Polarization Density

The following development is applicable to polarization phenomena having diverse microscopic origins. Whether representative of atoms, molecules, groups of ordered atoms or molecules (domains), or even macroscopic particles, the dipoles are pictured as opposite charges

qseparated by a vector distancedirected from the negative to the positive charge. Thus, the individual dipoles, represented as in Sec. 4.4, have momentsddefined asp

Because

is generally smaller in magnitude than the size of the atom, molecule, or other particle, it is small compared with any macroscopic dimension of interest.d

Now consider a medium consisting of

Nsuch polarized particles per unit volume. What is the net chargeqcontained within an arbitrary volumeVenclosed by a surfaceS? Clearly, if the particles of the medium withinVwere unpolarized, the net charge inVwould be zero. However, now that they are polarized, some charge centers that were contained inVin their unpolarized state have moved out of the surfaceSand left behind unneutralized centers of charge. To determine the net unneutralized charge left behind inV, we will assume (without loss of generality) that the negative centers of charge are stationary and that only the positive centers of charge are mobile during the polarization process.

Consider the particles in the neighborhood of an element of area

don the surfaceaS, as shown in Fig. 6.1.1. All positive centers of charge now outsideSwithin the volumedV =have left behind negative charge centers. These contribute a net negative charge toddaV. Because there areNsuch negative centers of charge inddadV, the net charge left behind inVis

Figure 6.1.1Volume element containing positive charges which have left negative charges on the other side of surfaceS.

Note that the integrand can be either positive or negative depending on whether positive centers of charge are leaving or entering

Vthrough the surface elementd. Which of these possibilities occurs is reflected by the relative orientation ofaanddd. Ifahas a component parallel (anti-parallel) todd, then positive centers of charge are leaving (entering)aVthroughd.aThe integrand of (1) has the dimensions of dipole moment per unit volume and will therefore be defined as the

polarization density.

Also by definition, the net charge in

Vcan be determined by integrating the polarization charge density over its volume.

Thus, we have two ways of calculating the net charge, the first by using the polarization density from (3) in the surface integral of (2).

Here Gauss' theorem has been used to convert the surface integral to one over the enclosed volume. The charge found from this volume integral must be the same as given by the second way of calculating the net charge, by (4). Because the volume under consideration is arbitrary, the integrands of the volume integrals in (4) and (5) must be identical.

In this way, the polarization charge density

has been related to the polarization density_{p}.P

It may seem that little has been accomplished in this development because, instead of the unknown

, the new unknown_{p}appeared. In some instances,Pis known. But even in the more common cases where the polarization density and hence the polarization charge density is not known a priori but is induced by the field, it is easier to directly linkPwithPthanEwith_{p}.E

In Fig. 6.0.1, the polarized sphere could acquire no net charge. Our representation of the polarization charge density in terms of the polarization density guarantees that this is true. To see this, suppose

Vis interpreted as the volume containing the entire polarized body so that the surfaceSenclosing the volumeVfalls outside the body. Becausevanishes onPS, the surface integral in (5) must vanish. Any distribution of charge density related to the polarization density by (6) cannot contribute a net charge to an isolated body.

We will often find it necessary to represent the polarization density by a discontinuous function. For example, in a material surrounded by free space, such as the sphere in Fig. 6.0.1, the polarization density can fall from a finite value to zero at the interface. In such regions, there can be a surface polarization charge density. With the objective of determining this density from

, (6) can be integrated over a pillbox enclosing an incremental area of an interface. With the substitutionP-andP_{o}E, (6) takes the same form as Gauss' law, so the proof is identical to that leading from (1.3.1) to (1.3.17). We conclude that where there is a jump in the normal component of_{p}, there is aPsurface polarization charge density

Just as (6) tells us how to determine the polarization charge density for a given distribution of

in the volume of a material, this expression serves to evaluate the singularity in polarization charge density (the surface polarization charge density) at an interface.P

Note that according to (6),

originates on negative polarization charge and terminates on positive charge. This contrasts with the relationship betweenPand the charge density. For example, according to (6) and (7), the uniformly polarized cylinder of material shown in Fig. 6.1.2 withEpointing upward has positivePon the top and negative on the bottom._{sp}

Figure 6.1.2Polarization surface charge due to uniform polarization of right cylinder.