Page 226 - Engineering Electromagnetics, 8th Edition

208 ENGINEERING ELECTROMAGNETICS

No such source has ever been discovered for the lines of magnetic flux. In the
example of the infinitely long straight filament carrying a direct current I, the H field
formed concentric circles about the filament. Because B = µ 0 H, the B field is of the
same form. The magnetic flux lines are closed and do not terminate on a “magnetic
charge.” For this reason Gauss’s law for the magnetic field is

B · dS = 0 (35)
S

and application of the divergence theorem shows us that

∇ · B = 0 (36)

Equation (36) is the last of Maxwell’s four equations as they apply to static
electric fields and steady magnetic fields. Collecting these equations, we then have
for static electric fields and steady magnetic fields

∇ · D = ρ ν
∇× E = 0
(37)
∇× H = J
∇ · B = 0

To these equations we may add the two expressions relating D to E and B to H
in free space,

D = 0 E (38)

B = µ 0 H (39)

We have also found it helpful to define an electrostatic potential,

E =−∇V (40)

and we will discuss a potential for the steady magnetic field in the following section. In
addition, we extended our coverage of electric fields to include conducting materials
and dielectrics, and we introduced the polarization P. A similar treatment will be
applied to magnetic fields in the next chapter.
Returning to (37), it may be noted that these four equations specify the divergence
and curl of an electric and a magnetic field. The corresponding set of four integral