Ordinary objects—tables, chairs, beakers of water—seem to our senses to be continuous distributions of matter. There is no obvious evidence for atoms, even though we have good reasons to believe that we would find atoms if we subdivided the matter sufficiently far. Thus it is easier, for many practical purposes, to consider matter to be continuous and to talk about the density of matter. Density—the number of kilograms of matter per cubic meter—allows us to describe the distribution of matter as if the matter were continuous rather than atomic.

Much the same situation occurs with charge. If a charged object contains a large number of excess electrons—for example, ${10}^{12}$ extra electrons on a metal rod—it is not practical to track every electron. It makes more sense to consider the charge to be continuous and to describe how it is distributed over the object.

FIGURE 23.9a shows an object of length L, such as a plastic rod or a metal wire, with charge Q spread uniformly along it. (We will use an uppercase Q for the total charge of an object, reserving lowercase q for individual point charges.) The linear charge density $\text{λ}$ is defined to be

Linear charge density, which has units of $\text{C}\text{/}\text{m}$, is the amount of charge per meter of length. The linear charge density of a 20-cm-long wire with 40 nC of charge is $2.0\text{nC}\text{/}\text{c}\text{m}$ or $2.0\times {10}^{-7}\text{C}\text{/}\text{m}$.

FIGURE 23.9 One-dimensional and two-dimensional continuous charge distributions.

Full Alternative Text

We’ll also be interested in charged surfaces. FIGURE 23.9b shows a two-dimensional distribution of charge across a surface of area A. We define the surface charge density $\eta$ (lowercase Greek eta) to be

Surface charge density, with units of $\text{C}\text{/}{\text{m}}^2$, is the amount of charge per square meter. A $1.0\text{mm}\times 1.0\text{mm}$ square on a surface with $\eta =2.0\times {10}^{-4}\text{C}\text{/}{\text{m}}^2$ contains $2.0\times {10}^{-10}\text{C}$ or 0.20 nC of charge. (The volume charge density $\rho =Q\text{/}V$, measured in ${\text{C/m}}^3$, will be used in Chapter 24.)

Figure 23.9 and the definitions of Equations 23.12 and 23.13 assume that the object is uniformly charged, meaning that the charges are evenly spread over the object. We will assume objects are uniformly charged unless noted otherwise.