Page 95 - Engineering Electromagnetics, 8th Edition

CHAPTER 4 Energy and Potential 77

where the path must be specified before the integral can be evaluated. The charge is
assumed to be at rest at both its initial and final positions.
This definite integral is basic to field theory, and we shall devote the following
section to its interpretation and evaluation.

1
2
2
D4.1. Given the electric field E = z 2 (8xyza x + 4x za y − 4x ya z ) V/m, find
the differential amount of work done in moving a 6-nC charge a distance of
6
2 µm, starting at P(2, −2, 3) and proceeding in the direction a L = (a) − a x +
7
6
2
3 a y + a z ;(b) a x − a y − a z ;(c) a x + a y .
6
3
3
2
7 7 7 7 7 7 7
Ans. −149.3 fJ; 149.3 fJ; 0
4.2 THE LINE INTEGRAL
The integral expression for the work done in moving a point charge Q from one
position to another, Eq. (3), is an example of a line integral, which in vector-analysis
notation always takes the form of the integral along some prescribed path of the dot
product of a vector field and a differential vector path length dL.Without using vector
analysis we should have to write
final
W =−Q E L dL
init
where E L = component of E along dL.
A line integral is like many other integrals which appear in advanced analysis,
including the surface integral appearing in Gauss’s law, in that it is essentially de-
scriptive. We like to look at it much more than we like to work it out. It tells us to
choose a path, break it up into a large number of very small segments, multiply the
component of the field along each segment by the length of the segment, and then
add the results for all the segments. This is a summation, of course, and the integral
is obtained exactly only when the number of segments becomes infinite.
This procedure is indicated in Figure 4.1, where a path has been chosen from
1
an initial position B to a final position A and a uniform electric field is selected
for simplicity. The path is divided into six segments, L 1 , L 2 ,... , L 6 , and the
components of E along each segment are denoted by E L1 , E L2 ,... , E L6 . The work
involved in moving a charge Q from B to A is then approximately
W =−Q(E L1 L 1 + E L2 L 2 +· · ·+ E L6 L 6 )

or, using vector notation,
W =−Q(E 1 · L 1 + E 2 · L 2 + ··· + E 6 · L 6 )

1 The final position is given the designation A to correspond with the convention for potential
difference, as discussed in the following section.