Show that the interaction energy of two dipoles separated by...

Show that the interaction energy of two dipoles separated by a displacement $\mathbf{r}$ is $$U=\frac{1}{4 \pi \epsilon_{0}} \frac{1}{r^{3}}\left[\mathbf{p}_{1} \cdot \mathbf{p}_{2}-3\left(\mathbf{p}_{1} \cdot \hat{\mathbf{r}}\right)\left(\mathbf{p}_{2} \cdot \hat{\mathbf{r}}\right)\right].$$ [Hint: use Prob. 4.7 and Eq. 3.104.]

Key Concepts

Multipole Expansion

Multipole expansion is a mathematical technique used to approximate the potential at a distance from a localized charge distribution, breaking it down into contributions from monopoles, dipoles, quadrupoles, etc. In the case of dipole-dipole interactions, only the dipole term is retained when the interacting objects are sufficiently far apart, simplifying the energy expression.

Unit Vector and Displacement Vector

The unit vector in the direction of the displacement vector defines the orientation between the two dipoles. It is crucial for isolating the directional dependence of the interaction energy, particularly in separating out the components of the dipole moments that are aligned with the line joining the two dipoles.

Dipole-Dipole Interaction Energy

Dipole-dipole interaction energy describes how two electric dipoles influence each other when placed in proximity. It is derived from the work done by the electric field of one dipole on the other and involves terms accounting for both the magnitudes of the dipole moments and their orientations relative to the line connecting them.

Electric Dipole Moment

An electric dipole moment is a vector quantity that characterizes the separation of positive and negative charges within a system. It is fundamental in understanding how such a charge configuration interacts with external electric fields, and it quantifies the strength and orientation of the dipole's field.

Vector Dot Product

The dot product is a mathematical operation that projects one vector onto another and quantifies the degree of alignment between the two. In the context of dipole interactions, it is used to express both the direct interaction of the dipole moments and the projection of these moments along the direction of the displacement vector.

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