembly of atoms; matter is by nature discontinuous. But there is not a counterpart of that in fields.

    So where does the quantization come in? It has to do with the fact that electromagnetic disturbances can keep propagating when the original source has long gone. When such electromagnetic energy hits a scintillation counter the interaction is like a particle-particle interaction. So we say a photon has hit the scintillation counter. That photon wasn's mediating electromagnetic interaction, you might say it was independently propagating energy

    In the case of actual electromagnetic interaction, such as the attraction between two oppositely charged particles, the field is not independent from its source. When the interaction is described in terms of exchange, the exchanged entities are referred to as 'virtual photons'. Such a 'virtual photon' is not an observable entity; only photons that are on their own, not mediating interaction, are observable at all.

  7. Oct 2, 2009 #6
    There is another law - that of Hook's. The elastic force is -kX for many materials. Materials differ with k-values and limits where the law holds. At too large deformations the materials behave differently.

    Similarly the Coulomb and the Newton laws: they are alike in a certain range of distances and are not outside it.
    Last edited: Oct 2, 2009
  8. Oct 3, 2009 #7
    Potentials have a back-and-forth relationship with scattering amplitudes (leading to scattering probabilities and their distributions) in quantum mechanics. Look at the Breit-Wigner scattering formula. You can learn things about a potential based on the scattering distribution of particles, or you go the other way, putting in a potential and getting out a scattering (probability) distribution.
    But the potential can be seen to arise from the fundamental interaction (probability) amplitudes in quantum field theory. In particular, an important contribution comes from the “propagator” of a virtual particle exchange between real interacting particles. The *dominant form* of this contribution to an interaction amplitude involving a massless exchange particle is such that the potential interpreted from the interaction (or scattering) amplitude is of a 1/r form. For an exchange particle with a mass, the dominant form is the Yuakawa potential, dropping off exponentially at a rate depending on the mass of that particle. In quantum electrodynamics the massless photon is the exchange particle. In gravitation, the hypothetical particle would be the massless graviton. For the weak interaction, the W and Z exchange particles have a mass. And for the strong nuclear interaction among protons and neutrons effectively involves a massive particle called the pion. [The strong interaction is fundamentally called quantum chromodynamics and involves quarks interacting via massless exchange particles called gluons. But due to the nature of the strong interaction, there is a “mass gap”, meaning that effectively we see quarks and gluons bound up in a quantum states that we see as massive composite particles, like pions, protons and neutrons. That’s why I talked about that *effective* picture above to describe the fast drop-off of the potential.]
    As implied by one of the previous posts, these forms are approximations (hence “dominant form”) and the corrections can be calculated in quantum field theory.
    Also, the virtual exchange particles can really only be talked about when there are real particles interacting. In the classical picture, we say that a charge is a source of potential, and that if we were to put another charged particle in that potential field, it would experience a force. By contrast, in quantum field theory, the potential arises as a derived thing based on the way two or more particles interact.
  9. Oct 3, 2009 #8
    By the way, atomic nucleus potential is a Coulomb-like only within a certain range of distances: at large distances it is screened by atomic electrons, at short distances it is smeared quantum mechanically due to motion around atomic center of inertia. The latter case is especially interesting as it is hardly known. The positive charge cloud size in an atom depends on atomic state ψn. If this state is a highly excited and quasi-stable one (n>>1, Rydberg atoms), the positive charge cloud is very large: ~an(me/Ma), an >> a0.
    Last edited: Oct 3, 2009
  10. Oct 3, 2009 #9


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    you might want to go to Wikipedia and look up the concepts of "flux" and "Gauss's Law" (as well as "inverse-square law"). but it sounds like you got the idea for why there are inverse square laws in the 3-dimensional space we exist it.
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