# 3.5 Divergence of a Vector Field

From our brief introduction of Coulomb’s law in Chapter 1, we know that an isolated, positive point charge q induces an electric field E in the space around it with the direction of E being outward away from the charge. Also, the strength (magnitude) of E is proportional to q and decreases with distance R from the charge as 1/R 2. In a graphical presentation, a vector field is usually represented by field lines, as shown in Fig. 3-21. The arrowhead denotes the direction of the field at the point where the field line is drawn, and the length of the line provides a qualitative depiction of the field’s magnitude.

# Figure 3-21 Flux lines of the electric field E due to a positive charge q.

Figure 3 21 Full Alternative Text

At a surface boundary, flux density is defined as the amount of outward flux crossing a unit surface d s:

Flux density of E=E⋅d s|d s|=E⋅n ˆ d s d s=E⋅n ˆ, (3.85)

where n ˆ is the normal to d s. The total flux outwardly crossing a closed surface S, such as the enclosed surface of the imaginary sphere outlined in Fig. 3-21, is

Total flux=∮S E⋅d s. (3.86)

Let us now consider the case of a differential rectangular parallelepiped, such as a cube, whose edges align with the Cartesian axes shown in Fig. 3-22. The edges are of lengths Δ x along x, Δ y along y, and Δ z along z. A vector field E(x,y,z) exists in the region of space containing the parallelepiped, and we wish to determine the flux of E through its total surface S. Since S includes six faces, we need to sum up the fluxes through all of them, and by definition, the flux through any face is the outward flux from the volume Δ υ through that face.

Let E be defined as

E=x ˆ E x+y ˆ E y+z ˆ E z. (3.87)

# Figure 3-22 Flux lines of a vector field E passing through a differential rectangular parallelepiped of volume Δ υ=Δ x Δ y Δ z.

Figure 3 22 Full Alternative Text

The area of the face marked 1 in Fig. 3-22 is Δ y Δ z, and its unit vector n ˆ 1=−x ˆ. Hence, the outward flux F 1 through face 1 is

F 1=∫Face 1 E⋅n^1 d s=∫Face 1(x^E x+y^E y+z^E z)⋅(−x^) d y d z≈−E x(1)Δ y Δ z, (3.88)

where E x(1) is the value of E x at the center of face 1. Approximating E x over face 1 by its value at the center is justified by the assumption that the differential volume under consideration is very small.

Similarly, the flux out of face 2 (with n ˆ 2=x ˆ) is

F 2=E x(2)Δ y Δ z, (3.89)

where E x(2) is the value of E x at the center of face 2. Over a differential separation Δ x between the centers of faces 1 and 2, E x(2) is related to E x(1) by

E x(2)=E x(1)+∂E x∂x Δ x, (3.90)

where we have ignored higher-order terms involving (Δ x)2 and higher powers because their contributions are negligibly small when Δ x is very small. Substituting Eq. (3.90) into Eq. (3.89) gives

F 2=E x(1)+∂E x∂x Δ x Δ y Δ z. (3.91)

The sum of the fluxes out of faces 1 and 2 is obtained by adding Eqs. (3.88) and (3.91),

F 1+F 2=∂E x∂x Δ x Δ y Δ z. (3.92a)

Repeating the same procedure to each of the other face pairs leads to

F 3+F 4=∂E y∂y Δ x Δ y Δ z, (3.92b)
F 5+F 6=∂E z∂z Δ x Δ y Δ z. (3.92c)

The sum of fluxes F 1 through F 6 gives the total flux through surface S of the parallelepiped:

∮S E⋅d s=∂E x∂x+∂E y∂y+∂E z∂z Δ x Δ y Δ z=(div E)Δ υ, (3.93)

where Δ υ=Δ x Δ y Δ z and div E is a scalar function called the divergence of E, specified in Cartesian coordinates as

div E=∂E x∂x+∂E y∂y+∂E z∂z. (3.94)

Thus, from Eq. (3.93), we have

div E≜lim Δ υ→0∮s E⋅d s Δ υ, (3.95)

where S encloses the elemental volume Δ υ. Instead of denoting the divergence of E by div E, it is common practice to denote it as ∇⋅E. That is,

for a vector E in Cartesian coordinates.

The divergence is a differential operator, it always operates on vectors, and the result of its operation is a scalar. This is in contrast with the gradient operator, which always operates on scalars and results in a vector. Expressions for the divergence of a vector in cylindrical and spherical coordinates are provided in Appendix C.

The divergence operator is distributive. That is, for any pair of vectors E 1 and E 2,

∇⋅(E 1+E 2)=∇⋅E 1+∇⋅E 2. (3.97)

If ∇⋅E=0, the vector field E is called divergenceless.

The result given by Eq. (3.93) for a differential volume Δ υ can be extended to relate the volume integral of ∇⋅E over any volume υ to the flux of E through the closed surface S that bounds υ. That is,

This relationship, known as the divergence theorem, is used extensively in electromagnetics.

# Example 3-13: Calculating the Divergence

Determine the divergence of each of the following vector fields and then evaluate them at the indicated points:

1. E=x ˆ 3 x 2+y ˆ 2 z+z ˆ x 2 z at (2,−2,0);

2. E=R ˆ(a 3 cos θ/R 2)−θ ˆ(a 3 sin θ/R 2) at (a/2,0,π).

Solution: (a)

∇⋅E=∂E x∂x+∂E y∂y+∂E z∂z=∂∂x(3 x 2)+∂∂y(2 z)+∂∂z(x 2 z)=6 x+0+x 2=x 2+6 x.

At (2,−2,0), ∇⋅E|(2,−2,0)=16.

(b) From the expression given in Appendix C for the divergence of a vector in spherical coordinates, it follows that

∇⋅E=1 R 2∂∂R(R 2 E R)+1 R sin θ∂∂θ(E θ sin θ) +1 R sin θ∂E ϕ∂ϕ=1 R 2∂∂R(a 3 cos θ)+1 R sin θ∂∂θ−a 3 sin 2 θ R 2=0−2 a 3 cos θ R 3=−2 a 3 cos θ R 3.

At R=a/2 and θ=0, ∇⋅E|(a/2,0,π)=−16.