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