Page 108 - Engineering Electromagnetics, 8th Edition

90 ENGINEERING ELECTROMAGNETICS

4.6 POTENTIAL GRADIENT
We now have two methods of determining potential, one directly from the electric field
intensity by means of a line integral, and another from the basic charge distribution
itself by a volume integral. Neither method is very helpful in determining the fields
in most practical problems, however, for as we will see later, neither the electric field
intensity nor the charge distribution is very often known. Preliminary information is
much more apt to consist of a description of two equipotential surfaces, such as the
statement that we have two parallel conductors of circular cross section at potentials
of 100 and −100 V. Perhaps we wish to find the capacitance between the conductors,
or the charge and current distribution on the conductors from which losses may be
calculated.
These quantities may be easily obtained from the potential field, and our im-
mediate goal will be a simple method of finding the electric field intensity from the
potential.
We already have the general line-integral relationship between these quantities,

E · dL (21)
V =−
but this is much easier to use in the reverse direction: given E, find V.
However, Eq. (21) may be applied to a very short element of length L along
which E is essentially constant, leading to an incremental potential difference V,

V ˙=−E · L (22)
Now consider a general region of space, as shown in Figure 4.5, in which E and
V both change as we move from point to point. Equation (22) tells us to choose an
incremental vector element of length L = L a L and multiply its magnitude by

Figure 4.5 A vector incremental element of
length L is shown making an angle of θ with an
E field, indicated by its streamlines. The sources
of the field are not shown.