Page 61 - Engineering Electromagnetics, 8th Edition

CHAPTER 2 Coulomb’s Law and Electric Field Intensity 43

Figure 2.10 The equation of a streamline is
obtained by solving the differential equation
E y /E x = dy/dx.

As an illustration of this method, consider the field of the uniform line charge
with ρ L = 2π 0 ,
1
E = a ρ
ρ
In rectangular coordinates,
x y
E = a x + a y
x + y 2 x + y 2
2
2
Thus we form the differential equation
dy E y y dy dx
dx = E x = x or y = x
Therefore,

ln y = ln x + C 1 or ln y = ln x + ln C
from which the equations of the streamlines are obtained,

y = Cx

If we want to find the equation of one particular streamline, say one passing
through P(−2, 7, 10), we merely substitute the coordinates of that point into our
equation and evaluate C. Here, 7 = C(−2), and C =−3.5, so y =−3.5x.
Each streamline is associated with a specific value of C, and the radial lines
shown in Figure 2.9d are obtained when C = 0, 1, −1, and 1/C = 0.
The equations of streamlines may also be obtained directly in cylindrical or
sphericalcoordinates.AsphericalcoordinateexamplewillbeexaminedinSection4.7.