Page 82 - Engineering Electromagnetics, 8th Edition

64 ENGINEERING ELECTROMAGNETICS

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3
4
2 4
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D3.6. In free space, let D = 8xyz a x +4x z a y +16x yz a z pC/m .(a) Find
the total electric flux passing through the rectangular surface z = 2, 0 <
x < 2, 1 < y < 3, in the a z direction. (b) Find E at P(2, −1, 3). (c) Find
an approximate value for the total charge contained in an incremental sphere
3
located at P(2, −1, 3) and having a volume of 10 −12 m .
Ans. 1365 pC; −146.4a x + 146.4a y − 195.2a z V/m; −2.38 × 10 −21 C
3.5 DIVERGENCE AND MAXWELL’S
FIRST EQUATION

We will now obtain an exact relationship from (7), by allowing the volume element
ν to shrink to zero. We write this equation as

D · dS Q
∂D x ∂D y ∂D z S
+ + = lim = lim = ρ ν (9)
∂x ∂y ∂z ν→0 ν ν→0 ν
in which the charge density, ρ ν ,is identified in the second equality.
The methods of the previous section could have been used on any vector A to
find S A · dS for a small closed surface, leading to

A · dS
∂A x ∂A y ∂A z = lim S (10)
∂x + ∂y + ∂z ν→0 ν
where A could represent velocity, temperature gradient, force, or any other vector
field.
This operation appeared so many times in physical investigations in the last cen-
tury that it received a descriptive name, divergence. The divergence of A is defined as

A · dS

Divergence of A = div A = lim S (11)
ν→0 ν
and is usually abbreviated div A. The physical interpretation of the divergence of a
vector is obtained by describing carefully the operations implied by the right-hand
side of (11), where we shall consider A to be a member of the flux-density family of
vectors in order to aid the physical interpretation.

The divergence of the vector flux density A is the outflow of flux from a small closed surface
per unit volume as the volume shrinks to zero.

The physical interpretation of divergence afforded by this statement is often
useful in obtaining qualitative information about the divergence of a vector field
without resorting to a mathematical investigation. For instance, let us consider the
divergence of the velocity of water in a bathtub after the drain has been opened. The
net outflow of water through any closed surface lying entirely within the water must
be zero, for water is essentially incompressible, and the water entering and leaving