 Solution

An electric dipole has equal and opposite charges separated by a small vector from negative to positive, with electric dipole moment $\mathbf{p}=q\mathbf{d}$. More generally, a localized neutral distribution has $\mathbf{p}=\int {\mathbf{r}}^{\prime }\rho ({\mathbf{r}}^{\prime })d^3{r}^{\prime }$, independent of origin. At distances large compared with its size, the leading nonzero term in the multipole expansion isThe field points along $\mathbf{p}$ on either axial side and opposite $\mathbf{p}$ in the equatorial plane; its axial magnitude is twice the equatorial magnitude at equal radius. Near the charges the finite-charge expression must replace the far-field approximation.

A permanent dipole in an externally imposed slowly varying electrostatic field has interaction energy $-\mathbf{p}\cdot \mathbf{E}$, torque $\mathbf{\tau }=\mathbf{p}\times \mathbf{E}$ and force $\nabla (\mathbf{p}\cdot \mathbf{E})$ for fixed $\mathbf{p}$. A uniform field aligns it but gives no net force. For a linearly induced dipole $\mathbf{p}=\alpha \mathbf{E}$, the formation-inclusive energy is $-\alpha E^2/2$, not $-\alpha E^2$: the moment itself grows while the field is applied. Dipole density gives electric polarization, whose bound charges are ${\rho }_b=-\nabla \cdot \mathbf{P}$ and ${\sigma }_b=\mathbf{P}\cdot \hat{\mathbf{n}}$.