Page 48 - Engineering Electromagnetics, 8th Edition

30 ENGINEERING ELECTROMAGNETICS

coulomb (J/C), or newton-meters per coulomb (N · m/C), we measure electric field
intensity in the practical units of volts per meter (V/m).
Now, we dispense with most of the subscripts in (6), reserving the right to use
them again any time there is a possibility of misunderstanding. The electric field of a
single point charge becomes:

Q
E = a R (8)
4π 0 R 2
We remember that R is the magnitude of the vector R, the directed line segment
from the point at which the point charge Q is located to the point at which E is desired,
and a R is a unit vector in the R direction. 3
We arbitrarily locate Q 1 at the center of a spherical coordinate system. The unit
vector a R then becomes the radial unit vector a r , and R is r. Hence
Q 1
E = a r (9)
4π 0 r 2
The field has a single radial component, and its inverse-square-law relationship is
quite obvious.
If we consider a charge that is not at the origin of our coordinate system, the
field no longer possesses spherical symmetry, and we might as well use rectangular
coordinates. For a charge Q located at the source point r = x a x + y a y + z a z ,as

illustrated in Figure 2.2, we find the field at a general field point r = xa x + ya y + za z
by expressing R as r − r , and then

Q r − r Q(r − r )

E(r) = =
3
2

4π 0 |r − r | |r − r | 4π 0 |r − r |
Q[(x − x )a x + (y − y )a y + (z − z )a z ]

= (10)
2
2 3/2
4π 0 [(x − x ) + (y − y ) + (z − z ) ]
2
Earlier, we defined a vector field as a vector function of a position vector, and this is
emphasized by letting E be symbolized in functional notation by E(r).
Because the coulomb forces are linear, the electric field intensity arising from
two point charges, Q 1 at r 1 and Q 2 at r 2 ,is the sum of the forces on Q t caused by
Q 1 and Q 2 acting alone, or
Q 1 Q 2
E(r) = a 1 + a 2
4π 0 |r − r 1 | 2 4π 0 |r − r 2 | 2
where a 1 and a 2 are unit vectors in the direction of (r − r 1 ) and (r − r 2 ), respectively.
The vectors r, r 1 , r 2 , r − r 1 , r − r 2 , a 1 , and a 2 are shown in Figure 2.3.
3 We firmly intend to avoid confusing r and a r with R and a R . The first two refer specifically to the
spherical coordinate system, whereas R and a R do not refer to any coordinate system—the choice is
still available to us.