# 4.2 Charge and Current Distributions

In electromagnetics, we encounter various forms of electric charge distributions. When put in motion, these charge distributions constitute current distributions. Charges and currents may be distributed over a volume of space, across a surface, or along a line.

# 4.2.1 Charge Densities

At the atomic scale, the charge distribution in a material is discrete, meaning that charge exists only where electrons and nuclei are and nowhere else. In electromagnetics, we usually are interested in studying phenomena at a much larger scale, typically three or more orders of magnitude greater than the spacing between adjacent atoms. At such a macroscopic scale, we can disregard the discontinuous nature of the charge distribution and treat the net charge contained in an elemental volume Δ υ as if it were uniformly distributed within. Accordingly, we define the volume charge density ρ v as

ρ v=lim Δ υ→0 Δ q Δ υ=d q d υ(C/m 3),(4.4)

where Δ q is the charge contained in Δ υ. In general, ρ v depends on spatial location (x, y, z) and t; thus, ρ v=ρ v(x, y, z, t). Physically, ρ v represents the average charge per unit volume for a volume Δ υ centered at (x, y, z) with Δ υ being large enough to contain a large number of atoms, yet it is small enough to be regarded as a point at the macroscopic scale under consideration. The variation of ρ v with spatial location is called its spatial distribution (or simply its distribution). The total charge contained in volume υ is

Q=∫υ ρ v d υ(C).(4.5)

In some cases, particularly when dealing with conductors, electric charge may be distributed across the surface of a material, where the quantity of interest is the surface charge density ρ s, which is defined as

ρ s=lim Δ s→0 Δ q Δ s=d q d s(C/m 2),(4.6)

where Δ q is the charge present across an elemental surface area Δ s. Similarly, if the charge is, for all practical purposes, confined to a line, which need not be straight, we characterize its distribution in terms of the line charge density ρ ℓ, defined as

ρ ℓ=lim Δ l→0 Δ q Δ l=d q d l(C/m).(4.7)

# Example 4-1: Line Charge Distribution

Calculate the total charge Q contained in a cylindrical tube oriented along the z axis, as shown in Fig. 4-1(a). The line charge density is ρ ℓ=2 z, where z is the distance in meters from the bottom end of the tube. The tube length is 10 cm.

# Figure 4-1 Charge distributions for Examples 4-1 and 4-2.

(a) Line charge distribution

(b) Surface charge distribution

Figure 4 1 Full Alternative Text

Solution: The total charge Q is

Q=∫0 0.1 ρ ℓ d z=∫0 0.1 2 z d z=z 2 0 0.1=10−2 C.

# Example 4-2: Surface Charge Distribution

The circular disk of electric charge shown in Fig. 4-1(b) is characterized by an azimuthally symmetric surface charge density that increases linearly with r from zero at the center to 6 C/m 2 at r=3 cm. Find the total charge present on the disk surface.

Solution: Since ρ s is symmetrical with respect to the azimuth angle ϕ, it depends only on r and is given by

ρ s=6 r 3×10−2=2×10 2 r(C/m 2),

where r is in meters. In polar coordinates, an elemental area is d s=r d r d ϕ, and for the disk shown in Fig. 4-1(b), the limits of integration are from 0 to 2 π (rad) for ϕ and from 0 to 3×10−2 m for r. Hence,

Q=∫S ρ s d s=∫ϕ=0 2 π∫r=0 3×10−2(2×10 2 r)r d r d ϕ=2 π×2×10 2 r 3 3 0 3×10−2=11.31(mC).

# 4.2.2 Current Density

Consider a tube with volume charge density ρ v (Fig. 4-2(a)). The charges in the tube move with velocity u along the tube axis. Over a period Δ t, the charges move a distance Δ l=u Δ t. The amount of charge that crosses the tube’s cross-sectional surface Δ s′ in time Δ t is therefore

Δ q′=ρ v Δ υ=ρ v Δ l Δ s′=ρ v u Δ s′Δ t. (4.8)

# Figure 4-2 Charges with velocity u moving through a cross section Δ s′ in (a) and Δ s in (b).

Figure 4 2-a Full Alternative Text
Figure 4 2-b Full Alternative Text

Now consider the more general case where the charges are flowing through a surface Δ s with normal n ˆ not necessarily parallel to u (Fig. 4-2(b)). In this case, the amount of charge Δ q flowing through Δ s is

Δ q=ρ v u⋅Δ s Δ t,(4.9)

where Δ s=n ˆ Δ s and the corresponding total current flowing in the tube is

Δ I=Δ q Δ t=ρ v u⋅Δ s=J⋅Δ s.(4.10)

Here

is defined as the current density in ampere per square meter. Generalizing to an arbitrary surface S, the total current flowing through it is

A wind-driven charged cloud, for example, gives rise to a convection current. In some cases, the charged matter constituting the convection current consists solely of charged particles, such as the electron beam of a scanning electron microscope or the ion beam of a plasma propulsion system.

When a current is due to the movement of charged particles relative to their host material, J is called a conduction current density. In a metal wire, for example, there are equal amounts of positive charges (in atomic nuclei) and negative charges (in the electron shells of the atoms). None of the positive charges and few of the negative charges can move; only those electrons in the outermost electron shells of the atoms can be pushed from one atom to the next if a voltage is applied across the ends of the wire.

Conduction current, which is discussed in more detail in Section 4-6, obeys Ohm’s law, whereas convection current does not.

Concept Question 4-1: What happens to Maxwell’s equations under static conditions?

Concept Question 4-2: How is the current density J related to the volume charge density ρ v?

Concept Question 4-3: What is the difference between convection and conduction currents?