Electric Dipole

An electric dipole is a distribution of electric charge with separated positive and negative components. Its simplest model consists of two equal and opposite point charges a finite distance apart. Although their total charge is zero, their electric field does not vanish. The electric dipole moment describes the magnitude and direction of this separation and provides a useful approximation to the distant field of many neutral charge distributions. Electric dipoles are important in electromagnetism, molecular physics, and the electrical response of materials. (feynmanlectures.caltech.edu)

Dipole moment and idealization

For charges $+q$ and $-q$, the dipole moment is

where $\mathbf d$ points from the negative charge to the positive charge. Its magnitude is $p=qd$. This convention distinguishes the moment’s direction from the field direction between the charges, which points from positive to negative. The SI unit of dipole moment is the coulomb metre, $\mathrm{C\,m}$. Molecular moments are also commonly expressed in debye, with $1\,\mathrm D\approx3.33564\times10^{-30}\,\mathrm{C\,m}$. (old.goldbook.iupac.org)

For several point charges, or a continuous distribution with charge density $\rho$,

If the coordinate origin is displaced by $\mathbf a$, the moment becomes $\mathbf p'=\mathbf p-Q\mathbf a$, where $Q$ is the total charge. Thus the moment is independent of origin for a neutral system, but not generally for a charged one. (iupac.org)

An ideal point dipole is obtained mathematically by letting the separation approach zero while holding $\mathbf p$ fixed. It represents the leading dipolar behavior outside a sufficiently small source, rather than a literal pair of finite charges at one point. (farside.ph.utexas.edu)

Electrostatic potential and field

In electrostatics, the superposition principle combines the two charges’ potentials. For a dipole centered at the origin, at distances $r$ much greater than its separation, the electric potential in vacuum is

Here $\varepsilon_0$ is the vacuum permittivity and $\hat{\mathbf r}$ is the radial unit vector. The corresponding field, valid away from the origin for a point dipole, is

These expressions follow from Coulomb’s law and $\mathbf E=-\nabla V$. (farside.ph.utexas.edu)

The potential decreases as $r^{-2}$, while the field decreases as $r^{-3}$, faster than the field of a single net charge. On the dipole axis the field points along $\mathbf p$; in the equatorial plane it points opposite to $\mathbf p$, and the potential is zero. In a multipole expansion, the dipole term follows the monopole term. For a neutral localized distribution with nonzero dipole moment, it dominates the sufficiently distant potential; if the moment also vanishes, higher multipoles determine the leading behavior. (farside.ph.utexas.edu)

Interaction with an external field

A fixed dipole in a uniform external field experiences a torque

and has orientation-dependent potential energy

The torque tends to align the moment with the field. Parallel alignment minimizes this energy, whereas antiparallel alignment maximizes it. Equal and opposite forces on the charges give zero net force in a uniform field, even when the torque is nonzero. (old.goldbook.iupac.org)

A nonuniform electrostatic field can exert a net force. For a small dipole with fixed moment,

This depends on spatial variation of the external field. An induced dipole requires additional care because its moment changes with the field; the fixed-moment energy formula alone does not describe the work required to induce it. (feynmanlectures.caltech.edu)

Molecular dipoles and dielectrics

A molecule has a permanent dipole when the centers of its positive and negative charge distributions do not coincide. Water is a familiar example: its nonsymmetric charge arrangement produces a permanent molecular moment. A symmetric molecule such as oxygen has no permanent moment, but an applied field can distort its charge distribution and induce one. (feynmanlectures.caltech.edu)

In an atom, this distortion involves displacement of the negative electron distribution relative to the positive nucleus. For weak fields, the induced moment is often approximately proportional to the local field,

where $\alpha$ is the electric polarizability in this convention. Other conventions factor out $\varepsilon_0$. Permanent moments can also acquire partial orientational alignment, opposed by thermal motion and therefore dependent on temperature. (feynmanlectures.caltech.edu)

The electric polarization $\mathbf P$ of a dielectric is its dipole moment per unit volume. In a linear isotropic material, $\mathbf P=\varepsilon_0\chi_e\mathbf E$, where $\chi_e$ is the electric susceptibility. Spatial variation and boundaries of polarization produce bound charges, connecting microscopic dipoles to macroscopic material fields. (feynmanlectures.caltech.edu)

Radiation and quantum transitions

A time-varying dipole can emit electromagnetic radiation. For a localized sinusoidally oscillating dipole much smaller than the wavelength, radiation is strongest perpendicular to its axis and vanishes along the axis. Its far-zone field decreases as $r^{-1}$, unlike the static dipole field’s $r^{-3}$ dependence. (farside.ph.utexas.edu)

In quantum mechanics, electric-dipole transitions depend on the transition moment

This is distinct from a state’s permanent dipole moment. A vanishing transition moment forbids the transition within the electric-dipole approximation, giving selection rules important in spectroscopy. Such transitions may still occur through weaker magnetic-dipole, electric-quadrupole, or other processes. (farside.ph.utexas.edu)