Page 88 - Engineering Electromagnetics, 8th Edition

70 ENGINEERING ELECTROMAGNETICS

3 2 3 2
=− (D x ) x=0 dy dz + (D x ) x=1 dy dz
0 0 0 0
3 1 3 1
− (D y ) y=0 dx dz + (D y ) y=2 dx dz
0 0 0 0
However, (D x ) x=0 = 0, and (D y ) y=0 = (D y ) y=2 , which leaves only

3 2 3 2
D · dS = (D x ) x=1 dy dz = 2ydydz
S 0 0 0 0
3
4 dz = 12
=
0
Since
∂ ∂ 2
∇ · D = (2xy) + (x ) = 2y
∂x ∂y
the volume integral becomes

3 2 1 3 2
2ydydz
∇ · D dv = 2ydx dydz =
vol 0 0 0 0 0
3
= 4 dz = 12
0
and the check is accomplished. Remembering Gauss’s law, we see that we have also
determined that a total charge of 12 C lies within this parallelepiped.

1
1
2
D3.9. Given the field D = 6ρ sin φ a ρ +1.5ρ cos φ a φ C/m , evaluate both
2
2
sides of the divergence theorem for the region bounded by ρ = 2, φ = 0,
φ = π, z = 0, and z = 5.
Ans. 225; 225
REFERENCES

1. Kraus, J. D., and D. A. Fleisch. Electromagnetics. 5th ed. New York: McGraw-Hill,
1999. The static electric field in free space is introduced in Chapter 2.
2. Plonsey, R., and R. E. Collin. Principles and Applications of Electromagnetic Fields.
New York: McGraw-Hill, 1961. The level of this text is somewhat higher than the one we
are reading now, but it is an excellent text to read next. Gauss’s law appears in the second
chapter.
3. Plonus, M. A. Applied Electromagnetics.New York: McGraw-Hill, 1978. This book
contains rather detailed descriptions of many practical devices that illustrate
electromagnetic applications. For example, see the discussion of xerography on
pp. 95–98 as an electrostatics application.
4. Skilling, H. H. Fundamentals of Electric Waves.2d ed. New York: John Wiley & Sons,
1948. The operations of vector calculus are well illustrated. Divergence is discussed on
pp. 22 and 38. Chapter 1 is interesting reading.