Page 99 - Engineering Electromagnetics, 8th Edition

CHAPTER 4 Energy and Potential 81

Figure 4.2 (a)A circular path and (b)a radial path along which a charge of Q is carried
in the field of an infinite line charge. No work is expected in the former case.

As a final example illustrating the evaluation of the line integral, we investigate
several paths that we might take near an infinite line charge. The field has been
obtained several times and is entirely in the radial direction,
ρ L
E = E ρ a ρ = a ρ
2π 0 ρ
First we find the work done in carrying the positive charge Q about a circular
path of radius ρ b centered at the line charge, as illustrated in Figure 4.2a.Without
lifting a pencil, we see that the work must be nil, for the path is always perpendicular
to the electric field intensity, or the force on the charge is always exerted at right
angles to the direction in which we are moving it. For practice, however, we will set
up the integral and obtain the answer.
The differential element dL is chosen in cylindrical coordinates, and the circular
path selected demands that dρ and dz be zero, so dL = ρ 1 dφ a φ . The work is then
final ρ L
W =−Q a ρ · ρ 1 dφ a φ
init 2π 0 ρ 1
2π ρ L
=−Q dφ a ρ · a φ = 0
0 2π 0
We will now carry the charge from ρ = a to ρ = b along a radial path
(Figure 4.2b). Here dL = dρ a ρ and
final ρ L b ρ L d ρ
W =−Q a ρ · dρ a ρ =−Q
init 2π 0 ρ a 2π 0 ρ
or
Qρ L b
W =− ln
2π 0 a
Because b is larger than a,ln (b/a)is positive, and the work done is negative,
indicating that the external source that is moving the charge receives energy.