What Determines the Dependence of Dipole-Dipole Interactions on Distance?

[LeoYard]

What is the basis for the sixth root dependancy on the inverse of the distance between the dipoles (in any dipole-dipole interaction)? Is it empirical or can it be mathematically derived?

 

[genneth]

http://en.wikipedia.org/wiki/Lennard-Jones_potential

 

[genneth]

Though I should add that it is possible to derive the long distance attraction between dipoles theoretically. It's an elementary exercise in electrostatics -- if you get stuck, post your efforts here and I'm sure PF members will guide you.

 

[LeoYard]

Thank you, genneth. Coulomb's law states that the electric field from a point charge drops as the square of the radius. Put two charges at the same place and you get zero electric field, so the two charges need to be slightly displaced. However, as you go to large radii, the separation between the two charges becomes irrelevant, so that starts to look more like a zero-charge object.

 

[genneth]

Not quite. A dipole field falls off as 1/r^3 at large distances -- it's a tedious calculation, but you should learn to do it anyway, since it's quite typical of mathematical analysis of physics. Another dipole would feel 1/r^4 attraction. However, the usual law assumes induced dipole moments, so you get another 1/r^2 factor.

(I think -- all this is off the top of my head.)

 

[Meir Achuz]

What is the basis for the sixth root dependancy on the inverse of the distance between the dipoles (in any dipole-dipole interaction)? Is it empirical or can it be mathematically derived?

I assume you mean the the potential energy falls like 1/r^6.
The interaction energy of a dipole in an electric field = -p.E.
E of a dipole varies like 1/r^3, so the energy of two permanent dipoles varies like 1/r^3.
The 1/r^6 results if the dipole moments are induced dipoles, that is each dipole moment is caused by the E field of the other dipole. This gives another factor of 1/r^3,
resulting in the 1/r^6 for the energy of two induced dipoles.

 

[Loren Booda]

Four of the six radial orders in the field's denominator represent the permutations for Coulomb forces between the charges in the two dipoles. Binomial expansion with dipole separation approaching zero accounts for the extra two orders of displacement.