Page 68 - Engineering Electromagnetics, 8th Edition

50 ENGINEERING ELECTROMAGNETICS

Figure 3.1 The electric flux in the region between a
pair of charged concentric spheres. The direction and
magnitude of D are not functions of the dielectric
between the spheres.

field intensity E. The direction of D at a point is the direction of the flux lines at that
point, and the magnitude is given by the number of flux lines crossing a surface normal
to the lines divided by the surface area.
Referring again to Figure 3.1, the electric flux density is in the radial direction
and has a value of
Q

D = a r (inner sphere)

4πa 2
r=a
Q
D = a r (outer sphere)
4πb 2
r=b
and at a radial distance r, where a ≤ r ≤ b,
Q
D = a r
4πr 2
If we now let the inner sphere become smaller and smaller, while still retaining a
charge of Q,it becomes a point charge in the limit, but the electric flux density at a
point r meters from the point charge is still given by

Q
D = a r (1)
4πr 2
for Q lines of flux are symmetrically directed outward from the point and pass through
2
an imaginary spherical surface of area 4πr .
This result should be compared with Section 2.2, Eq. (9), the radial electric field
intensity of a point charge in free space,

Q
E = a r
4π 0 r 2