Page 103 - Engineering Electromagnetics, 8th Edition

CHAPTER 4 Energy and Potential 85

or, as there is no reason to identify this point with the A subscript,

Q
V = (14)
4π 0 r

This expression defines the potential at any point distant r from a point charge Q
at the origin, the potential at infinite radius being taken as the zero reference. Returning
to a physical interpretation, we may say that Q/4π 0 r joules of work must be done
in carrying a unit charge from infinity to any point r meters from the charge Q.
A convenient method to express the potential without selecting a specific zero
reference entails identifying r A as r once again and letting Q/4π 0 r B be a constant.
Then

Q
V = + C 1 (15)
4π 0 r
and C 1 may be selected so that V = 0atany desired value of r.We could also select
the zero reference indirectly by electing to let V be V 0 at r = r 0 .
It should be noted that the potential difference between two points is not a func-
tion of C 1 .
Equations (14) and (15) represent the potential field of a point charge. The po-
tential is a scalar field and does not involve any unit vectors.
We now define an equipotential surface as a surface composed of all those points
having the same value of potential. All field lines would be perpendicular to such a
surface at the points where they intersect it. Therefore, no work is involved in moving
a unit charge around on an equipotential surface. The equipotential surfaces in the
potential field of a point charge are spheres centered at the point charge.
An inspection of the form of the potential field of a point charge shows that it
is an inverse-distance field, whereas the electric field intensity was found to be an
inverse-square-law function. A similar result occurs for the gravitational force field
of a point mass (inverse-square law) and the gravitational potential field (inverse
distance). The gravitational force exerted by the earth on an object one million miles
from it is four times that exerted on the same object two million miles away. The
kinetic energy given to a freely falling object starting from the end of the universe
with zero velocity, however, is only twice as much at one million miles as it is at two
million miles.

D4.5. A 15-nC point charge is at the origin in free space. Calculate V 1 if point
P 1 is located at P 1 (−2, 3, −1) and (a) V = 0at(6, 5, 4); (b) V = 0at infinity;
(c) V = 5Vat(2, 0, 4).
Ans. 20.67 V; 36.0 V; 10.89 V