Page 69 - Engineering Electromagnetics, 8th Edition

CHAPTER 3 Electric Flux Density, Gauss’s Law, and Divergence 51

In free space, therefore,

D = 0 E (free space only) (2)

Although (2) is applicable only to a vacuum, it is not restricted solely to the field of
a point charge. For a general volume charge distribution in free space,

ρ ν dv

E = a R (free space only) (3)
vol 4π 0 R 2

where this relationship was developed from the field of a single point charge. In a
similar manner, (1) leads to

ρ ν dv

D = a R (4)
vol 4πR 2
and (2) is therefore true for any free-space charge configuration; we will consider (2)
as defining D in free space.
As a preparation for the study of dielectrics later, it might be well to point out now
that, for a point charge embedded in an infinite ideal dielectric medium, Faraday’s
results show that (1) is still applicable, and thus so is (4). Equation (3) is not applicable,
however, and so the relationship between D and E will be slightly more complicated
than (2).
Because D is directly proportional to E in free space, it does not seem that it
should really be necessary to introduce a new symbol. We do so for a few reasons.
First, D is associated with the flux concept, which is an important new idea. Second,
the D fields we obtain will be a little simpler than the corresponding E fields, because
0 does not appear.

D3.1. Given a 60-µC point charge located at the origin, find the total electric
flux passing through: (a) that portion of the sphere r = 26 cm bounded by
π π
0 <θ < 2 and 0 <φ < 2 ;(b) the closed surface defined by ρ = 26 cm and
z =±26 cm; (c) the plane z = 26 cm.

Ans. 7.5 µC; 60 µC; 30 µC

D3.2. Calculate D in rectangular coordinates at point P(2, −3, 6) produced
by: (a)a point charge Q A = 55 mC at Q(−2, 3, −6); (b)a uniform line
charge ρ LB = 20 mC/m on the x axis; (c)a uniform surface charge density
2
ρ SC = 120 µC/m on the plane z =−5m.
2
2
Ans. 6.38a x − 9.57a y + 19.14a z µC/m ; −212a y + 424a z µC/m ;60a z µC/m 2