Page 54 - Engineering Electromagnetics, 8th Edition

36 ENGINEERING ELECTROMAGNETICS

Figure 2.6 The contribution dE = dE ρ a ρ +
dE z a z to the electric field intensity produced by an
element of charge dQ = ρ L dz located a distance

z from the origin. The linear charge density is

uniform and extends along the entire z axis.
Symmetry should always be considered first in order to determine two specific
factors: (1) with which coordinates the field does not vary, and (2) which compo-
nents of the field are not present. The answers to these questions then tell us which
components are present and with which coordinates they do vary.
Referring to Figure 2.6, we realize that as we move around the line charge,
varying φ while keeping ρ and z constant, the line charge appears the same from
every angle. In other words, azimuthal symmetry is present, and no field component
may vary with φ.
Again, if we maintain ρ and φ constant while moving up and down the line charge
by changing z, the line charge still recedes into infinite distance in both directions
and the problem is unchanged. This is axial symmetry and leads to fields that are not
functions of z.
If we maintain φ and z constant and vary ρ, the problem changes, and Coulomb’s
law leads us to expect the field to become weaker as ρ increases. Hence, by a process
of elimination we are led to the fact that the field varies only with ρ.
Now, which components are present? Each incremental length of line charge
acts as a point charge and produces an incremental contribution to the electric field
intensity which is directed away from the bit of charge (assuming a positive line
charge). No element of charge produces a φ component of electric intensity; E φ is
zero. However, each element does produce an E ρ and an E z component, but the
contribution to E z by elements of charge that are equal distances above and below
the point at which we are determining the field will cancel.