5.3 Charge Distributions

In principle, the smallest unit of electric charge that can be isolated is the charge of a single electron, which is \(\cong -1.60 \times 10^{-19}\) C. This is very small, and we rarely deal with electrons one at a time, so it is usually more convenient to describe charge as a quantity that is continuous over some region of space. In particular, it is convenient to describe charge as being distributed in one of three ways: along a curve, over a surface, or within a volume.

Line Charge Distribution. Imagine that charge is distributed along a curve \({\mathcal C}\) through space. Let \(\Delta q\) be the total charge along a short segment of the curve, and let \(\Delta l\) be the length of this segment. The line charge density \(\rho_l\) at any point along the curve is defined as

\begin{equation} \rho_l \triangleq \lim_{\Delta l \to 0} \frac{\Delta q}{\Delta l} = \frac{dq}{dl} \tag{5.12}\end{equation}

which has units of C/m. We may then define \(\rho_l\) to be a function of position along the curve, parameterized by \(l\); e.g., \(\rho_l(l)\). Then, the total charge \(Q\) along the curve is

\begin{equation} Q = \int_{\mathcal C} \rho_l(l)~dl \tag{5.13}\end{equation}

which has units of C. In other words, line charge density integrated over length yields total charge.

Surface Charge Distribution. Imagine that charge is distributed over a surface. Let \(\Delta q\) be the total charge on a small patch on this surface, and let \(\Delta s\) be the area of this patch. The surface charge density \(\rho_s\) at any point on the surface is defined as

\begin{equation} \rho_s \triangleq \lim_{\Delta s \to 0} \frac{\Delta q}{\Delta s} = \frac{dq}{ds} \tag{5.14}\end{equation}

which has units of C/m\(^2\). Let us define \(\rho_s\) to be a function of position on this surface. Then the total charge over a surface \({\mathcal S}\) is

\begin{equation} Q = \int_{\mathcal S} \rho_s~ds \tag{5.15}\end{equation}

In other words, surface charge density integrated over a surface yields total charge.

Volume Charge Distribution. Imagine that charge is distributed over a volume. Let \(\Delta q\) be the total charge in a small cell within this volume, and let \(\Delta v\) be the volume of this cell. The volume charge density \(\rho_v\) at any point in the volume is defined as

\begin{equation} \rho_v \triangleq \lim_{\Delta v \to 0} \frac{\Delta q}{\Delta v} = \frac{dq}{dv} \tag{5.16}\end{equation}

which has units of C/m\(^3\). Since \(\rho_v\) is a function of position within this volume, the total charge within a volume \({\mathcal V}\) is

\begin{equation} Q = \int_{\mathcal V} \rho_v~dv \tag{5.17}\end{equation}

In other words, volume charge density integrated over a volume yields total charge.