Electric dipole moment

The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system: that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb-metre. The debye is a CGS unit of measurement used in atomic physics and chemistry.
Theoretically, an electric dipole is defined by the first-order term of the multipole expansion; it consists of two equal and opposite charges that are infinitesimally close together, although real dipoles have separated charge.

Elementary definition

Often in physics, the dimensions of an object can be ignored so it can be treated as a point-like object, i.e. a point particle. Point particles with electric charge are referred to as point charges. Two point charges, one with charge and the other one with charge separated by a distance, constitute an electric dipole. For this case, the electric dipole moment has a magnitude and is directed from the negative charge to the positive one.
A stronger mathematical definition is to use vector algebra, since a quantity with magnitude and direction, like the dipole moment of two point charges, can be expressed in vector form where is the displacement vector pointing from the negative charge to the positive charge. The electric dipole moment vector also points from the negative charge to the positive charge. With this definition the dipole direction tends to align itself with an external electric field. Note that this sign convention is used in physics, while the opposite sign convention for the dipole, from the positive charge to the negative charge, is used in chemistry.
An idealization of this two-charge system is the electrical point dipole consisting of two charges only infinitesimally separated, but with a finite. This quantity is used in the definition of polarization density.

Energy and torque

An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. The torque tends to align the dipole with the field. A dipole aligned parallel to an electric field has lower potential energy than a dipole making some non-zero angle with it. For a spatially uniform electric field across the small region occupied by the dipole, the energy U and the torque are given by
The scalar dot "" product and the negative sign shows the potential energy minimizes when the dipole is parallel with the field, maximizes when it is antiparallel, and is zero when it is perpendicular. The symbol "" refers to the vector cross product. The E-field vector and the dipole vector define a plane, and the torque is directed normal to that plane with the direction given by the right-hand rule. A dipole in such a uniform field may twist and oscillate, but receives no overall net force with no linear acceleration of the dipole. The dipole twists to align with the external field.
However, in a non-uniform electric field a dipole may indeed receive a net force since the force on one end of the dipole no longer balances that on the other end. It can be shown that this net force is generally parallel to the dipole moment.

Expression (general case)

More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is:
where r locates the point of observation and ³r′ denotes an elementary volume in V. For an array of point charges, the charge density becomes a sum of Dirac delta functions:
where each r_i is a vector from some reference point to the charge q_i. Substitution into the above integration formula provides:
This expression is equivalent to the previous expression in the case of charge neutrality and. For two opposite charges, denoting the location of the positive charge of the pair as r₊ and the location of the negative charge as r₋:
showing that the dipole moment vector is directed from the negative charge to the positive charge because the position vector of a point is directed outward from the origin to that point.
The dipole moment is particularly useful in the context of an overall neutral system of charges, such as a pair of opposite charges or a neutral conductor in a uniform electric field.
For such a system, visualized as an array of paired opposite charges, the relation for electric dipole moment is:
where r is the point of observation and, r_i being the position of the negative charge in the dipole i, and r_i the position of the positive charge.
This is the vector sum of the individual dipole moments of the neutral charge pairs. Thus, the value of p is independent of the choice of reference point, provided the overall charge of the system is zero.
When discussing the dipole moment of a non-neutral system, such as the dipole moment of the proton, a dependence on the choice of reference point arises. In such cases it is conventional to choose the reference point to be the center of mass of the system, not some arbitrary origin. This choice is not only a matter of convention: the notion of dipole moment is essentially derived from the mechanical notion of torque, and as in mechanics, it is computationally and theoretically useful to choose the center of mass as the observation point. For a charged molecule the center of charge should be the reference point instead of the center of mass. For neutral systems the reference point is not important, and the dipole moment is an intrinsic property of the system.

Potential and field of an electric dipole

An ideal dipole consists of two opposite charges with infinitesimal separation. We compute the potential and field of such an ideal dipole starting with two opposite charges at separation, and taking the limit as.
Two closely spaced opposite charges ±q have a potential of the form:
corresponding to the charge density
by Coulomb's law,
where the charge separation is:
Let R denote the position vector relative to the midpoint, and the corresponding unit vector:
Taylor expansion in expresses this potential as a series.
where higher order terms in the series are vanishing at large distances, R, compared to d. Here, the electric dipole moment p is, as above:
The result for the dipole potential also can be expressed as:
which relates the dipole potential to that of a point charge. A key point is that the potential of the dipole falls off faster with distance R than that of the point charge.
The electric field of the dipole is the negative gradient of the potential, leading to:
Thus, although two closely spaced opposite charges are not quite an ideal electric dipole, at distances much larger than their separation, their dipole moment p appears directly in their potential and field.
As the two charges are brought closer together, the dipole term in the multipole expansion based on the ratio d/''R becomes the only significant term at ever closer distances R'', and in the limit of infinitesimal separation the dipole term in this expansion is all that matters. As d is made infinitesimal, thus, the dipole charge must be made to increase to hold p constant. This limiting process results in a "point dipole".

Dipole moment density and polarization density

The dipole moment of an array of charges,
determines the degree of polarity of the array, but for a neutral array it is simply a vector property of the array with no information about the array's absolute location. The dipole moment density of the array p contains both the location of the array and its dipole moment. 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 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. As explained below, sometimes it is sufficiently accurate to take. Sometimes a more detailed description is needed and sometimes even more elaborate versions of P are necessary.
It now is explored just in what way the polarization density P that enters Maxwell's equations is related to the dipole moment p of an overall neutral array of charges, and also to the dipole moment density p. Only static situations are considered in what follows, so P has no time dependence, and there is no displacement current. First is some discussion of the polarization density P. That discussion is followed with several particular examples.
A formulation of Maxwell's equations based upon division of charges and currents into "free" and "bound" charges and currents leads to introduction of the D- and P-fields:
where P is called the polarization density. In this formulation, the divergence of this equation yields:
and as the divergence term in E is the total charge, and ρ_f is "free charge", we are left with the relation:
with ρ_b as the bound charge, by which is meant the difference between the total and the free charge densities.
As an aside, in the absence of magnetic effects, Maxwell's equations specify that
which implies
Applying Helmholtz decomposition:
for some scalar potential φ, and:
Suppose the charges are divided into free and bound, and the potential is divided into
Satisfaction of the boundary conditions upon φ may be divided arbitrarily between φ_f and φ_b because only the sum φ must satisfy these conditions. It follows that P is simply proportional to the electric field due to the charges selected as bound, with boundary conditions that prove convenient. In particular, when no free charge is present, one possible choice is.
Next is discussed how several different dipole moment descriptions of a medium relate to the polarization entering Maxwell's equations.