Page 53 - Engineering Electromagnetics, 8th Edition

CHAPTER 2 Coulomb’s Law and Electric Field Intensity 35

Finally,
0.01
e e
−2000ρ −4000ρ
Q =−10 −10 π −
−2000 −4000 0
1 1 −π
Q =−10 −10 π − = = 0.0785 pC
2000 4000 40
where pC indicates picocoulombs.
The incremental contribution to the electric field intensity at r produced by an
incremental charge Q at r is

Q r − r ρ ν ν r − r
E(r) = 2 = 2

4π 0 |r − r | |r − r | 4π 0 |r − r | |r − r |
If we sum the contributions of all the volume charge in a given region and let the
volume element ν approach zero as the number of these elements becomes infinite,
the summation becomes an integral,
r − r
ρ ν (r ) dν
E(r) = 2 (15)

vol 4π 0 |r − r | |r − r |
This is again a triple integral, and (except in Drill Problem 2.4) we shall do our best
to avoid actually performing the integration.
The significance of the various quantities under the integral sign of (15) might
stand a little review. The vector r from the origin locates the field point where E is
being determined, whereas the vector r extends from the origin to the source point

where ρ v (r )dν is located. The scalar distance between the source point and the

field point is |r − r |, and the fraction (r − r )/|r − r | is a unit vector directed from

source point to field point. The variables of integration are x , y , and z in rectangular

coordinates.
D2.4. Calculatethetotalchargewithineachoftheindicatedvolumes:(a)0.1 ≤
1
|x|, |y|, |z|≤ 0.2: ρ ν = ;(b)0 ≤ ρ ≤ 0.1, 0 ≤ φ ≤ π,2 ≤ z ≤ 4; ρ ν =
3 3 3
x y z
2 2
2
ρ z sin 0.6φ;(c) universe: ρ ν = e −2r /r .
Ans. 0; 1.018 mC; 6.28 C
2.4 FIELD OF A LINE CHARGE
Up to this point we have considered two types of charge distribution, the point charge
3
andchargedistributedthroughoutavolumewithadensityρ ν C/m .Ifwenowconsider
a filamentlike distribution of volume charge density, such as a charged conductor of
very small radius, we find it convenient to treat the charge as a line charge of density
ρ L C/m.
We assume a straight-line charge extending along the z axis in a cylindrical
coordinate system from −∞ to ∞,as shown in Figure 2.6. We desire the electric
field intensity E at any and every point resulting from a uniform line charge density ρ L .