Two-Dipole Interaction

Two classical dipoles with dipole moments μ₁ and μ₂ are separated by a distance R so that only the orientation of the magnetic moments is free. They are in thermal equilibrium at a temperature Compute the mean force ⟨f⟩ between the dipoles for the high-temperature limit μ₁ μ₂/τR³<< 1.

Hint: The potential energy of interaction of two dipoles is

SOLUTION

Introduce spherical coordinates with the ẑ axis along the line of the separation between the dipoles. Then the partition function reads

(1)

The potential energy of the interaction can be rewritten in the form

(2)

Since

(3)

(2) becomes

(4)

and

(5)

We can expand the exponential at high temperatures μ₁μ₂/τr³ << 1 so that

(6)

where A ≡ μ²₁μ²₂/τ²r⁶. The first-order terms are all zero upon integration, and we have

(7)

where the cross term also vanishes, and we find

(8)

The average force is given by

where F is the free energy. So,

(9)

The minus sign indicates an average attraction between the dipoles.