Page 106 - Engineering Electromagnetics, 8th Edition

88 ENGINEERING ELECTROMAGNETICS

Figure 4.3 The potential field of a ring of uniform line

charge density is easily obtained from V = ρ L (r ) dL /

(4π 0 |r − r |).
In other words, the expression for potential (zero reference at infinity),
A
E · dL
V A =−
∞
or potential difference,
A
E · dL
V AB = V A − V B =−
B
is not dependent on the path chosen for the line integral, regardless of the source of
the E field.
This result is often stated concisely by recognizing that no work is done in
carrying the unit charge around any closed path,or

E · dL = 0 (20)

A small circle is placed on the integral sign to indicate the closed nature of the
path. This symbol also appeared in the formulation of Gauss’s law, where a closed
surface integral was used.
Equation (20) is true for static fields, but we will see in Chapter 9 that Faraday
demonstrated it was incomplete when time-varying magnetic fields were present. One
of Maxwell’s greatest contributions to electromagnetic theory was in showing that a
time-varying electric field produces a magnetic field, and therefore we should expect
to find later that Eq. (20) is not correct when either E or the magnetic field varies
with time.
Restricting our attention to the static case where E does not change with time,
consider the dc circuit shown in Figure 4.4. Two points, A and B, are marked, and