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 $$\rho_l \triangleq \lim_{\Delta l \to 0} \frac{\Delta q}{\Delta l} = \frac{dq}{dl} \nonumber$$ 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 $$Q = \int_{\mathcal C} \rho_l(l)~dl \nonumber$$ 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 $$\rho_s \triangleq \lim_{\Delta s \to 0} \frac{\Delta q}{\Delta s} = \frac{dq}{ds} \nonumber$$ 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 $$Q = \int_{\mathcal S} \rho_s~ds \nonumber$$ 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 $$\rho_v \triangleq \lim_{\Delta v \to 0} \frac{\Delta q}{\Delta v} = \frac{dq}{dv} \nonumber$$ 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 $$Q = \int_{\mathcal V} \rho_v~dv \nonumber$$ In other words, volume charge density integrated over a volume yields total charge.