Electric dipole moment

In physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a system of charges, that is, a measure of the charge system's overall polarity. In the simple case of two point charges, one with charge {+}q and one with charge {-}q, the electric dipole moment p is: \boldsymbol{p} = q \, \boldsymbol{d} where d is the displacement vector pointing from the negative charge to the positive charge. Thus, the electric dipole moment vector p points from the negative charge to the positive charge. There is no inconsistency here, because the electric dipole moment has to do with orientation of the dipole, that is, the positions of the charges, and does not indicate the direction of the field originating in these charges. 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 = q d.

General case

More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is: \boldsymbol{p}(\boldsymbol{r}) = \int_{V} \rho(\boldsymbol{r_0})\, (\boldsymbol{r_0}-\boldsymbol{r}) \ d^3 \boldsymbol{r_0}, where r locates the point of observation and d3r0 denotes an elementary volume in V. For an array of point charges, the charge density becomes a sum of Dirac delta functions: \rho (\boldsymbol{r}) = \sum_{i=1}^N \, q_i \, \delta (\mathbf{r} - \mathbf{r}_i ) \ , where each \mathbf{r}_i is a vector from some reference point to the charge q_i. Substitution into the above integration formula provides: \boldsymbol{p}(\boldsymbol{r}) = \sum_{i=1}^N \, q_i \int\delta (\mathbf{r_0} - \mathbf{r}_i )\, (\boldsymbol{r_0}-\boldsymbol{r}) \ d^3 \boldsymbol{r_0}  = \sum_{i=1}^N \, q_i (\boldsymbol{r_i}-\boldsymbol{r}), This expression is equivalent to the previous expression in the case of charge neutrality and N = 2. For two opposite charges, denoting the location of the positive charge of the pair as \boldsymbol {r_+} and the location of the negative charge as \boldsymbol {r_-} : \boldsymbol{p}(\boldsymbol{r}) =q_1(\boldsymbol{r_1}-\boldsymbol{r})+q_2(\boldsymbol{r_2}-\boldsymbol{r})  = q(\boldsymbol{r_+}-\boldsymbol{r})-q(\boldsymbol{r_-}-\boldsymbol{r})  =q (\boldsymbol{r_+} - \boldsymbol{r_-})=q\boldsymbol d \ , 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 most easily understood when the system has an overall neutral charge; for example, a pair of opposite charges, or a neutral conductor in a uniform electric field. For a system of charges with no net charge, visualized as an array of paired opposite charges, the relation for electric dipole moment is: \boldsymbol{p}(\boldsymbol{r}) = \sum_{i=1}^{N} \, \int q_i \left( \delta (\mathbf{r_0} - (\mathbf{r}_i + \boldsymbol{d_i}) )- \delta ( \mathbf{r_0} - \mathbf{r}_i ) \right)\, (\boldsymbol{r_0}-\boldsymbol{r}) \ d^3 \boldsymbol{r_0}  = \sum_{i=1}^{N} \, q_i \left( \boldsymbol{r_i +d_i}-\boldsymbol{r} -(\boldsymbol{r_i }-\boldsymbol{r}) \right) = \sum_{i=1}^{N} q_i\boldsymbol{d}_i \, = \sum_{i=1}^{N} \boldsymbol{p}_i \ , which is the vector sum of the individual dipole moments of the neutral charge pairs. (Because of overall charge neutrality, the dipole moment is independent of the observer's position r.) 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 or the center of charge, not some arbitrary origin. This convention ensures that 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. The potential and field of such an ideal dipole are found next as a limiting case of an example of two opposite charges at non-zero separation. Two closely spaced opposite charges have a potential of the form: \phi ( \boldsymbol{r} )=\frac {q}{4 \pi \varepsilon _0 | \boldsymbol{ r}- \boldsymbol{r}_+ |} -\frac {q}{4 \pi \varepsilon _0 | \boldsymbol{ r}- \boldsymbol{r}_- | } \ , with charge separation, d, defined as \boldsymbol d = \boldsymbol{r}_+ - \boldsymbol{r}_- \ , The radius to the center of charge, R, and the unit vector in the direction of R are given by: {\boldsymbol {R}} = \boldsymbol{r} - \frac{\boldsymbol{r}_+ + \boldsymbol{r}_-}{2}\ ; \ \boldsymbol {\hat{R}} = \frac {\boldsymbol {R}}{R} \ , Taylor expansion in d/r (see multipole expansion and quadrupole) allows this potential to be expressed as a series. \phi ( \boldsymbol{R} )=\frac {1}{4 \pi \varepsilon _0} \frac {q\boldsymbol {d \cdot \hat{R}}}{R^2} + O(\frac{d^2}{R^2}) \approx \frac {1}{4 \pi \varepsilon _0} \frac {\boldsymbol {p\cdot \hat{R}}}{R^2} \ , where higher order terms in the series are vanishing at large distances, R, compared to d. Each succeeding term provides a more detailed view of the distribution of charge, and falls off more rapidly with distance. For example, the quadrupole moment is the basis for the next term: Q_{ij} = \int d^3 \boldsymbol{r_0} \left( 3x_i x_j -r_0^2 \delta_{ij} \right) \rho( \boldsymbol{r_0}) \ , with ro = (x1, x2, x3). See Here, the electric dipole moment p is, as above: \boldsymbol p = q \boldsymbol d \ . The result for the dipole potential also can be expressed as: \phi ( \boldsymbol{R} )=- \boldsymbol {p\cdot \nabla}\frac {1}{4 \pi \varepsilon _0 R}\ , 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 field of the dipole is the gradient of the potential, leading to: \boldsymbol E = \frac {3 \boldsymbol {p \cdot \hat{R}}}{4 \pi \varepsilon_0 R^3} \boldsymbol { \hat{R}}-\frac {\boldsymbol{p}}{4 \pi \varepsilon_0 R^3} \ . Thus, although two closely spaced opposite charges are not an ideal electric dipole (because their potential at close approach is not that of a 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 (d is made smaller), 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, however, 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 charge, \boldsymbol p = \sum_{i=1}^N \ q_i \boldsymbol {d_i} \ , determines the degree of polarity of the array, but for a neutral array it is simply a vector property of the array with no directions about where the array happens to be located. The dipole moment density of the array p(r) 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(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). As explained below, sometimes it is sufficiently accurate to take P(r) = p(r). Sometimes a more detailed description is needed (for example, supplementing the dipole moment density with an additional quadrupole density) and sometimes even more elaborate versions of P(r) are necessary. It now is explored just in what way the polarization density P(r) 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(r) (which describes not only the dipole moment, but also the array location). 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(r). 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: \boldsymbol{D} = \varepsilon _0 \boldsymbol{E} + \boldsymbol{P}\ , where P is called the polarization density. In this formulation, the divergence of this equation yields: \nabla \cdot \boldsymbol{D} = \rho_f = \varepsilon _0 \nabla \cdot \boldsymbol{E} +\nabla \cdot \boldsymbol{P}\ , and as the divergence term in E is the total charge, and ρf is "free charge", we are left with the relation: \nabla \cdot \boldsymbol{P} = -\rho_b \ , with ρb 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 curl E = 0, which implies curl (D − P) = 0. Applying Helmholtz decomposition: \boldsymbol{ (D-P) = -\nabla } \varphi \ , for some scalar potential φ, and: \boldsymbol {\nabla \cdot (D-P)} =\varepsilon_0 \boldsymbol {\nabla \cdot E}=\rho_f +\rho_b = -\nabla ^2 \varphi \ . Suppose the charges are divided into free and bound, and the potential is divided into φ = φf + φb. Satisfaction of the boundary conditions upon φ may be divided arbitrarily between φf and φb 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. For example, one could place the boundary around the bound charges at infinity. Then φb falls off with distance from the bound charges. If an external field is present, and zero free charge, the field can be accounted for in the contribution of φf, which would arrange to satisfy the boundary conditions and Laplace's equation ∇2φf = 0. In principle, one could add the same arbitrary curl to both D and P, which would cancel out of the difference D − P. However, assuming D and P originate in in a simple division of charges into free and bound, they are at bottom electric fields and so have zero curl. In particular, when no free charge is present, one possible choice is P = ε0 E. Next is discussed how several different dipole-moment descriptions of a medium relate to the polarization entering Maxwell's equations.

Medium with charge and dipole densities

As described next, a model for polarization moment density p(r) results in a polarization P(r) = p(r) restricted to the same model. For a smoothly varying dipole moment distribution p(r), the corresponding bound charge density is simply ∇·p(r) = −ρb. However, in the case of a p(r) that exhibits an abrupt step in dipole moment at a boundary between two regions, ∇·p(r) exhibits a surface charge component of bound charge. This surface charge can be treated through a surface integral, or by using discontinuity conditions at the boundary, as illustrated in the various examples below. As a first example relating dipole moment to polarization, consider a medium made up of a continuous charge density ρ(r) and a continuous dipole moment distribution p(r). This medium can be seen as an idealization growing from the multipole expansion of the potential of an arbitrarily complex charge distribution, truncation of the expansion, and the forcing of the truncated form to apply everywhere. The result is a hypothetical medium. See The potential at a position r is: