Page 51 - Engineering Electromagnetics, 8th Edition

CHAPTER 2 Coulomb’s Law and Electric Field Intensity 33

5 m 4 m
1 + (−1) (0.1) + 1
D2.3. Evaluate the sums: (a) 2 ;(b) 2 1.5
m=0 m + 1 m=1 (4 + m )
Ans. 2.52; 0.176

2.3 FIELD ARISING FROM A CONTINUOUS
VOLUME CHARGE DISTRIBUTION

If we now visualize a region of space filled with a tremendous number of charges
separated by minute distances, we see that we can replace this distribution of very
small particles with a smooth continuous distribution described by a volume charge
3
density, just as we describe water as having a density of 1 g/cm (gram per cubic
centimeter) even though it consists of atomic- and molecular-sized particles. We can
do this only if we are uninterested in the small irregularities (or ripples) in the field
as we move from electron to electron or if we care little that the mass of the water
actually increases in small but finite steps as each new molecule is added.
This is really no limitation at all, because the end results for electrical engineers
are almost always in terms of a current in a receiving antenna, a voltage in an elec-
tronic circuit, or a charge on a capacitor, or in general in terms of some large-scale
macroscopic phenomenon. It is very seldom that we must know a current electron by
electron. 4
We denote volume charge density by ρ ν ,having the units of coulombs per cubic
3
meter (C/m ).
The small amount of charge Q in a small volume ν is

Q = ρ ν ν (12)
and we may define ρ ν mathematically by using a limiting process on (12),

Q
ρ ν = lim (13)
ν→0 ν
The total charge within some finite volume is obtained by integrating throughout that
volume,

Q = ρ ν dν (14)
vol
Only one integral sign is customarily indicated, but the differential dν signifies inte-
gration throughout a volume, and hence a triple integration.

4 A study of the noise generated by electrons in semiconductors and resistors, however, requires just
such an examination of the charge through statistical analysis.