Force with different distance dependence Recently, I have learnt about the central force motion. I came across some forces with different distance dependence, for example, k/r^2(common) one, k/r^4, or even the combined one k/r^2 + k'/r^4 (k, k' are force constant). 
I remembered when I first learnt the inverse square law, I was convinced that the inverse square law came from the density of the field line on a surface. However, how the forces with 1/r^3, 1/r^4 exists? Can they be derived by classical means?
I have searched for different information but I cannot get a good answer. I hope someone can help me, or relevant reference will do.
Thank you!
 A: While electric forces are 1/R^2, that's only between two charges.  If you look at distributions of mixed charges, the force laws include dipole, quadrupole,
octopole... and while those are a bit more complex than simple central forces (because they depend on orientation), the scaling with distance includes 1/R^N
terms (N being two or more).
Yukawa's potential for the strong atomic force came from an insight, that central forces were mediated by virtual particles (the photon, for electric force), and those particles made the Faraday-style 'lines of force' picture into a special case that depicts 1/R^2 forces due to massless virtual particles.  He then calculated the mass of the pion, based on the known properties of nuclear binding, and on the uncertainty principle linking mass to lifetime.   If that insight is correct, all fundamental forces in 3-dimensional space are likely to be inverse square law with an (optional) exponential decay factor.
Yukawa's particle, the pion, was found.   His insight seems correct, so far.
