Page 174 - Engineering Electromagnetics, 8th Edition

156 ENGINEERING ELECTROMAGNETICS

by definition, is everywhere tangent to the electric field intensity or to the electric flux
density. Because the streamline is tangent to the electric flux density, the flux density
is tangent to the streamline, and no electric flux may cross any streamline. In other
words, if there is a charge of 5 µCon the surface between A and B (and extending
1m into the paper), then 5 µCof flux begins in this region, and all must terminate
between A and B . Such a pair of lines is sometimes called a flux tube, because it

physically seems to carry flux from one conductor to another without losing any.
We next construct a third streamline, and both the mathematical and visual in-
terpretations we may make from the sketch will be greatly simplified if we draw this
line starting from some point C chosen so that the same amount of flux is carried in
the tube BC as is contained in AB.How do we choose the position of C?
The electric field intensity at the midpoint of the line joining A to B may be
found approximately by assuming a value for the flux in the tube AB, say , which
allows us to express the electric flux density by / L t , where the depth of the tube
into the paper is 1 m and L t is the length of the line joining A to B. The magnitude
of E is then
1
E =
L t
We may also find the magnitude of the electric field intensity by dividing the
potential difference between points A and A 1 , lying on two adjacent equipotential
surfaces, by the distance from A to A 1 .If this distance is designated L N and an
increment of potential between equipotentials of V is assumed, then
V
E =
L N
This value applies most accurately to the point at the middle of the line segment
from A to A 1 , while the previous value was most accurate at the midpoint of the line
segment from A to B. If, however, the equipotentials are close together ( V small)
and the two streamlines are close together ( small), the two values found for the
electric field intensity must be approximately equal,
1 V
= (18)
L t L N
Throughout our sketch we have assumed a homogeneous medium ( constant), a
constant increment of potential between equipotentials ( V constant), and a constant
amount of flux per tube ( constant). To satisfy all these conditions, Eq. (18) shows
that
L t 1
= constant = (19)
L N V

A similar argument might be made at any point in our sketch, and we are therefore
led to the conclusion that a constant ratio must be maintained between the distance
between streamlines as measured along an equipotential, and the distance between
equipotentials as measured along a streamline. It is this ratio that must have the same
value at every point, not the individual lengths. Each length must decrease in regions
of greater field strength, because V is constant.