Page 102 - Engineering Electromagnetics, 8th Edition

84 ENGINEERING ELECTROMAGNETICS

to understand that the cathode is not the zero reference, but that all potentials in that
circuit are customarily measured with respect to the metallic shield about the tube.
The cathode may be several thousands of volts negative with respect to the shield.
Perhaps the most universal zero reference point in experimental or physical po-
tential measurements is “ground,” by which we mean the potential of the surface
region of the earth itself. Theoretically, we usually represent this surface by an infinite
plane at zero potential, although some large-scale problems, such as those involving
propagation across the Atlantic Ocean, require a spherical surface at zero potential.
Another widely used reference “point” is infinity. This usually appears in theo-
retical problems approximating a physical situation in which the earth is relatively far
removed from the region in which we are interested, such as the static field near the
wing tip of an airplane that has acquired a charge in flying through a thunderhead, or
the field inside an atom. Working with the gravitational potential field on earth, the
zero reference is normally taken at sea level; for an interplanetary mission, however,
the zero reference is more conveniently selected at infinity.
Acylindrical surface of some definite radius may occasionally be used as a zero
reference when cylindrical symmetry is present and infinity proves inconvenient. In a
coaxial cable the outer conductor is selected as the zero reference for potential. And,
of course, there are numerous special problems, such as those for which a two-sheeted
hyperboloid or an oblate spheroid must be selected as the zero-potential reference,
but these need not concern us immediately.
If the potential at point A is V A and that at B is V B , then

(13)
V AB = V A − V B
where we necessarily agree that V A and V B shall have the same zero reference point.

2
D4.4. An electric field is expressed in rectangular coordinates by E = 6x a x +
6ya y +4a z V/m. Find: (a) V MN if points M and N are specified by M(2, 6, −1)
and N(−3, −3, 2); (b) V M if V = 0at Q(4, −2, −35); (c) V N if V = 2at
P(1, 2, −4).

Ans. −139.0V; −120.0V; 19.0 V

4.4 THE POTENTIAL FIELD
OF A POINT CHARGE
In Section 4.3 we found an expression Eq. (12) for the potential difference between
two points located at r = r A and r = r B in the field of a point charge Q placed
at the origin. How might we conveniently define a zero reference for potential? The
simplest possibility is to let V = 0at infinity. If we let the point at r = r B recede to
infinity, the potential at r A becomes
Q
V A =
4π 0 r A