Page 127 - Electromagnetics

3.1.1 Decoupling of the electric and magnetic fields
For the remainder of this chapter we shall assume that there is no coupling between
E and H or between D and B in the constitutive relations. Then the static equations
decouple into two independent sets of equations in terms of two independent sets of fields.
The static electric field set (E,D)is described by the equations
∇× E(r) = 0, (3.5)
∇· D(r) = ρ(r). (3.6)

Integrating these over a stationary contour and surface, respectively, we have the large-
scale forms

E · dl = 0, (3.7)

D · dS = ρ dV. (3.8)
S V
The static magnetic field set (B,H)is described by

∇× H(r) = J(r), (3.9)
∇· B(r) = 0, (3.10)

or, in large-scale form,

H · dl = J · dS, (3.11)
S

B · dS = 0. (3.12)
S
We can also specialize the Maxwell–Boffi equations to static form. Assuming that the
fields, sources, and equivalent sources are time-invariant, the electrostatic field E(r) is
described by the point-form equations
∇× E = 0, (3.13)
1
∇· E = (ρ −∇ · P) , (3.14)
0
or the equivalent large-scale equations

E · dl = 0, (3.15)

1

E · dS = (ρ −∇ · P) dV. (3.16)
S 0 V
Similarly, the magnetostatic field B is described by
∇× B = µ 0 (J +∇ × M) , (3.17)
∇· B = 0, (3.18)

or

(J +∇ × M) · dS, (3.19)
B · dl = µ 0
S

B · dS = 0. (3.20)
S

© 2001 by CRC Press LLC