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An electric quadrupole is a charge distribution whose total charge and electric dipole moment vanish while its second charge moment does not. Its potential decays as $r^{-3}$.

The free-space Green function of the Laplacian for the Poisson equation gives the electric potentialFor $r=|\mathbf{x}|\gg R$, its electric multipole expansion begins withDefine the total electric charge and electric dipole moment byThen

The analogous solution for the magnetic vector potential isIts apparent $r^{-1}$ coefficient is ${\mu }_0(4\pi {)}^{-1}\int \mathbf{J}{d}^3{x}^{\prime }$. For each component, the supplied identity and the divergence-free vector field condition giveThe divergence theorem turns this into a surface integral outside the compact support of the current density, where $\mathbf{J}=0$. Hence $\int \mathbf{J}{d}^3{x}^{\prime }=0$, so the $1/r$ term vanishes.

The electric potential of a point charge $Q$ at the origin isFor the two charges forming the dipole,The first-order Taylor expansion at large $|\mathbf{x}|$ isand therefore, with the electric dipole moment $\mathbf{p}=Q\mathbf{d}$,to leading order. Taking the interaction of a second dipole with the corresponding electric field gives the stated electric dipole-dipole interaction

Let the lattice spacing be $d$ and write the central dipole asIts two horizontal neighbours have moment $p(\cos \theta ,-\sin \theta )$. For either one, the expression in parentheses, after extracting $d^{-3}$, isIts two vertical neighbours have moment $p(-\cos \theta ,\sin \theta )$, and each contributes insteadAdding all four nearest-neighbour interactions and using the Pythagorean trigonometric identity givesThe angle has cancelled, so the energy is independent of $\theta$.

Successive reflections in the two grounded planes require four charges:Their total charge and electric dipole moment vanish. Applying the second-order expansion from part (a), the terms proportional to $a^2{x}^2$, $b^2{y}^2$, and $a^2+b^2$ also cancel, while the mixed terms add. ThusThis is an electric quadrupole potential.

An anomalous chiral redefinition moves phase between the bare QCD angle and the quark mass matrix, so the measurable parameter isThe neutron electric dipole moment requires $|\bar{\theta }|$ to be extremely small, although no generic Standard Model symmetry enforces this. This is the Strong CP problem.

In Peccei–Quinn theory, an anomalous spontaneously broken global symmetry replaces $\bar{\theta }$ by a dynamical axion field. Nonperturbative QCD generates an axion potential whose minimum lies at the CP-conserving value, dynamically relaxing the effective angle to zero.

The far field of a dipole is $[3(p\cdot \hat{x})\hat{x}-p]/(4\pi {\epsilon }_0{r}^3)$. Comparison yields the induced electric dipole moment

The hydrogen ground-state charge cloud is spherically symmetric and has even parity, so it has no permanent electric dipole moment. A uniform electric field couples through a perturbation proportional to $\mathbf{E}\cdot \mathbf{r}$, which is odd under parity. Its expectation in the ground state therefore vanishes, giving no first-order energy correction.

The field can nevertheless polarize the atom: it admixes odd-parity excited states and creates an induced dipole. Such mixing contributes from second order onward, producing the quadratic Stark effect of the hydrogen ground state.

The electric dipole moment of a localized charge density isDifferentiate and use the continuity equation $\dot{\rho }+\nabla \cdot \mathbf{J}=0$:Localization makes the boundary integral vanish. Component by component,This is the time derivative of the electric dipole moment.

Let $\theta$ be the angle between $r$ and the positive $z$-axis. By superposition of the electric potential of point charges,For $r>d$, the Legendre polynomial expansion givesThereforeThe three displayed terms are respectively the monopole, dipole, and quadrupole terms of the electric multipole expansion. In particular, the total charge is $Q(N-M-1)$ and the electric dipole moment is $Qd(M-1)\hat{z}$. This is the multipole expansion of three collinear charges.