3.4.1 Nonempty spaces

The phenomena we have studied in Sections 3.1 to 3.3 were all produced by known charges lying in the vacuum. We were told where the charges were, and from Coulomb's law, Gauss's law or the potential we computed the field they produce. The universe, however, is not empty. It contains bodies of very different physical properties (gases, liquids, glass, plastics, metals, the human body, the Earth itself), and every one of them is made of an enormous number of positive nuclei and negative electrons.

In an ordinary body these charges stay in balance: in every small volume there is as much positive as negative charge, and the body as a whole is electrically neutral. This balance persists as long as the body is left alone. When an external electric field is applied, however, the field pushes the positive charges one way and the negative charges the other. The charges are displaced (by a tiny distance in an insulator, freely in a metal), and the balance is broken locally: inner charges appear inside the body and on its surface.

These inner charges are charges like any other, so they too produce an electric field. The field we observe in and around a material body is therefore the sum of two contributions: the field of the charges we placed (the "external" sources) and the field of the inner charges that the body itself has formed (Figure 3.4.1).

Gauss's law with inner charges

Gauss's law in differential form (Section 3.2) states that the divergence of ${\epsilon }_{0}E$ equals the total charge density at a point, whatever the origin of the charge. Let us split this total into two parts: ${\rho }_v$, the density of the charges that we place and control (often called free charges), and $\bar{\rho }$, the density of the inner charges formed by the medium. Then

The inner charge density $\bar{\rho }$ is not given in advance. It is a function of the electric field itself, $\bar{\rho }=\bar{\rho }(E)$. To make progress we need a model for this function. We define it in the form

where the dimensionless quantity $\chi$ (Greek "chi") is called the electric susceptibility of the medium. It measures how susceptible the material is to the field: $\chi =0$ means no response at all (vacuum), a large $\chi$ a strong response.

Why a divergence? Two properties make this form natural.

• No field, no inner charges: $\bar{\rho }=0$ wherever $E=0$.

• A neutral body stays neutral. Integrate Eq. (3.4.2) over a volume $v$ that contains the whole body and use the divergence theorem (Section 2.4):

  $$\begin{array}{rl}{\int }_v\bar{\rho }dv& =-{\int }_v\nabla \cdot \left({\epsilon }_0\chi E\right)dv\\ & =-{\oint }_S{\epsilon }_0\chi E\cdot dS=0,\end{array}$$

  because the bounding surface $S$ lies outside the body, where there is no matter and $\chi =0$.

The total inner charge is therefore zero automatically: the field only separates charges, it cannot create them. The physical meaning of ${\epsilon }_0\chi E$ will become clear in Section 3.4.3, where we shall recognise it as the density of electric dipoles in the material.

Substituting Eq. (3.4.2) into Eq. (3.4.1) and moving the inner-charge term to the left-hand side,

that is, collecting the two divergences into one,

The inner charges have disappeared from the right-hand side: they are now hidden in the vector field on the left. This vector field deserves a name of its own.

Electric flux density and permittivity

With this definition, Eq. (3.4.3) takes the compact form

This is Gauss's law in a material medium: the sources of $D$ are the free charges only, whereas the sources of ${\epsilon }_{0}E$ are all charges, free and inner. In vacuum $\chi =0$, ${\epsilon }_r=1$, $\epsilon ={\epsilon }_0$ and $D={\epsilon }_{0}E$, so everything we did in Section 3.2 is the special case of an empty medium. The unit of $D$, coulomb per square metre, is that of a surface charge density, a fact we shall use constantly at boundaries.

The susceptibility $\chi$ differs from material to material. It reflects the atomic and molecular structure of the substance and, for our purposes, can only be determined experimentally. Typical values of ${\epsilon }_r$ are collected in Table 3.4.1.