Electric dipole moment

[1] An idealization of this two-charge system is the electrical point dipole consisting of two (infinite) charges only infinitesimally separated, but with a finite p. This quantity is used in the definition of polarization density.

For a spatially uniform electric field across the small region occupied by the dipole, the energy U and the torque

The scalar dot "⋅" product and the negative sign shows the potential energy minimises when the dipole is parallel with the field, maximises when it is antiparallel, and is zero when it is perpendicular.

More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is:

For such a system, visualized as an array of paired opposite charges, the relation for electric dipole moment is:

In such cases it is conventional to choose the reference point to be the center of mass of the system, not some arbitrary origin.

We compute the potential and field of such an ideal dipole starting with two opposite charges at separation d > 0, and taking the limit as d → 0.

where higher order terms in the series are vanishing at large distances, R, compared to d.[notes 2] Here, the electric dipole moment p is, as above:

When it comes time to calculate the electric field in some region containing the array, Maxwell's equations are solved, and the information about the charge array is contained in the polarization density P(r) of Maxwell's equations.

Depending upon how fine-grained an assessment of the electric field is required, more or less information about the charge array will have to be expressed by P(r).

It follows that P is simply proportional to the electric field due to the charges selected as bound, with boundary conditions that prove convenient.

For a smoothly varying dipole moment distribution p(r), the corresponding bound charge density is simply

Notice, p(r) has a non-zero divergence equal to the bound charge density (as modeled in this approximation).

The right side vanishes as the volume shrinks, inasmuch as ρb is finite, indicating a discontinuity in E, and therefore a surface charge.

The above general remarks about surface charge are made more concrete by considering the example of a dielectric sphere in a uniform electric field.

[25][26] The sphere is found to adopt a surface charge related to the dipole moment of its interior.

If observation is confined to regions sufficiently remote from a system of charges, a multipole expansion of the exact polarization density can be made.

The simplest approximation is to replace the charge array with a model of ideal (infinitesimally spaced) dipoles.

In particular, as in the example above that uses a constant dipole moment density confined to a finite region, a surface charge and depolarization field results.

[27][28] A related approach is to divide the charges into those nearby the point of observation, and those far enough away to allow a multipole expansion.

[34] Therefore, values for these EDMs place strong constraints upon the scale of CP-violation that extensions to the standard model of particle physics may allow.

Experiments have been performed to measure the electric dipole moment of various particles like the electron and the neutron.

Instanton corrections from a nonzero θ term in quantum chromodynamics predict a nonzero electric dipole moment for the neutron and proton, which have not been observed in experiments (where the best bounds come from analysing neutrons).

Dipole moments in molecules are responsible for the behavior of a substance in the presence of external electric fields.

This effect forms the basis of a modern experimental technique called dielectric spectroscopy.

[37] By means of the total dipole moment of some material one can compute the dielectric constant which is related to the more intuitive concept of conductivity.

In general the total dipole moment have contributions coming from translations and rotations of the molecules in the sample,

[38] It is possible to calculate dipole moments from electronic structure theory, either as a response to constant electric fields or from the density matrix.

[39] Such values however are not directly comparable to experiment due to the potential presence of nuclear quantum effects, which can be substantial for even simple systems like the ammonia molecule.

[43] The dipole moment of a molecule can also be calculated based on the molecular structure using the concept of group contribution methods.

Quantities defining the electric dipole moment of two point charges.
Electric dipole p and its torque τ in a uniform E field.
Potential map of a physical electric dipole. Negative potentials are in blue; positive potentials, in red.
A uniform array of identical dipoles is equivalent to a surface charge.
Field lines of the D -field in a dielectric sphere with greater susceptibility than its surroundings, placed in a previously uniform field. [ notes 9 ] The field lines of the E -field (not shown) coincide everywhere with those of the D -field, but inside the sphere, their density is lower, corresponding to the fact that the E -field is weaker inside the sphere than outside. Many of the external E -field lines terminate on the surface of the sphere, where there is a bound charge.