The polarization vector $\FLPP$
If we follow the above analysis further, we discover that the idea of regions of perfect conductivity and insulation is not essential. Each of the small spheres acts like a dipole, the moment of which is induced by the external field. The only thing that is essential to the understanding of dielectrics is that there are many little dipoles induced in the material. Whether the dipoles are induced because there are tiny conducting spheres or for any other reason is irrelevant. Why should a field induce a dipole moment in an atom if the atom is not a conducting sphere? This subject will be discussed in much greater detail in the next chapter, which will be about the inner workings of dielectric materials.  However, we give here one example to illustrate a possible mechanism. An atom has a positive charge on the nucleus, which is surrounded by negative electrons.  In an electric field, the nucleus will be attracted in one direction and the electrons in the other. The orbits or wave patterns of the electrons (or whatever picture is used in quantum mechanics) will be distorted to some extent, as shown in Fig. 10–4; the center of gravity of the negative charge will be displaced and will no longer coincide with the positive charge of the nucleus. We have already discussed such distributions of charge. If we look from a distance, such a neutral configuration is equivalent, to a first approximation, to a little dipole. It seems reasonable that if the field is not too enormous, the amount of induced dipole moment will be proportional to the field. That is, a small field will displace the charges a little bit and a larger field will displace them further—and in proportion to the field—unless the displacement gets too large. For the remainder of this chapter, it will be supposed that the dipole moment is exactly proportional to the field. We will now assume that in each atom there are charges $q$ separated by a distance $\FLPdelta$, so that $q\FLPdelta$ is the dipole moment per atom. (We use $\FLPdelta$ because we are already using $d$ for the plate separation.) If there are $N$ atoms per unit volume, there will be a dipole moment per unit volume equal to $Nq\FLPdelta$. This dipole moment per unit volume will be represented by a vector, $\FLPP$. Needless to say, it is in the direction of the individual dipole moments, i.e., in the direction of the charge separation $\FLPdelta$: \begin{equation} \label{Eq:II:10:4} \FLPP=Nq\FLPdelta. \end{equation}  In general, $\FLPP$ will vary from place to place in the dielectric. However, at any point in the material, $\FLPP$ is proportional to the electric field $\FLPE$. The constant of proportionality, which depends on the ease with which the electrons are displaced, will depend on the kinds of atoms in the material. What actually determines how this constant of proportionality behaves, how accurately it is constant for very large fields, and what is going on inside different materials, we will discuss at a later time. For the present, we will simply suppose that there exists a mechanism by which a dipole moment is induced which is proportional to the electric field.