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As I humbly asked on Astronomy SE, I was wondering if there any forces or similar phenomena that scale inversely proportionally to the fourth power of the distance.

There are plenty of things that obey the inverse-square law and thus are inversely proportional to the square of the distance (light, Newtonian gravity), and while fewer there also exists some that inversely scale to the cube of the distance (tidal forces).

Are there any phenomenon, not necessarily gravity-related, that scale to the fourth power of the distance involved?

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  • $\begingroup$ See e.g. physics.stackexchange.com/q/50142 for the gist of your question with regards to "fundamental" forces. Off the top of my head I would think that dipole-dipole interaction is fourth order, but it's not isotropic. Maybe I am wrong. $\endgroup$ Commented Jan 22 at 23:03

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The force of attraction between two permanent electric dipoles scales like this. The electric field of one permament dipole scales like $1/r^3$, and then the force on one dipole from the electric field of the other is $\mathbf{F}=\mathbf{P}\cdot\nabla \mathbf{E}\propto (\partial_r)1/r^3\propto1/r^4$. Indeed there is plenty of complication in there with regards to the directions that each dipole faces, but if you assume that the two dipoles align themselves (as they would in many reasonable circumstances), then you do get this force law.

The potential due to a neutral atom (not a molecule with a constant dipole moment) and a charged particle scales like $1/r^4$, and has no such directionality issue, and the force scales like $1/r^5$. The electric field of the charged particle, which scales like $1/r^2$, induces a dipole moment in the neutral atom, which is proportional to the electric field, $\mathbf{P}=\alpha\mathbf{E}$, where $\alpha$ is the atom's polarizability. Then the force on a dipole from the electric field is $$ \mathbf{F}=\mathbf{P}\cdot\nabla \mathbf{E}\propto \frac{1}{r^2}\frac{\partial}{\partial r}\frac{1}{r^2}\propto\frac{1}{r^5} $$ This is relevant in my line of work, where antiprotons are stored in a cryogenic Penning trap, and they are lost on the timescale of an hour to collisions with background gas. If you wanted to calculate a theoretical rate for this process, you would need to use this power law. Similarly, you can get all kinds of odd power laws using different combinations of molecules with dipole moments, atoms without permanent dipole moments, and charged particles.

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