Page 104 - Engineering Electromagnetics, 8th Edition

86 ENGINEERING ELECTROMAGNETICS

4.5 THE POTENTIAL FIELD OF A SYSTEM OF
CHARGES: CONSERVATIVE PROPERTY
The potential at a point has been defined as the work done in bringing a unit positive
charge from the zero reference to the point, and we have suspected that this work, and
hence the potential, is independent of the path taken. If it were not, potential would
not be a very useful concept.
Let us now prove our assertion. We do so by beginning with the potential field
of the single point charge for which we showed, in Section 4.4, the independence
with regard to the path, noting that the field is linear with respect to charge so that
superposition is applicable. It will then follow that the potential of a system of charges
has a value at any point which is independent of the path taken in carrying the test
charge to that point.
Thus the potential field of a single point charge, which we shall identify as Q 1
and locate at r 1 ,involves only the distance |r − r 1 | from Q 1 to the point at r where
we are establishing the value of the potential. For a zero reference at infinity, we have

Q 1
V (r) =
4π 0 |r − r 1 |
The potential arising from two charges, Q 1 at r 1 and Q 2 at r 2 ,isa function only of
|r − r 1 | and |r − r 2 |, the distances from Q 1 and Q 2 to the field point, respectively.

Q 1 Q 2
V (r) = +
4π 0 |r − r 1 | 4π 0 |r − r 2 |
Continuing to add charges, we find that the potential arising from n point charges is

n

Q m
V (r) = (16)
m=1 4π 0 |r − r m |
If each point charge is now represented as a small element of a continuous volume
charge distribution ρ ν ν, then

ρ ν (r 1 ) ν 1 ρ ν (r 2 ) ν 2 ρ ν (r n ) ν n
V (r) = + +· · ·+
4π 0 |r − r 1 | 4π 0 |r − r 2 | 4π 0 |r − r n |
As we allow the number of elements to become infinite, we obtain the integral
expression

ρ ν (r ) dv

V (r) = (17)

vol 4π 0 |r − r |
We have come quite a distance from the potential field of the single point charge,
and it might be helpful to examine Eq. (17) and refresh ourselves as to the meaning of
each term. The potential V (r)is determined with respect to a zero reference potential
at infinity and is an exact measure of the work done in bringing a unit charge from