Page 113 - Engineering Electromagnetics, 8th Edition

CHAPTER 4 Energy and Potential 95

Next, we may use the gradient operation to obtain the electric field intensity,
2
E =−∇V =−4xya x − 2x a y + 5a z V/m
The value of E at point P is

E P = 48a x − 32a y + 5a z V/m
and
2 2 2
|E P |= 48 + (−32) + 5 = 57.9 V/m
The direction of E at P is given by the unit vector
a E,P = (48a x − 32a y + 5a z )/57.9

= 0.829a x − 0.553a y + 0.086a z
If we assume these fields exist in free space, then

2
D = 0 E =−35.4xy a x − 17.71x a y + 44.3 a z pC/m 3
Finally, we may use the divergence relationship to find the volume charge density that
is the source of the given potential field,

ρ ν =∇ · D =−35.4y pC/m 3
3
At P, ρ ν =−106.2pC/m .
D4.7. A portion of a two-dimensional (E z = 0) potential field is shown in
Figure 4.7. The grid lines are 1 mm apart in the actual field. Determine approx-
imate values for E in rectangular coordinates at: (a) a;(b) b;(c) c.

Ans. −1075a y V/m; −600a x − 700a y V/m; −500a x − 650a y V/m

100
D4.8. Giventhepotentialfieldincylindricalcoordinates, V = z + 1 ρ cos φV,
2
and point P at ρ = 3m, φ = 60 , z = 2m, find values at P for (a) V ;(b) E;
◦
(c) E;(d) dV/dN;(e) a N ;( f ) ρ ν in free space.

Ans. 30.0 V; −10.00a ρ +17.3a φ +24.0a z V/m; 31.2 V/m; 31.2 V/m; 0.32a ρ −0.55a φ
− 0.77a z ; −234 pC/m 3
4.7 THE ELECTRIC DIPOLE

The dipole fields that we develop in this section are quite important because they
form the basis for the behavior of dielectric materials in electric fields, as discussed
in Chapter 6, as well as justifying the use of images, as described in Section 5.5 of
Chapter 5. Moreover, this development will serve to illustrate the importance of the
potential concept presented in this chapter.
An electric dipole,or simply a dipole,is the name given to two point charges of
equal magnitude and opposite sign, separated by a distance that is small compared to