Statistical Mechanics · Problem 3.36

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · dipole-dipole-interaction-force

Problem

Consider a pair of electric dipoles

\mu

and

\mu'

, oriented in the directions

(\theta, \phi)

and

(\theta', \phi')

, respectively; the distance

R

between their centers is assumed to be fixed. The potential energy in this orientation is given by

-\frac{\mu\mu'}{R^3}\{2\cos\theta\cos\theta' - \sin\theta\sin\theta'\cos(\phi - \phi')\}.

Now, consider the pair to be in thermal equilibrium, their orientations being governed by a canonical distribution. Show that the mean force between the dipoles, at high temperatures, is given by

-2\frac{\mu\mu'}{kT}\frac{\hat{R}}{R^7},

\hat{R}

being a unit vector in the direction of the line of centers.

Related theory

Estimating Dipole Dipole Energy From Critical Temperature

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