Page 101 - Engineering Electromagnetics, 8th Edition

CHAPTER 4 Energy and Potential 83

We have to agree on the direction of movement, and we do this by stating that
V AB signifies the potential difference between points A and B and is the work done in
moving the unit charge from B (last named) to A (first named). Thus, in determining
V AB , B is the initial point and A is the final point. The reason for this somewhat
peculiar definition will become clearer shortly, when it is seen that the initial point B
is often taken at infinity, whereas the final point A represents the fixed position of the
charge; point A is thus inherently more significant.
Potential difference is measured in joules per coulomb, for which the volt is
defined as a more common unit, abbreviated as V. Hence the potential difference
between points A and B is

A
V AB =− E · dL V (10)
B

and V AB is positive if work is done in carrying the positive charge from B to A.
From the line-charge example of Section 4.2 we found that the work done in
taking a charge Q from ρ = b to ρ = a was
Qρ L b
W = ln
2π 0 a
Thus, the potential difference between points at ρ = a and ρ = b is
W ρ L b
V ab = = ln (11)
Q 2π 0 a
We can try out this definition by finding the potential difference between points
A and B at radial distances r A and r B from a point charge Q. Choosing an origin at Q,
Q
E = E r a r = a r
4π 0 r 2
and

dL = dr a r
we have
A r A Q Q 1 1
V AB =− E · dL =− 2 dr = − (12)
B r B 4π 0 r 4π 0 r A r B
If r B > r A , the potential difference V AB is positive, indicating that energy is
expended by the external source in bringing the positive charge from r B to r A . This
agrees with the physical picture showing the two like charges repelling each other.
It is often convenient to speak of the potential,or absolute potential,ofa point,
rather than the potential difference between two points, but this means only that we
agree to measure every potential difference with respect to a specified reference point
that we consider to have zero potential. Common agreement must be reached on
the zero reference before a statement of the potential has any significance. A person
having one hand on the deflection plates of a cathode-ray tube that are “at a potential
of 50 V” and the other hand on the cathode terminal would probably be too shaken up