Page 71 - Engineering Electromagnetics, 8th Edition

CHAPTER 3 Electric Flux Density, Gauss’s Law, and Divergence 53

Figure 3.2 The electric flux density D S at P arising
from charge Q. The total flux passing through S is
D S · S.

At any point P, consider an incremental element of surface S and let D S make
an angle θ with S,as shown in Figure 3.2. The flux crossing S is then the product
of the normal component of D S and S,
= flux crossing S = D S,norm S = D S cos θ S = D S · S
where we are able to apply the definition of the dot product developed in Chapter 1.
The total flux passing through the closed surface is obtained by adding the dif-
ferential contributions crossing each surface element S,

= d = D S · dS
closed
surface
The resultant integral is a closed surface integral, and since the surface element
dS always involves the differentials of two coordinates, such as dx dy, ρ dφ dρ,
2
or r sin θ dθ dφ, the integral is a double integral. Usually only one integral sign is
used for brevity, and we will always place an S below the integral sign to indicate
a surface integral, although this is not actually necessary, as the differential dS is
automatically the signal for a surface integral. One last convention is to place a small
circle on the integral sign itself to indicate that the integration is to be performed over
a closed surface. Such a surface is often called a gaussian surface.We then have the
mathematical formulation of Gauss’s law,

= D S · dS = charge enclosed = Q (5)
S

The charge enclosed might be several point charges, in which case
Q = Qn
or a line charge,

Q = ρ L dL