Page 109 - Engineering Electromagnetics, 8th Edition

CHAPTER 4 Energy and Potential 91

the component of E in the direction of a L (one interpretation of the dot product) to
obtain the small potential difference between the final and initial points of L.
If we designate the angle between L and E as θ, then

V ˙=−E L cos θ
We now pass to the limit and consider the derivative dV/dL.Todo this, we need
to show that V may be interpreted as a function V (x, y, z). So far, V is merely the
result of the line integral (21). If we assume a specified starting point or zero reference
and then let our end point be (x, y, z), we know that the result of the integration is a
unique function of the end point (x, y, z) because E is a conservative field. Therefore
V is a single-valued function V (x, y, z). We may then pass to the limit and obtain
dV
=−E cos θ
dL
In which direction should L be placed to obtain a maximum value of V ?
Remember that E is a definite value at the point at which we are working and is
independent of the direction of L. The magnitude L is also constant, and our
variable is a L , the unit vector showing the direction of L.Itisobvious that the
maximum positive increment of potential, V max , will occur when cos θ is −1, or
L points in the direction opposite to E.For this condition,

dV

= E
dL
max
This little exercise shows us two characteristics of the relationship between E
and V at any point:
1. The magnitude of the electric field intensity is given by the maximum value of
the rate of change of potential with distance.
2. This maximum value is obtained when the direction of the distance increment is
opposite to E or, in other words, the direction of E is opposite to the direction in
which the potential is increasing the most rapidly.

We now illustrate these relationships in terms of potential. Figure 4.6 is intended
to show the information we have been given about some potential field. It does this by
showing the equipotential surfaces (shown as lines in the two-dimensional sketch).
We desire information about the electric field intensity at point P. Starting at P,we lay
off a small incremental distance L in various directions, hunting for that direction
in which the potential is changing (increasing) the most rapidly. From the sketch, this
direction appears to be left and slightly upward. From our second characteristic above,
the electric field intensity is therefore oppositely directed, or to the right and slightly
downward at P. Its magnitude is given by dividing the small increase in potential by
the small element of length.
It seems likely that the direction in which the potential is increasing the most
rapidly is perpendicular to the equipotentials (in the direction of increasing potential),
and this is correct, for if L is directed along an equipotential, V = 0by our