Page 59 - Engineering Electromagnetics, 8th Edition

CHAPTER 2 Coulomb’s Law and Electric Field Intensity 41

from the edges. The field outside the capacitor, while not zero, as we found for the
preceding ideal case, is usually negligible.

D2.6. Three infinite uniform sheets of charge are located in free space as
2
2
2
follows: 3 nC/m at z =−4, 6 nC/m at z = 1, and −8 nC/m at z = 4.
Find E at the point: (a) P A (2, 5, −5); (b) P B (4, 2, −3); (c) P C (−1, −5, 2); (d)
P D (−2, 4, 5).
Ans. −56.5a z ; 283a z ; 961a z ;56.5a z all V/m

2.6 STREAMLINES AND SKETCHES
OF FIELDS

We now have vector equations for the electric field intensity resulting from several
different charge configurations, and we have had little difficulty in interpreting the
magnitude and direction of the field from the equations. Unfortunately, this simplicity
cannot last much longer, for we have solved most of the simple cases and our new
charge distributions must lead to more complicated expressions for the fields and
more difficulty in visualizing the fields through the equations. However, it is true that
one picture would be worth about a thousand words, if we just knew what picture to
draw.
Consider the field about the line charge,
ρ L
E = a ρ
2π 0 ρ
Figure 2.9a shows a cross-sectional view of the line charge and presents what might
be our first effort at picturing the field—short line segments drawn here and there
having lengths proportional to the magnitude of E and pointing in the direction of E.
The figure fails to show the symmetry with respect to φ,sowe try again in Figure 2.9b
with a symmetrical location of the line segments. The real trouble now appears—the
longest lines must be drawn in the most crowded region, and this also plagues us
if we use line segments of equal length but of a thickness that is proportional to E
(Figure 2.9c). Other schemes include drawing shorter lines to represent stronger fields
(inherently misleading) and using intensity of color or different colors to represent
stronger fields.
For the present, let us be content to show only the direction of E by drawing
continuous lines, which are everywhere tangent to E, from the charge. Figure 2.9d
shows this compromise. A symmetrical distribution of lines (one every 45 ) indicates
◦
azimuthal symmetry, and arrowheads should be used to show direction.
These lines are usually called streamlines, although other terms such as flux lines
and direction lines are also used. A small positive test charge placed at any point in
this field and free to move would accelerate in the direction of the streamline passing
through that point. If the field represented the velocity of a liquid or a gas (which,
incidentally, would have to have a source at ρ = 0), small suspended particles in the
liquid or gas would trace out the streamlines.