Electric Dipole Moment - Dipole Moment Density and Polarization Density - Medium With Charge and Dipole Densities

Medium With Charge and Dipole Densities

As described next, a model for polarization moment density p(r) results in a polarization

restricted to the same model. For a smoothly varying dipole moment distribution p(r), the corresponding bound charge density is simply

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). The potential at a position r is:

where ρ(r) is the unpaired charge density, and p(r) is the dipole moment density. Using an identity:

the polarization integral can be transformed:

The first term can be transformed to an integral over the surface bounding the volume of integration, and contributes a surface charge density, discussed later. Putting this result back into the potential, and ignoring the surface charge for now:

where the volume integration extends only up to the bounding surface, and does not include this surface.

The potential is determined by the total charge, which the above shows consists of:

showing that:

In short, the dipole moment density p(r) plays the role of the polarization density P for this medium. Notice, p(r) has a non-zero divergence equal to the bound charge density (as modeled in this approximation).

It may be noted that this approach can be extended to include all the multipoles: dipole, quadrupole, etc. Using the relation:

the polarization density is found to be:

where the added terms are meant to indicate contributions from higher multipoles. Evidently, inclusion of higher multipoles signifies that the polarization density P no longer is determined by a dipole moment density p. For example, in considering scattering from a charge array, different multipoles scatter an electromagnetic wave differently and independently, requiring a representation of the charges that goes beyond the dipole approximation.